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\begin{document}
% slide 1
%slide 1 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf Schwinger functions}
\end{center}
\vfill
\begin{center}
{\bf Moments of a Euclidean path integral}
\end{center}
\vfill
\[
S_n (x_1,\cdots , x_n) := \int D[\phi] e^{-A [\phi]} \phi (x_1) \cdots \phi (x_n) 
\]
\vfill

\begin{center}
{\bf Euclidean invariance}
\end{center}
\vfill

\[
S_n (x_1,\cdots x_n) = 
S_n (Ox_1-a ,\cdots O x_n-a )
\]

\vfill
\[
OO^t = I; \qquad a \qquad \mbox{\bf constant Euclidean 4 vector}
\]

\end{frame}

\begin{frame}
\begin{center}
{\bf 
Relativity and Euclidean invariance
}
\end{center}
\[ 
\mbox{det}(A) = \mbox{det}(B) =1
\]
\vfill
\[ 
X =
\left (
\begin{array}{cc}
t +z & x-iy \\
x+iy & t-z 
\end{array}
\right ) 
\qquad 
{\cal X} =
\left (
\begin{array}{cc}
i\tau +z & x-iy \\
x+iy & i\tau-z 
\end{array}
\right ) 
\]
\vfill
\[
\mbox{det}(X)= \mbox{det}(AXB^t)= t^2 -\mathbf{x}^2
\]
\[
\mbox{det}({\cal X})= \mbox{det}(A{\cal X}B^t) =
- (\tau^2 +\mathbf{x}^2)
\]
\vfill
\[
(A,B) \in SL(2,\mathbb{C}) \times SL(2,\mathbb{C}) 
\]
\begin{itemize}
{\bf 
\item[] Complex Lorentz group = Complex O(4) 
\vfill
\item[] Real O(4) $=$ Subgroup of Complex Lorentz group
}
\end{itemize}


\end{frame}


\begin{frame}
\begin{center}
{\bf 
Analytic properties - spectral condition $p^0>0$
}
\end{center}
\[
\langle 0 \vert \phi (x_n) \cdots \phi (x_1) \vert 0 \rangle_c = 
\]
\vfill
\begin{center}
{\bf 
Insert positive energy intermediate states\\
(no vacuum in truncated functions)
}
\end{center}
\vfill
\[
\sumint 
_c\langle 0 \vert \phi (x_1) \cdots \phi (0) \vert \mathbf{p}_n \rangle 
e^{i p_n\cdot (x_k-x_{k-1})} d\mathbf{p}_n
\langle \mathbf{p}_n \vert \phi (0) \cdots   
\phi (x_n) \vert 0 \rangle_c  
\]
\vfill
\[
i p_n\cdot (x_k-x_{k-1})  = - i p_n^0 (t_k-t_{k-1}) + 
i \mathbf{p}_n \cdot (\mathbf{x}_k - \mathbf{x}_{k-1})
\]
\vfill
\[
p_n^0>0 \quad \mbox{{\color{red}\bf analytic for }} t_k-t_{k-1}  - i \tau \qquad \tau > 0
\]
\end{frame}

\begin{frame}
\begin{center}
{\bf
Analytic continuation
}
\end{center}
\[
\mbox{\bf Tube: } \mbox{Im}( p \cdot z) >0    
%\qquad
%\mbox{Im} (p\cdot (x_k-x_{k-1}+z))> 0
\]
\vfill
\begin{center}
{\bf
Covariance w.r.t. complex Lorentz group
}
\end{center}
\vfill
\[
\mbox{\bf Extended Tube: } 
z_k \in \mbox{\bf tube } z_k' = \Lambda_c z_k
\]
\vfill
\[
\langle 0 \vert \phi (z_n') \cdots \phi (z_1') \vert 0 \rangle_c := 
\langle 0 \vert \phi (\Lambda_c z_n) \cdots \phi (\Lambda_c z_1) \vert 0 \rangle_c  
\]
\vfill
\[
\mbox{\bf Extended permuted tube: }\{x_n\} \in 
\mbox{\bf extended tube } x_k-x_l 
\,\mbox{\bf   spacelike} 
\]
\vfill
\begin{center}
{\bf
Use locality to permute variables
}
\end{center}
\vfill
\[
\langle 0 \vert \phi (z_1) \cdots \phi (z_n)\vert 0 \rangle_c = 
\langle 0 \vert \phi (z_{\sigma (1)}) \cdots 
\phi (z_{\sigma (n)})\vert 0 \rangle_c  
\]
\vfill
\begin{center}
{\bf
Schwinger functions symmetric!
}
\end{center}

\end{frame}


\begin{frame}

\begin{center}
{\bf Schwinger continuation} 
\end{center}
\vfill  
\[
G_n (x_1, \cdots ,x_n ) 
= \lim_{\theta \to \pi/2} S_n (x_1(\phi), \cdots , x_n(\phi) )
\]
\vfill
\[
x_k(\phi) = (\tau_k e^{i\phi},\mathbf{x}_k) \qquad 
0\leq \phi < \pi/2
\]
\vfill
\begin{center}
{\bf
Euclidean time-order $=$ Minkowski time-order
}
\end{center}
\vfill
\begin{center}
{\bf Osterwalder - Schrader continuation} 
\end{center}  
\vfill
\[
\langle 0\vert (\phi_1(x_1) \cdots \phi_n ({x}_n) ) \vert 0 \rangle =
\lim_{0 <\tau_1 < \cdots < \tau_n \to 0}
S_n (\tau_1+i t_1,\mathbf{x}_1 ,  \cdots \tau_n+i t_n ,\mathbf{x}_n )
\]
\vfill
\begin{center}
{\bf 
$n!$ different possible time orderings  $\to$ \,
$n!$ field orderings
}
\end{center}

\end{frame}

\begin{frame}
\begin{itemize}
{\bf
\item [${\color{red}\bullet}$] Locality: Single Schwinger function gives time-ordered Green 
function and $n!$ Wightman functions.
\vfill
\item [${\color{red}\bullet}$] Analytic continuation in $x$-space has no obstructions.    
\vfill
\item [${\color{red}\bullet}$] Euclidean Green functions satisfy 
Schwinger-Dyson equations.
\vfill
\item [${\color{red}\bullet}$] {\color{red}Euclidean Green functions can be used to 
construct quantum
mechanics} (${\cal H}, U(\Lambda,a):{\cal H}\to {\cal H}$) 
\vfill
\item [${\color{red}\bullet}$] Euclidean Green functions related to 
Lagrangian via path integral.
} 
\end{itemize}

\end{frame}

\begin{frame} 

\begin{center}
{\bf Quantum Mechanics: Hilbert space vectors}
\end{center}
\vfill
\[
f := (f_1 (x_{11}), f_2(x_{21},x_{22}), 
f_3 (x_{31},x_{32},x_{33}), \cdots ) \in {\cal S}_+
\]
\vfill
\[
{\cal S}_+: \qquad f = 0 \qquad \mbox{\bf unless}  
\qquad 0 < x_{n1}^0 < x_{n2}^0 < \cdots
< x_{nn}^0
\]
\vfill
\[
\Theta f := (f_1 (\theta x_{11}), f_2(\theta x_{21},\theta x_{22}), 
f_3 (\theta x_{31},\theta x_{32},\theta x_{33}), \cdots )
\]
\vfill
\[
\theta x = \theta (x^0,\mathbf{x}) = (-x^0,\mathbf{x})  
\]

\end{frame}

\begin{frame}
\begin{center}
{\bf Quantum mechanics: Hilbert space inner product}
\end{center}
\vfill
\[
\langle f \vert g \rangle = (\Theta f , S g)_e = ( f,\Theta S g)_e =  
\]
\vfill
\[
\sum_{mn} \int d^{4n}x \, d^{4m}y \,
f_n^* ( \x_{n1}, \x_{n2},\cdots , \x_{nn}) \times
\]
\[
S_{m+n}
(\theta \x_{nn}, \cdots , \theta \x_{1n},\y_{1m} , \cdots , \y_{mm})
\times
\]
\[
g_m (\y_{m1}, \y_{m2},\cdots , \y_{mm}) .
\]
\vfill
\begin{center}
{\bf Reflection positivity}
\end{center}
\vfill
\[
f \in {\cal S}_+ \qquad \langle f \vert f \rangle \geq 0
\]
\vfill
\begin{center}
{\bf No Analytic continuation!}
\end{center}

\end{frame}
\begin{frame}

\begin{center}
{\bf Comments on reflection positivity}
\end{center}


\vfill
\begin{center}
{\bf The support conditions are necessary}
\end{center}
\vfill
\[
\Theta f = - f \quad \rightarrow \quad \langle f \vert f \rangle 
< 0
\]
\vfill

\begin{center}
{\bf Euclidean time support must be disjoint - }
\end{center}
\begin{center}
{\bf the order is not important (locality) }
\end{center}
\vfill
\[
S_n (\x_1, \cdots ,\x_n) = 
S_n (\x_{\sigma(1)}, \cdots ,\x_{\sigma (n)}) 
\]
\vfill
\begin{center}
{\bf Sequential support needed for cluster properties}
\end{center}
\vfill

\begin{center}
{\bf Not easy to verify, \\
not stable with respect to small perturbations, \\
can be satisfied on lattice\\
preserved under limits}
\end{center}


\end{frame}

\begin{frame}
\begin{center}
{\bf 
$\langle f \vert g \rangle = $ Minkowski inner product !
}
\end{center}

\vfill

\begin{center}
{\bf
Example: \qquad  $\langle f \vert g \rangle = (f ,\Theta S_2 g)$     
}
\end{center}

\vfill

\begin{center}
{\bf 
General $S_2$: K\"allen-Lehmann representation
}
\end{center}

\vfill

\[
S_2 (x-y) = {1 \over (2\pi)^4} \int {e^{i p \cdot (x-y) }\rho (m) \over
p^2 + m^2} d^4 p dm
\]

\vfill

\begin{center}
{\bf 
$\rho(m) = $ Lehmann weight
}
\end{center}

\vfill
\[
\rho (m) = \sum_n z_n \delta (m-m_n) + \rho_c(m)
\]

\end{frame}

\begin{frame}

\[
\langle f\vert g \rangle =  
%\int f^*(\theta \x) S_2(\x-\y) g(\y) d^4\x d^4\y=
\int f^*(\x) S_2(\theta \x-\y) g(\y) d^4\x d^4\y=
\]
\vfill 
\[
{1 \over (2 \pi)^4} \int
f^*(\x)
{ e^{i p^0 (-x^0-\y^0) +i \mathbf{p} \cdot (\mathbf{x}-\mathbf{y})}
\rho (m)
\over (p^0)^2 + \mathbf{p}^2 + m^2 }  g(\y)  dm 
d^4 p d^4\x d^4\y=
\]
\vfill
\begin{center}
{\bf 
Close contour LHP
}
\end{center}
\vfill
\[
{1 \over (2 \pi)^3} \int
f^*(\x)
{ e^{- \omega_m(\mathbf{p}) (x^0+\y^0) +i \mathbf{p} \cdot (\mathbf{x}-\mathbf{y})}
\rho (m)
\over 2 \omega_m(\mathbf{p})  }  g(\y)  dm 
d\mathbf{p} d^4\x d^4\y=
\]
\vfill
\[
\int \chi^*(\mathbf{p}) 
{ \rho (m)  \over 2 \omega_m(\mathbf{p})} dm  d\mathbf{p}
\psi (\mathbf{p}) 
\label{d.1a}
\]
\vfill 
\begin{center}
{\bf
Lorentz invariant measure / no analytic continuation!
}   
\end{center}
\end{frame}
\begin{frame}

\begin{itemize}
\item[] {\bf The momentum-space wave functions are
}
\end{itemize}
\vfill 
\[
\chi(\mathbf{p}) =
\int {d\mathbf{x} \over (2 \pi)^{3/2} }
f (\x^0,\mathbf{x}) e^{- \omega_m (\mathbf{p}) \x^0
  - i \mathbf{p} \cdot \mathbf{x} } 
\]
\vfill
\[ 
\phi (\mathbf{p})=
\int {d\mathbf{y} \over (2 \pi)^{3/2} }g (\x^0,\mathbf{x}) e^{- \omega_m (\mathbf{p}) \x^0
- i \mathbf{p} \cdot \mathbf{y} }
\label{d.2}
\]
\vfill
\begin{itemize}
\item[] {\bf The Hamiltonian and (mass)$^2$ operators are 
}
\end{itemize}
\vfill
\[
H = {\partial \over \partial \tau}  \qquad M^2 =
{\partial^2 \over \partial \tau^2} + \nabla^2_{\mathbf{x}} = \nabla^2_x
\]

\end{frame}

\begin{frame}
\begin{center}
{\bf Widder's theorem/reflection positivity\\
{\color{red} most general $g(t)$ satisfying:}}
\end{center}
\vfill
\[
g(t) : \int f(t) g(t+t')f(t')dt' dt \geq 0
\]
\[
\Downarrow
\]
\[
g(t)= \int dm \rho(m) e^{- m t} 
\]
\begin{center}
{\bf Lehmann version
}
\end{center}
\[
{1 \over \pi} \int_{- \infty}^{\infty}d\lambda \int_0^\infty dm
{e^{i \lambda t} m \rho(m) \over \lambda^2 + m^2}  = 
\]
%\[
%\theta (t) \int  \rho(m) e^{-mt} dm +
%\theta (-t) \int  \rho(m) e^{mt} dm =
%\]
\[
\theta (t) g(t) + \theta (-t) g(-t)
\]
\end{frame}

\begin{frame}
\begin{center}
{\bf For positive-time support 
}
\end{center}
\vfill
\[
g(t)= \int_0^\infty dm {m \rho(m)\over \pi} e^{- m t} 
=
{1 \over \pi}
\int_{- \infty}^{\infty}d\lambda \int_0^\infty dm
{e^{i \lambda t} m \rho (m)  \over \lambda^2 + m^2}  . 
\]
\vfill
\begin{itemize}
{\bf   
\item[] Widder's theorem implies that the most general reflection
positive operator in 1 dimension is essentially given by a
Lehmann representation. 
}
\end{itemize}

\end{frame}  

\begin{frame}

\begin{center}
{\bf
Relation to positive self-adjoint operators, $P$: 
}
\end{center}
\[
(f',f) = \int_0^\infty f^*(p) g(p) d \mu (p) 
\]
\vfill
\begin{center}
{\bf
Spectral decomposition 
}
\end{center}
\vfill
\[
f(p) = \int_0^\infty  e^{-ps} k(s)ds =
\int_0^\infty  ds \int_{-\infty}^{\infty} d\lambda e^{-ps+i\lambda s } h(\lambda) =
\]
\vfill
\[
\int_{-\infty}^{\infty} {1 \over p -i \lambda} h(\lambda) d \lambda
\]
\vfill
\[
(f',f) =
\int_{-\infty}^{\infty} d\lambda \int_{0}^{\infty} dp 
k'(\lambda) {p d\mu (p)\over \pi} { d\lambda
\over \lambda^2 + p^2} k(\lambda)=
\]  
\[
\int_{-\infty}^{\infty} d\lambda \int_{0}^{\infty} 
h'(\lambda) {p d\mu (p)\over \pi} { 1
\over \lambda^2 + p^2} h(\lambda) 
\]
\end{frame}

\begin{frame}
\begin{itemize}
{\bf
\item [] Spectral decomposition of positive operator (energy) has the
same form as most general reflection positive operator!
\[
m\rho(m) \leftrightarrow p d\mu(p)/\pi
\]
\item [] Energy poles appear in the Fourier transform variables.
\vfill
\item [] Sequential time support because positive energy intermediate
states between each adjacent pair of field operators?
\vfill
\item [] Positive energy colorless intermediate states - reflection
positivity for singlets.
}
\end{itemize}
\end{frame}


\begin{frame}

\begin{center}
{\bf Quantum mechanics: Poincar\'e generators}
\end{center}
\vfill
\begin{center}
{\bf Euclidean group $=$ complex subgroup of Poincar\'e group}
\end{center}
\vfill
\[
H f_n (\x_{n1}, \x_{n2},\cdots , \x_{nn})
=
\sum_{k=1}^n {\partial \over \partial \x_{nk}^0} f_n (\x_{n1}, \x_{n2},\cdots , \x_{nn})
\]
\vfill
\[
\mathbf{P}  f_n (\x_{n1},\x_{n2},\cdots , \x_{nn})
=
-i \sum_{k=1}^n {\partial \over \partial \mathbf{x}_{nk}} f_n (\x_{n1}, \x_{n2},\cdots ,\x_{nn})
\]
\vfill
\[
\mathbf{J} f_n ( \x_{n1}, \x_{n2},\cdots ,
\x_{nn}) = -i \sum_{k=1}^n \mathbf{x}_{nk} \times {\partial \over
\partial \mathbf{x}_{nk}} f_n (\x_{n1},
\x_{n2},\cdots ,\x_{nn})
\]
\vfill
\[
\mathbf{K} f_n ( \x_{n1}, \x_{n2},\cdots ,
\x_{nn}) = \sum_{k=1}^n (\mathbf{x}_{nk} \times {\partial \over
\partial {x}^0_{nk}}-  {x}^0_{nk}  {\partial \over
\partial \mathbf{x}_{nk}}) f_n ( \x_{n1},
\x_{n2},\cdots , \x_{nn}).
\]
\end{frame}


\begin{frame}
\begin{center}
{\bf
Hermiticity of $H$ on ${\cal H}$
}
\end{center}
\[
H = {\partial \over \partial \tau}
\]
\[
\int f^*(\tau') S_2(\theta (\tau') - \tau ) {\partial \over \partial \tau}
f(\tau) d\tau d\tau' =
\]
\[
- \int f^*(\tau') {\partial \over \partial \tau} S_2(-\tau' - \tau )
f(\tau) d\tau d\tau' =
\]
\[
- \int f^*(\tau') {\partial \over \partial \tau'} S_2(-\tau' - \tau )
f(\tau) d\tau d\tau' =
\]
\[
\int (({\partial \over \partial \tau'}) f(\tau'))^*  S_2(\theta \tau' - \tau )
f(\tau) d\tau d\tau' 
\]
\vfill
\[
\langle f \vert H f \rangle = \langle H f \vert f \rangle
\qquad H = H^{\dagger}
\]

\begin{center}
{\bf
Similarly $\mathbf{K} = \mathbf{K}^{\dagger} $ on ${\cal H}$
}
\end{center}
\end{frame}

\begin{frame}

\begin{center}
{\bf
Self-adjointness of generators
} 
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [] $H$ generator of contractive semigroup on ${\cal H}$
\vfill
\item [] $\mathbf{P}$,$\mathbf{J}$ generators of unitary one-parameter groups
on ${\cal H}$
\vfill
\item [] $\mathbf{K}$ generator of local symmetric seimgroups
\vfill
\item [] In all cases generators are self adjoint!
\vfill
\item [] Generators satisfy Poincar\'e Lie algebra
}
\end{itemize} 
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{5.0in}{!}{
\includegraphics{support.pdf}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}



\begin{frame}

\begin{center}
{\bf
Spectral condition
}
\end{center}

\[
\Vert e^{-\beta H} \vert f \rangle\Vert^2 =
\langle e^{-\beta H}  f  \vert
e^{-\beta H} f \rangle  =
\]
\[
\langle  f \vert
e^{-2\beta H} f \rangle \leq 
\Vert e^{-2\beta H} \vert f \rangle \Vert
\Vert  \vert f \rangle \Vert 
\]
\[
\Vert e^{-\beta H} \vert f \rangle \Vert \leq
\Vert e^{-2^n\beta H} \vert f \rangle \Vert^{{1 \over 2^n}}
\Vert  \vert f \rangle \Vert^{1 - {1 \over 2^n}}
\leq
( f ,S f)_e^{{1 \over 2^{n+1}}}
\Vert  \vert f \rangle \Vert^{1 - {1 \over 2^n}}
\]
\[
( f ,S f)_e < \infty
\]
\[
\Downarrow
\]
\[
\Vert e^{-\beta H} \vert f \rangle \Vert \leq 
\Vert \vert f \rangle \Vert 
\]
\[
\Downarrow
\]
\[
H \geq 0
\]

\end{frame}

\begin{frame}
\begin{center}
{\bf Scattering - general considerations}
\end{center}
\[
P = \vert \langle f_{+} \vert f_{-} \rangle \vert^2 = 
\vert \langle f_{0+} \vert S\vert f_{0-} \rangle \vert^2  
\] 
\begin{center}
{\bf where}
\end{center}
\[
\lim_{t \to \pm \infty }
\Vert \vert f_{\pm} (t) \rangle - J \vert f_{0\pm} (t) \rangle \Vert =0
\]

\[
\lim_{t \to \pm \infty }
\Vert \vert e^{-iHt} f_{\pm} (0) \rangle - 
J e^{-i H_f t} \vert f_{0\pm} (0) \rangle \Vert=0  
\]

\[
\lim_{t \to \pm \infty }
\Vert \vert f_{\pm} (0) \rangle - 
e^{iHt}J e^{-i H_f t} \vert f_{0\pm} (0) \rangle \Vert=0  
\]

\[
S = \lim_{t \to \infty} e^{i H_f t}J^{\dagger}_f e^{-i2Ht}J_i  e^{i H_f t}
\]

\[
J = \int \vert (m_1,j_1) \mathbf{p}_1,\mu_1  \rangle \times 
\cdots \times
\vert (m_n,j_n) \mathbf{p}_n,\mu_n  \rangle
%{d^{3^n x} \over (2 \pi)^{3n/2}}
%e^{- i \sum \mathbf{p}_n \cdot \mathbf{x}_n}
\]

\[
H_f = \sum_n \sqrt{\mathbf{p}_n^2 +m_n^2}
\]


\end{frame}

\begin{frame}
\begin{itemize}
{\bf
\item[] $J$ includes internal structure of composite particles\\
and contributions due to self interactions
\vfill
\item[] $J$ separates observable properties of asymptotic 
particles (mass,
momentum, spin) from internal structure
\vfill
\item[] $J$ maps tensor products of irreducible representations
of Poincar\'e group (particles) into ${\cal H}$
}
\end{itemize}

\end{frame}


\begin{frame}
\[
\lim_{t \to \pm \infty } 
e^{iHt} J e^{-i H_f t} \vert f_{0\pm} (0) \rangle =
\]
\vfill
\[
\vert f_{0\pm} (0) \rangle + 
\int_0^{\pm \infty} {d \over dt} 
e^{iHt} J e^{-i H_f t} \vert f_{0\pm} (0) \rangle dt 
\]
\vfill
\begin{center}
{\bf 
sufficient condition for convergence\\
(Cook condition)
}
\end{center}
\vfill

\[
\int_c^{ \infty}  \Vert (H J - J H_f ) 
e^{\mp i H_f t} \vert f_{0\pm} (0) \rangle \Vert dt < \infty 
\]


\end{frame}

\begin{frame}

\begin{center}
{\bf 
Scattering - Euclidean case $(2 \to 2)$
}
\end{center}
\vfill
  
\[
S_4(\x_1, \x_2, \y_2, \y_1) =
S_2(\x_1,\y_1) S_2(\x_2, \y_2 ) + S_c (\x_1, \x_2, \y_2, \y_1)  
\]
\vfill
\[
S_2(\x_1,\y_1) = {1 \over (2\pi)^4} 
\int {e^{i{p} \cdot ({x}-y }\rho (m)
\over m^2 + p^2 } d^4p dm  
\]
\vfill
\[
\rho (m) = \sum_{i} z_i \delta (m-m_i) + \rho_c (m) 
\]
\vfill
\[
{\color{red}
\langle x_1, x_2  \vert  J \vert \mathbf{p}_1, \mathbf{p}_2 \rangle
= h_1 (\nabla_1^2) h_2 (\nabla_2^2)
\delta (\x_1^0 - \tau_1 )\delta (\x_2^0 - \tau_2 )
{1 \over (2 \pi)^3} e^{i \mathbf{p}_1\cdot  \mathbf{x}_1
+ i \mathbf{p}_2\cdot  \mathbf{x}_2}
}
\]
\vfill
\[
h(m_i^2) =1 \qquad \rho (m) h(m^2) = z_i \delta (m-m_i)
\]

\end{frame} 

\begin{frame}
\begin{itemize}
{\bf 
\item [] Cluster properties of Euclidean Green functions 
essential for scattering.
\vfill
\item [] $g(x) = f(\mathbf{x})\delta (x^0-\tau )$ is square integrable on
${\cal H}$ in spite of delta function!
\vfill
\item [] Simple way to satisfy the support condition.
\vfill
\item [] We use the fact that $m^2$ on the two-point function 
is represented by Laplacian.
\vfill
\item [] $h$ selects mass of scattering asymptote.
\vfill
\item [] {\color{red}We need to be sure that $h (\nabla^2)$ preserves
the relative Euclidean time support condition}.
}
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
{\bf 
Existence (Cook condition - Euclidean representation)
}
\end{center}
\vfill
\[
\int_0^\infty  \Vert (H J - J H_f ) 
e^{-i H_f t} \vert f_{0\pm} (0) \rangle \Vert dt < \infty 
\]
\vfill
\[
\Vert (H J - J H_f ) 
e^{-i H_f t} \vert f_{0\pm} (0) \rangle \Vert^2 =
\]
\vfill
\[
((HJ^{\dagger}-J^{\dagger}H_f) e^{-i H_f t} f \Theta (S_2S_2+S_c) (HJ-JH_f) e^{-i H_f t}f)  
\]

\end{frame}

\begin{frame}

\[
\Vert (H J - J H_{f} )
e^{i H_f t} \vert f_0 \rangle \Vert^2 =
\]
\vfill
\[
\int f_1^* (\mathbf{p}_1) f_2^* (\mathbf{p}_2) 
e^{i  (\omega_{m_1}(\mathbf{p}_1) + \omega_{m_2}(\mathbf{p}_2))t}
d\mathbf{p}_1
d\mathbf{p}_2
\]
\vfill
\[
(
{\partial \over \partial x^{0}_1} + {\partial \over \partial x^{0}_2}
- 
\omega_{m_1}(\mathbf{p}_1) - \omega_{m_2}(\mathbf{p}_2))
\langle \mathbf{p}_1 , \mathbf{p}_2  \vert J^{\dagger} 
\vert \x_1 , \x_2 \rangle
\times 
\]
\vfill
\[ 
d^4x_1 d^4 x_2 
S_{2} (\theta \x_1, \y_1)
S_{2} (\theta \x_2, \y_2)+
S_{4c} (\theta \x_1, \theta \x_2, \y_2, \y_1)
d^4 y_1 d^4 y_2 \times
\]
\vfill
\[
(
{\partial \over \partial y^{0}_1} + {\partial \over \partial y^{0}_2}
- 
\omega_{m_1}(\mathbf{p}_1') - \omega_{m_2}(\mathbf{p}_2'))
\langle \y_1 , \y_2 \vert J  \vert \mathbf{p}_1' , \mathbf{p}_2'  \rangle
\]
\vfill
\[
e^{-i  (\omega_{m_1}(\mathbf{p}_1') + \omega_{m_2}(\mathbf{p}_2'))t}
f_1 (\mathbf{p}_1') f_2 (\mathbf{p}_2') d\mathbf{p}_1'
d\mathbf{p}_2'
\]

\end{frame}
\begin{frame}

\[
{\partial \over \partial x^0} \to \omega_m(\mathbf{p})
\]
\vfill
\begin{itemize}
{\bf 
\item[] $h(m^2)$ in $J$ picks out single $m_i$ so the coefficients of the 
$S_2$ terms vanish.
\vfill
\item[] This {\color{red}eliminates the Maiani Testa} problem!
\vfill
\item[] The strong limit would {\color{red}not exist} without the $h(\nabla^2)$
factors.
\vfill
\item[] Since $h(m^2) = h(\nabla^2)$ it is not automatic that $hf $ and 
$f$ have the same support condition (0 for $\tau<0$): 
}
\end{itemize}
\vfill
\[
e^{a {\partial \over \partial \tau}} f(\tau,\mathbf{x}) = 
f(\tau+a,\mathbf{x})
\]
 
\end{frame}

\begin{frame}

\[
\Vert (H J - J H_{f} )
e^{i H_f t} \vert f_0 \rangle \Vert^2 =
\]
\vfill
\[
\int f_1^* (\mathbf{p}_1) f_2^* (\mathbf{p}_2) 
e^{i  (\omega_{m_1}(\mathbf{p}_1) + \omega_{m_2}(\mathbf{p}_2))t}
d\mathbf{p}_1
d\mathbf{p}_2
\]
\vfill
\[
(
{\partial \over \partial x^{0}_1} + {\partial \over \partial x^{0}_2}
- 
\omega_{m_1}(\mathbf{p}_1) - \omega_{m_2}(\mathbf{p}_2))
\langle \mathbf{p}_1 , \mathbf{p}_2  \vert J^{\dagger} 
\vert \x_1 , \x_2 \rangle
\times 
\]
\vfill
\[ 
d^4x_1 d^4 x_2 S_{4c} (\theta \x_1, \theta \x_2, \y_2, \y_1)
d^4 y_1 d^4 y_2 \times
\]
\vfill
\[
(
{\partial \over \partial y^{0}_1} + {\partial \over \partial y^{0}_2}
- 
\omega_{m_1}(\mathbf{p}_1') - \omega_{m_2}(\mathbf{p}_2'))
\langle \y_1 , \y_2 \vert J  \vert \mathbf{p}_1' , \mathbf{p}_2'  \rangle
\]
\vfill
\[
e^{-i  (\omega_{m_1}(\mathbf{p}_1') + \omega_{m_2}(\mathbf{p}_2'))t}
f_1 (\mathbf{p}_1') f_2 (\mathbf{p}_2') d\mathbf{p}_1'
d\mathbf{p}_2'
\]

\begin{center}
{\bf
Integral converges for good $S_{4c}$
}
\end{center}
\end{frame}  

\begin{frame}
\begin{center}
{\bf
Lorentz invariance of $S$-matrix
}
\end{center}

\vfill
\[
\lim_{t \to \pm \infty} \Vert (\mathbf{K} J - J \mathbf{K}_{f} )
e^{i H_f t} \vert f_0 \rangle \Vert = 0
\]
\vfill

\[
\int_c^{ \infty}  \Vert (\mathbf{K} J - J \mathbf{K}_f ) 
e^{\mp i H_f t} \vert f_{0\pm} (0) \rangle \Vert dt < \infty 
\]

\end{frame}


\begin{frame}

\begin{center}
{\bf
Does $h(\nabla^2)$ preserve the support condition?
}
\end{center}

\[
f(x) = 0 \quad x^0<0 \to h(\nabla_4^2) f(x) =0 \quad x^0<0 ? 
\]

\begin{itemize}
{\bf 
\item [] Yes {\color{red}if} polynomials in $m^2$ are complete 
on ${\cal H}$, 
$h(m^2 ) \approx p(m^2)$ 
\vfill
\item [] Sufficient condition - Carleman
\vfill
\[
\sum_{n=1}^\infty \gamma_n^{-{1 \over 2 n}} > \infty
\]
\vfill
\[
\gamma_n := \langle \psi  \vert \Theta S_2 
(\nabla^2)^n  \vert \psi \rangle =
\]
\vfill
\[
\int_0^\infty \psi (\mathbf{p},x_0) {e^{- \omega_m (\mathbf{p})(x_0 +y_0)}
\over 2 \omega_m (\mathbf{p}^2)} \rho(m^2) 
m^{2n} \psi (\mathbf{p},y_0)
d\mathbf{p} dx_0 dy_0 dm . 
\] 
\vfill
\item [] Satisfied if $\rho_c (m)$ is polynomially bounded.
}
\end{itemize}

\end{frame}

\begin{frame}

\begin{center}
{\bf
Discrete Lehmann weight (not physical)
}
\end{center}

\vfill

\[
h_j (x) = \prod_{i\not=j}^N {x-m_i^2 \over m_j^2-m_i^2} 
\]

\vfill

\begin{center}
{\bf
Continuous part of Lehmann weight has compact support\\
(not physical)
}
\end{center}

\vfill

\[
h_j (x) \approx p_j(m^2)
\]

\end{frame}

\begin{frame}

\begin{center}
{\bf
Continuous part of Lehmann weight semi infinite\\
(physical case)
}
\end{center}

\begin{center}
{\bf
Completeness depends on growth of moments
}
\end{center}

\[
\gamma_n := \int_0^\infty {e^{- \sqrt{m^2 + \mathbf{p}^2} \tau} 
\over 2 \sqrt{m^2 + \mathbf{p}^2}} \rho(m) 
m^{2n} dm 
\]

\[
\gamma_n \to \gamma_n' = 
\int_0^\infty {e^{- \sqrt{m^2 + \mathbf{p}^2} \tau} 
\over 2 \sqrt{m^2 + \mathbf{p}^2}}  
m^{2n+k} dm 
\]


\[
{1 \over 2} \int_0^\infty
{e^{-p\tau \cosh (\eta)} \over \cosh (\eta)} 
( p \sinh (\eta))^{2n+k} \cosh (\eta) d\eta 
\leq
{1 \over2} \tau^{-2n-k} \Gamma (2n+k-2)
\]

\[
\Gamma (x+1) = \sqrt{2 \pi} x^{x+1/2} e^{-x+\theta/12x}
\]

\[
({1 \over \gamma_n })^{{1\over 2n}} \geq 
\sqrt{2 \over \pi}^{1\over 2n}\tau^{1+{k \over 2n}} (2n+k-2)^{-1-(k-3/2)/2n}
e^{(1+(k-2)/2n)-{\theta \over 1+(k-2)/2n}}
\]

\end{frame}

\begin{frame}

\[
\sum_{n=1}^\infty \gamma_n^{-{1 \over 2 n}} > 
\sum_{n=0}^\infty  {c \over 2n +k-2} > \infty
\]

\vfill
\begin{center} 
{\bf 
The inequality shows that $h(m^2)$ can be approximated by 
a polynomial with controlled error. 
}
\end{center} 

\vfill

\begin{center} 
{\bf 
We still need a computational strategy to compute $S$-matrix elements.
}
\end{center} 


\end{frame}

\begin{frame}

\begin{center}
{\bf
Computations
}
\end{center}
  
\[
\lim_{t \to \infty}
(e^{iH_0t} f_{0+}  \Pi \Theta S_4 e^{-2iHt} \Pi e^{iH_0t} f_{0-})   
\]

\begin{center}
{\bf
Invariance principle: $H \to f(H) = e^{-\beta H}$
}
\end{center}

\[
= \lim_{t \to \infty}
(e^{-ine^{-\beta H_0}} f_{0+}  \Pi \Theta S_4 e^{i2ne^{-\beta H}} \Pi
e^{-ine^{-\beta H_0}}f_{0-})   
\]

\end{frame}


\begin{frame}
\begin{center} 
{\bf 
Leads to following expression for $S$-matrix elements
}
\end{center}
\[
\langle f^0_{+} \vert S \vert f^0_{-} \rangle = 
\langle f'_{1} f_2'  \vert S \vert f_1 f_2 \rangle = 
\]
\vfill
\[
\lim_{s \to \infty}
\int
f_1^{\prime *} (\mathbf{p}_1') f_2^{\prime *}
(\mathbf{p}_2')
h_1'(-p_{1}^{\prime 2})
h_2'(-p_{2}^{\prime 2}) d4p_1'd4p_2'd^4p_1d^4p_2 \times
\]
\vfill
\[
e^{-is e^{-\beta
(\omega'_1 (\mathbf{p}'_1) + \omega_2' (\mathbf{p}'_2))}}
e^{2is e^{-i\beta (p_{10} + p_{20})}} 
e^{-is e^{-\beta (\omega_1 (\mathbf{p}_1) + \omega_2 (\mathbf{p}_4))}}
\times 
\]
\vfill
\[ 
e^{-i ((p_{10}+p_{10}')\tau_1 + (p_{20}+p_{20}')\chi \tau_2)} \times
\]
\vfill
\[
{\color{red}\tilde{S}_{4}(-p_2',-p_1',p_1,p_2)}
h_1(-p_{1}^2)
h_2(-p_{2}^2) f_1 (\mathbf{p}_1) f_2
(\mathbf{p}_2)
\]

\begin{center} 
{\bf 
No analytic continuation !\\
existence of limit requires smearing
}
\end{center}

\end{frame}


\begin{frame}

\begin{center}
{\bf 
To calculate $\langle \mathbf{p}_1', \mathbf{p}_2' \vert T \vert 
\mathbf{p}_1, \mathbf{p}_2 \rangle$ assume that it varies slowly 
on the width of the momentum-space wave packets. 
}
\end{center}
\vfill
\[
\Downarrow
\]
\vfill

\[
\langle \mathbf{p}_1', \mathbf{p}_2' \vert T \vert 
\mathbf{p}_1, \mathbf{p}_2 \rangle \approx
{i \over 2 \pi} {\langle f_1' f_2' \vert (S-I) f_1 f_2 \rangle \over 
\langle f_1' f_2' \delta (E-E') f_1 f_2 \rangle}
\]
\vfill
\begin{center}
{\bf 
for sharply peaked wave packets
}
\end{center}
\end{frame}

\begin{frame}
\begin{itemize}
{\bf 
\item[] Entire calculation can be performed without analytic
continuation.
\vfill
\item[] Straightforward generaliztion to arbitrary initial and final states.
\vfill
\item[] Needs Schwinger functions as input.
\vfill
\item[] Explicit formula - no analytic continuation.
\vfill
\item[] Reflection positivity essential.
\vfill
\item[] QCD only need reflection positivity for singlets.
}
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
{\bf 
\item[] Calculations? How can we perform 12 dimensional interals accurately?
\vfill
\item[] Reflection positivity; structure theorem ?
\vfill
\item[] Implementation on lattice?
\vfill
\item[] Current matrix elements, final state interactions?
\vfill
\item[] Timelike momentum transfers?  
}
\end{itemize}
\end{frame}


\end{document}


\begin{frame} 

\[
( \psi , \Pi \Theta S_2 \Pi h(\nabla^2) \psi )
%\langle \psi \vert \Theta S_2 h(\nabla^2 ) \psi \rangle 
=
\]
\[
\int \psi (\mathbf{p},x_0) {e^{- \omega_m (\mathbf{p})(x_0 +y_0)}
\over 2 \omega_m (\mathbf{p}^2)} \rho(m^2) 
h({d^2 \over dy_0^2}- p^2)\psi (\mathbf{p},y_0)
d\mathbf{p} dx_0 dy_0 dm 
=  
\]
\beq
\int \psi (\mathbf{p},x_0) {e^{- \omega_m (\mathbf{p})(x_0 +y_0)}
\over 2 \omega_m (\mathbf{p}^2)} \rho(m^2) 
h(m^2)\psi (\mathbf{p},y_0)
d\mathbf{p} dx_0 dy_0 dm 
\label{g.3}
\eeq


\beq
\langle \x_1, \x_2  \vert  J \vert \mathbf{p}_1, \mathbf{p}_2 \rangle
= h_1 (\nabla_1^2) h_2 (\nabla_2^2)
\delta (\x_1^0 - \tau_1 )\delta (\x_2^0 - \tau_2 )
{1 \over (2 \pi)^3} e^{i \mathbf{p}_1\cdot  \mathbf{x}_1
+ i \mathbf{p}_2\cdot  \mathbf{x}_2}
\label{f.9}
\eeq

\end{frame}


\begin{frame}
 

\beq
\lim_{s \to \pm \infty}
\Vert e^{ise^{-\beta H}} \vert f^{\pm} \rangle - 
J e^{i se^{-\beta H_0} } \vert  f^0 \rangle \Vert =0 .   
\label{h.1}
\eeq

\end{frame}

%slide 2 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Outline
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item[{\color{red}$\bullet$}] Physics questions
\vfill
\item[{\color{red}$\bullet$}] Scales and applicable theories
\vfill
\item[{\color{red}$\bullet$}] Quantum mechanics
\vfill
\item[{\color{red}$\bullet$}] Relativity
\vfill
\item[{\color{red}$\bullet$}] Challenges
}
\end{itemize}
\end{frame}

%slide 3 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Structure of matter
}
\end{center}
\vfill
\begin{center}
{\bf 
molecules
}
\end{center}
\[
\Downarrow
\]
\begin{center}
{\bf 
atoms
}
\end{center}
\[
\Downarrow
\]
\begin{center}
{\bf 
nuclei
}
\end{center}
\[
\Downarrow
\]
\begin{center}
{\bf 
protons and neutrons
}
\end{center}
\[
\Downarrow
\]
\begin{center}
{\bf 
quarks \qquad p $=$ (uud) \qquad n $=$ (ddu)
}
\end{center}
\end{frame}

%slide 4 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Quarks have never been observed!
}
\end{center}
\vfill
\begin{center}
{\bf 
6 flavors, 3 colors, 2 spin states
}
\end{center}
\vfill
\begin{center}
{\bf 
Their existence classifies the properties of hundreds of 
elementary particles.
}
\end{center}
\vfill
\begin{center}
{\bf 
How do we understand the physics of particles that cannot be seen?
}
\end{center}

\vfill
\begin{center}
{\bf 
How do we understand the forces between 
nucleons in terms of the forces between quarks?
}
\end{center}

\end{frame}

%slide 5 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Limitations from quantum mechanics: \\
Heisenberg uncertainty principle
}
\end{center}
\vfill
\begin{center}
{\bf 
Radius of proton $= .6 \times 10^{-15}$ meters $= .6 \,fm$
}
\end{center}
\vfill
\[
\Delta (pc) \Delta x > \hbar c  
\]
\[
\Downarrow
\]
\[
\Delta (pc) > {\hbar c \over \Delta x } = {.2\, GeV\,- fm\over .6\,fm} = 1/3\, GeV
\]
\vfill
\begin{center}
{\bf 
The uncertainty principle implies that sensitivity to structure 
that is a fraction of the proton radius requires energy transfers
of a few GeV.
}
\end{center}
\end{frame}

%slide 6 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Limitations from special relativity: \\
needed for $E/mc^2 \gtrsim 1$
}
\end{center}

\[
m_p c^2 = .94 GeV 
\]
\vfill
\begin{center}
{\bf 
Since the energy required to see the inside of a proton is 
comparable to the rest mass of the proton special relativity 
becomes important. 
}
\end{center}
\vfill

\begin{center}
{\bf 
Quantum theory $+$ special relativity needed to 
study structure of proton.
}
\end{center}
\end{frame}

%slide 7 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Quantum mechanics and mathematics
}
\end{center}
\vfill

\begin{center}
{\bf 
State of system $=$ ray in Hilbert space
}
\end{center}
\vfill
\begin{center}
{\bf 
Probability of measuring system prepared in state $A$ to be
found in state $B$:
}
\end{center}
\vfill
\[
P_{AB} = {(A,B)(B,A) \over (A,A)(B,B)} = \vert (\hat{A},\hat{B})\vert^2 
\]

\begin{center}
{\bf 
Measured using statistically significant number of identical 
measurements.
}
\end{center}

\end{frame}

%slide 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Quantum mechanics and mathematics
}
\end{center}
\vfill

\begin{center}
{\bf 
Expectation values of linear operators, $X$
}
\end{center}
\vfill
\[
\langle X_A \rangle := (\hat{A},X\hat{A})  = \sum_n x_n \vert (\hat{n},\hat{A})\vert ^2 
= \sum_n x_n P_{n,A}
\]
\vfill
\begin{center}
{\bf 
Self-adjointness of $X \Rightarrow$ every measurement of $X$ gives
an eigenvalue $x_n$ of $X$ with probability 1.
}
\end{center}
\vfill
\[
\sum_n P_{n,A} = 1
\]

\end{frame}

%slide 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Quantum mechanics
}
\end{center}
\vfill
\[
\Updownarrow
\]
\vfill
\begin{center} 
{\bf 
Linear algebra on a complex Hilbert space
}
\end{center}

\end{frame}

%slide 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Symmetries in quantum mechanics
}
\end{center}
\vfill
\begin{center}
{\bf 
$T$: correspondence relating states
}
\end{center}
\vfill
\[
T: A,B \Leftrightarrow A', B'
\]
\vfill
\begin{center}
{\bf 
$T$: symmetry if:
}
\end{center}
\vfill
\[
P_{AB} = P_{A'B'} \qquad \forall \, A,B 
\]
\vfill
\[
A' = T(A) \qquad B' = T(B) 
\]
\vfill
\[
(A,B) = (T(A),T(B))
\qquad 
\mbox{or} 
\qquad 
(A,B) = \overline{(T(A),T(B))}
\]
\vfill
\begin{center}
{\bf 
T: unitary-linear or antiunitary-antilinear
}
\end{center}
\end{frame}

%slide 7 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
  
 
\begin{frame}
\begin{center}
{\bf 
Special relativity 
}
\end{center}
\vfill
\begin{center}
{\bf 
Inertial coordinate systems (preserve $c^2 (\Delta \tau)^2$ )
}
\end{center}
\vfill
\[
\mathbf{X} \qquad \mathbf{X}' \qquad \mbox{\bf inertial}
\qquad x=(ct,\mathbf{x}) \qquad x'=(ct',\mathbf{x}') 
\]
\vfill
\[
c^2 (\Delta \tau)^2 :=
c^2 (\Delta t)^2 - \Delta \mathbf{x} \cdot \Delta \mathbf{x} =
c^2 (\Delta t')^2 - \Delta \mathbf{x}' \cdot \Delta \mathbf{x}' 
\]
\vfill
\begin{center}
{\bf 
Poincar\'e group: $x \to x'$ preserving $c^2 (\Delta \tau)^2$
}
\end{center}


\end{frame}

%slide 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Poincar\'e group
}
\end{center}
\[
\mathbf{X} \qquad \mathbf{X}'\qquad \mbox{\bf inertial}
\]
\vfill
\[
x' = \Lambda x + r  
\qquad \eta = \Lambda^t \eta \Lambda  
\]
\vfill
\[
\eta = 
\left (
\begin{array}{cccc}
-1 & 0 & 0 & 0 \\
0  & 1 & 0 & 0 \\
0  & 0 & 1 & 0 \\
0  & 0 & 0 & 1 
\end{array}
\right )
\]
\vfill
\begin{center}
{\bf 
Group multiplication law
}
\end{center}
\[
(\Lambda_2 ,r_2) (\Lambda_1 ,r_1) =
(\Lambda_2\Lambda_1, \Lambda_2 r_1 + r_2)
\]
\end{frame}

\begin{frame}
\vfill 
\begin{center}
{\bf 
Relativistic invariance in quantum theory
}
\end{center}
\vfill
\[
\underbrace{A, B \to A', B'}_{(\Lambda ,r)}
\]
\vfill
\[
\Downarrow
\]
\vfill
\[
P_{AB} = P_{A'B'}
\]
\vfill
\[
\Updownarrow 
\]
\vfill
\[
A' = U(\Lambda ,r) A \qquad B' = U(\Lambda ,r ) B
\qquad U(\Lambda ,r)^{\dagger} = U(\Lambda ,r)^{-1}
\]
\vfill
\[
U(\Lambda_2 , r_2) U(\Lambda_1 ,r_1) = U(\Lambda_2 \Lambda_1 , \Lambda_2 r_1 + r_2)
\]
\end{frame}

%slide 9 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Quantum Theory + Relativity
}
\end{center}
\vfill
\begin{center}
{\bf 
Hilbert space ${\cal H}$
}
\end{center}
\vfill
\begin{center}
{\bf 
Unitary representation of the Poincar\'e group on ${\cal H}$
}
\end{center}
\vfill
\[
U(\Lambda ,r ) :{\cal H} \to {\cal H}
\]
\end{frame}

%slide 10 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Representations of ${\cal H}$ and $U(\Lambda,r)$ in terms of functions
}
\end{center}
\vfill
\begin{center}
{\bf 
$U(\Lambda ,a)$ has 10 one-parameter unitary subgroups $U_i(\lambda)$
}
\end{center}
\vfill
\[
U_i(0) = I \qquad 
U_i(\lambda_1) U_i(\lambda_2) = U_i(\lambda_1 + \lambda_2)
\]
\vfill
\begin{center}
{\bf 
Infinitesimal generators $G_i$ are defined by
}
\end{center}
\vfill
\[
G_j = i {d U_j(\lambda) \over d \lambda} U_j^{\dagger} (\lambda)
= G_j^{\dagger}
\]
\vfill
\end{frame}


\begin{frame}
\begin{center}
{\bf 
Poincar\'e Lie Algebra
}
\end{center}
\vfill
\[
[G_i, G_j ] = i \sum_k g_{ijk} G_k
\] 
\vfill 

\begin{center}
{\bf 
Lie algebra gives algebra of self-adjoint operators on 
${\cal H}$ with {\color{blue}physical interpretations}!
}
\end{center}
\vfill
\[
\{ G_i \} = \{ H,\mathbf{P}, \mathbf{J}, \mathbf{K} \}
\]
\vfill
\begin{center}
{\bf 
Energy, linear momentum, angular momentum, boosts
}
\end{center}

\end{frame}

%slide 11 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Unitary one-parameter groups
}
\end{center}
\vfill
\[
i {d U_j(\lambda) \over d \lambda} =  G_j U_j(\lambda )
\qquad U_j(0) = I 
\]
\vfill
\[
U_j (\lambda ) = e^{-i G_j \lambda}
\]
\vfill
\[
G= H 
\]
\vfill
\[
\Downarrow
\]
\vfill
\[
A' = A(t) = U_H (t) A(0) 
\]
\vfill
\[
i {d \over dt }A(t) = H A(t) 
\]
\vfill
\begin{center}
{\bf 
Schr\"odinger equation - time evolution of quantum states
}
\end{center} 

\end{frame}


%slide 11 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Hilbert space for a particle
}
\end{center}
\vfill 
\begin{center}
{\bf 
Representations of ${\cal H}$ and $U(\Lambda,r)$ in terms of functions
}
\end{center}
\vfill
\[
\{ G_i \} = \{ H,\mathbf{P}, \mathbf{J}, \mathbf{K} \}
\]
\vfill
\begin{center}
{\bf 
Energy, linear momentum, angular momentum, boosts
}
\end{center}
\vfill
\begin{center}
{\bf 
Enveloping algebra
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] 2 invariant functions of the $G_i$ (mass and 
spin of particle)
\vfill
\item [{\color{red}$\bullet$}] 4 mutually commuting self-adjoint functions of the $G_i$ 
(momentum, spin projection of particle)
\vfill
\item [{\color{red}$\bullet$}] 4 self-adjoint functions of $G_i$ that fix 
the spectrum of the momentum and spin-projection. 
}
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
{\bf 
Eigenvalue problems
}
\end{center}
\vfill
\[
M^2 A = m^2 A
\]
\vfill
\[
\mathbf{P} A = \mathbf{p} A 
\]
\vfill
\[
\mathbf{j} = \mathbf{J}_{\mathbf{P}=\mathbf{0}} \quad  \mbox{angular momentum of particle at rest} 
\]
\vfill
\[
\mathbf{j}^2 A = s(s+1) A
\]
\vfill
\[
\mathbf{j} \cdot \hat{\mathbf{z}} A = \mu A
\]
\vfill
\begin{center}
{\bf 
$\{M^2,\mathbf{P},\mathbf{j}^2,\mathbf{j} \cdot \hat{\mathbf{z}}\}$
all commute 
}
\end{center}
\vfill
\begin{center}
{\bf 
One solution to 6 eigenvalues problems
}
\end{center}

\end{frame}


%slide 12 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Eigenvalues: $m$, $s$ - particle of mass $m$ and spin $s$
}
\end{center}
\vfill
\[
m >0 
\]
\vfill
\[
\mathbf{j}^2:  s (s+1) \qquad  s = \{ 0, {1 \over 2}, 1, {3 \over 2},\cdots \}   
\]
\vfill
\[
\mathbf{j} \cdot \hat{\mathbf{z}}:  \mu \qquad \mu\in \{ -s, -s+1, \cdots , 
s-1, s \}
\]
\vfill
\[
\mathbf{P}: \mathbf{p} \qquad \mathbf{p} \in \{-\infty < p_x,p_y,p_z < \infty 
\} 
\]
\vfill
\begin{center}
{\bf
Representation of Hilbert space for  
structureless relativistic particle of {\color{red}mass $m$ and spin $s$}
}
\end{center}
\vfill
\[
A \to a(\mathbf{p},\mu) 
\qquad
B \to b(\mathbf{p},\mu) 
\]
\vfill
\[
(A,B) = 
\int_{\mathbb{R}^3}  d\mathbf{p} 
\sum_{\mu=-j}^j
\bar{a}(\mathbf{p},\mu)  b(\mathbf{p},\mu) 
\]
\end{frame}

%slide 13 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Square integrable functions of the eigenvalues of a maximal set of 
commuting self-adjoint operators over their spectrum  
}
\end{center}
\vfill
\[
{\cal H}_{mj} =\{ a(\mathbf{p},\mu) \, \vert \, 
\int_{\mathbb{R}^3}  d\mathbf{p} 
\sum_{\mu=-j}^j
\vert {a}(\mathbf{p},\mu)\vert^2 < \infty\}
\]
\vfill  
\begin{center}
{\bf 
$U(\Lambda ,r)$ uniquely determined  
on ${\cal H}_{mj}$ as product of one-parameter subgroups
}
\end{center}
\vfill
\[
(U(\Lambda , r)a)(\mathbf{p},\mu) = 
\int d\mathbf{p}' \sum_{\mu'} 
{\cal D}^{mj}_{\mathbf{p},\mu; \mathbf{p}',\mu'}[\Lambda ,r] 
a(\mathbf{p}' ,\mu')
\]
\end{frame}

\begin{frame}
\begin{center}
{\bf 
The matrices ${\cal D}^{mj}_{\mathbf{p},\mu; \mathbf{p}',\mu'}[\Lambda ,r]$
are known.
}
\end{center}
\vfill
\[
{\cal D} [\Lambda_2 , r_2] {\cal D} [\Lambda_1 , r_1] = 
{\cal D} [\Lambda_2 \Lambda_1  , 
\Lambda_2 r_1+ r_2]
\]
\vfill
\[
{\cal D} [\Lambda , r]^{\dagger} = {\cal D} [\Lambda , r]^{-1} = 
{\cal D} [\Lambda^{-1}  , - \Lambda^{_1} r] 
\]
\vfill

\begin{center}
{\bf 
${\cal D} [\Lambda , r]$: 
irreducible representations of the Poincar\'e group.
}
\end{center}
\end{frame} 

%slide 14 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Systems of particles (protons composed of quarks, $\cdots$) 
}
\end{center}
\vfill
\[
{\cal H} = \oplus (\otimes {\cal H}_{m_i j_i})
\]
\vfill
\begin{center}
{\bf 
$\oplus$: Orthogonal subspaces with different numbers of particles
}
\end{center}
\vfill
\begin{center}
{\bf 
$\otimes$: N-particle subspaces
}
\end{center}
\vfill
\[
U_0 (\Lambda ,r ) = \oplus (\otimes
U_{m_i j_i} (\Lambda ,r ))
\]
\end{frame}

%slide 15 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
What can we learn? 
}
\end{center}

\vfill

\begin{center}
{\bf 
Poincar\'e generators for {\color{red}interacting} system 
}
\end{center}
\vfill
\[
G_i = \sum_j G_i (j) + \sum_{jk} {\color{red}G_i(j,k)} +
\sum_{jkl} {\color{red}G_i(j,k,l)} + \cdots   
\]
\vfill
\begin{center}
{\bf
$\sum_j G_i (j) = G_i(0)$  known
}
\end{center}
\vfill
\begin{center}
{\bf 
Find/test  $G_i(j,k)$, $G_i(j,k,l)$, $\cdots$
}
\end{center}
\end{frame}

%slide 17 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Can't put detectors inside of proton! 
}
\end{center}
\vfill
\begin{center}
{\bf 
Scattering theory - measure particles long before and long after collision? 
}
\end{center}

\vfill
\begin{center}
{\bf 
Motivation for modern particle accelerators. 
}
\end{center}

\vfill
\begin{center}
{\bf 
Beam: statistical distribution of initial asymptotic states
}
\end{center}

\vfill
\begin{center}
{\bf 
Detectors: measure statistical distribution of final asymptotic states
}
\end{center}

\vfill
\begin{center}
{\bf 
Can we determine $G_i(j,k)$, $G_i(j,k,l)$, $\cdots$? from \\
these two distributions?
}
\end{center}

%\vfill
%\begin{center}
%{\bf 
%Can we determine $G_i(j,k)$, $G_i(j,k,l)$, $\cdots$? from \\
%states of particles long before and long after the collision?
%}
%\end{center}
%$\vfill
%\begin{center}
%{\bf 
%If not what can we learn? 
%}
%\end{center}

\end{frame}

%slide 18 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Scattering 
}
\end{center}
\vfill
\begin{center}
{\bf 
Identify states that look like free particles long before or 
long after collision
}
\end{center}
\vfill
\[ 
U(t) = e^{-iHt} \qquad U_0(t) = e^{-i H_0 t} \qquad H = H_0 +{\color{red}V}
\]
\vfill
\[
\lim_{t \to \pm \infty } \Vert U(t) A_{\pm}  - U_0(t) A_{0\pm} \Vert =0 
\] 
\vfill
\[
A_{\pm} = \lim_{t \to \pm \infty } U^{\dagger}(t) U_0(t) A_{0\pm}
:= \Omega_{\pm} A_{0\pm} 
\]
\vfill
\[
P_{A_+,A_-} = \vert (A_{0+},\Omega_{+}^{\dagger} \Omega_- A_{0-})\vert^2
= 
\vert (A_{0+},S A_{0-})\vert^2 
\]
\vfill
\begin{center}
{\bf 
Measurable in an accelerator
}
\end{center}
\end{frame} 

%slide 19 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
\begin{frame}
\begin{center}
{\bf 
How is $S:= \Omega_{+}^{\dagger} \Omega_-$ related to
$G_i(j,k)$, $G_i(j,k,l)$, $\cdots$? 
}
\end{center}
\vfill
\[
S = I - 2 \pi i \delta (E_f -E_i) T(E_i) 
\]
\vfill
\begin{center}
{\bf 
$T(E)$ can be calculated in terms of the interactions
$G_i(j,k)$, $G_i(j,k,l)$, $\cdots$ .
}
\end{center}
\vfill
\begin{center}
{\bf 
$G_i(j,k)$, $G_i(j,k,l)$, $\cdots$ are {\color{red}not} uniquely determined
by $T(E)$.
}
\end{center}

\end{frame}

%slide 19 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame} 

\begin{center}
{\bf 
Complication - free particles far away are bound systems
of quarks, not elementary particles
}
\end{center}
\vfill
\[
\Phi_\alpha = \otimes \Phi_{\alpha_i}
\]
\vfill
\begin{center}
{\bf 
Products of proton, neutron, $\cdots$ subsystem mass eigenstates.
}
\end{center}
\vfill
\[
A_{\pm} \to  \lim_{t \to \pm \infty } 
U^{\dagger}(t) \Phi_{\alpha} U_\alpha(t) A_{\alpha \pm}
:= \Omega_{\alpha \pm} A_{\alpha \pm} 
\]
\vfill

\begin{center}
{\bf 
Asymptotic condition requires solution of point spectrum eigenstates
of subsystem $M$ operator ($\Phi_{\alpha}$). 
}
\end{center}
\end{frame}

%slide 20 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Remaining dynamical generators
}
\end{center}
\vfill
\[
\Omega_pm \pm A_{\alpha\pm} = A_{\pm}
\]
\vfill
\[
G_i \Omega_{\pm} = \Omega_{\pm}  G_{0\alpha i}  
\]
\vfill
\[
G_{0\alpha i} = \sum_j G_{\alpha i}(j)
\]
\vfill
\[
G_i = \sum_\alpha 
\Omega_{\pm}  G_{0\alpha i} \Omega^{\dagger}_{\pm}
\]
\end{frame}


%slide 20 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx



%slide 21 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
QCD Candidate for theory of quarks 
}
\end{center}

\vfill

\begin{center}
{\bf 
Related to equations of electrodynamics by changing internal 
symmetry group $U(1) \to SU(3)$  
}
\end{center}

\vfill

\begin{center}
{\bf 
QCD $\Rightarrow   \, G_i$ quark-gluon 
}
\end{center}
\vfill
\[
\Downarrow
\]
\vfill
\begin{center}
{\bf 
$G_i$ quark-gluon  $\Rightarrow \, G_i$ nucleon-meson ? 
}
\end{center}

\end{frame}


\begin{frame} 

\begin{center}
{\bf 
Most physicists try to test QCD directly.
}
\end{center}

\vfill

\begin{center}
{\bf 
One the seven ``Millennium Prize Problems'' is to establish the existence of 
QCD with expected properties.  {\color{blue}Unknown how to make mathematically 
controlled approximations.}
}
\end{center}

\vfill

\begin{center}
{\bf 
Alternative - constructing realistic relativistic quantum mechanical models
with QCD degrees of freedom.  
}
\end{center}

\end{frame}

%slide 22 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Theoretical challenges in constructing mathematical models
}
\end{center}
\vfill
\[
H= \sum_i H_i + \sum_{ij} H_{ij} + \sum_{ijk} H_{ijk} +\cdots
\]
\[
\mathbf{P}= 
\sum_i \mathbf{P}_i + \sum_{ij} \mathbf{P}_{ij} + \sum_{ijk} \mathbf{P}_{ijk} +
\cdots
\]
\[
\mathbf{J}= 
\sum_i \mathbf{J}_i + \sum_{ij} \mathbf{J}_{ij} + \sum_{ijk} \mathbf{J}_{ijk} +
\cdots
\]
\[
\mathbf{K}= 
\sum_i \mathbf{K}_i + \sum_{ij} \mathbf{K}_{ij} + \sum_{ijk} \mathbf{K}_{ijk} +
\cdots
\]
\vfill
\begin{itemize}
{\bf
\item [{\color{red}$\bullet$}] Find interactions that preserve commutation relations
\vfill
\item [{\color{red}$\bullet$}] Preserve commutation relations for isolated subsystems
\vfill
\item [{\color{red}$\bullet$}] Provide realistic description of physics
}
\end{itemize} 

\end{frame}

%slide 22 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\vfill
\begin{itemize}
{\bf
\item [{\color{red}$\bullet$}] First problem is non-linear.
\vfill
\item [{\color{red}$\bullet$}] Second problem has only been solved for systems of fixed 
numbers of particles.
\vfill
\item [{\color{red}$\bullet$}] At the energy scale of interest reactions do not conserve 
particle number. 
}
\end{itemize} 

\end{frame}

%slide 22 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Numerical considerations
}
\end{center}
\vfill
\[
H = H_a + V^a  \qquad T(E) \to T^{ab}(E) 
\]
\vfill
\[
T(E)^{ab}  = V^b + V^a (E-H+i0^+)^{-1} V^b    
\]
\vfill
\begin{center}
{\bf 
Use resolvent equations 
}
\end{center}
\vfill
\[
(E-H+i0^+)^{-1} = (E-H_a+i0^+)^{-1} + (E-H_a+i0^+)^{-1} V^a  (E-H+i0^+)^{-1}
\]
\vfill
\begin{center}
{\bf 
to derive   
coupled integral equations for $T(E)^{ab}$ with compact 
iterated kernel.  
}
\end{center}

\end{frame}

%slide 22 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\[
T = D + K T   
\]
\vfill
\[
T = (1-K^m)^{-1} (1 +K + \cdots + K^{m-1})D
\]
\vfill
\[
K^m \quad \mbox{\bf compact}
\]
\vfill

\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] $3N-3$ continuous variables, $30-60$ points 
per dimension, $2^N$ spin channels.
\vfill

\item [{\color{red}$\bullet$}] Kernel has moving integrable singularities. 

\vfill

\item [{\color{red}$\bullet$}] Intervals infinite. 

\vfill

\item [{\color{red}$\bullet$}] Difficulty increases with energy.

\vfill

\item [{\color{red}$\bullet$}] $N=3,4$ about that largest that can be done supercomputing 
facilities.. 
}
\end{itemize}
\end{frame}

%slide 23 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
What can be learned
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] When and how do models with nucleons and 
mesons break down? 
\vfill
\item [{\color{red}$\bullet$}] Can we understand the relation between 
models with quark degrees of freedom and meson-nucleon degrees of freedom?
\vfill
\item [{\color{red}$\bullet$}] How do we separate effects due to 
relativity from
effect due to missing degrees of freedom?
\vfill
\item [{\color{red}$\bullet$}] How do we understand the dual roles of 
scattering and quark confinement?
\vfill
\item [{\color{red}$\bullet$}] How can be use QCD to understand these
questions?
}
\end{itemize}
\end{frame}


%slide 24 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Ongoing research
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Relativistic proton-deuteron scattering.
\vfill
\item [{\color{red}$\bullet$}] Models with equivalent quark and 
nucleon degrees of freedom (Peterson).
\vfill
\item [{\color{red}$\bullet$}] Scattering in Euclidean formulations of 
relativistic quantum mechanics (Kopp, Michlin).
\vfill
\item [{\color{red}$\bullet$}] The 0-body problem in light-front
QCD (Herrmann).
\vfill
\item [{\color{red}$\bullet$}] Poincar\'e symmetry in wavelet formulations of 
quantum field theory (Gelzer).
\vfill
\item  [{\color{red}$\bullet$}] Poincar\'e invariance of current operators
(Keister) .
}
\end{itemize}
\end{frame}
\end{document}

\end{frame}

\begin{frame}
\vfill
\begin{itemize}
{\bf 
\item [] H. Wita{\l}a, J. Golak, R. Skib\'nski, W. Gl\"ockle, 
H. Kamada, and W. N. Polyzou,  Phys. Rev. C83,044001(2011),
Data from: (135)
K. Sekiguchi et al., Phys. Rev. C65, 034003(2002)-(250 MeV),
K. Hatanaka et al., Phys. Rev. C66, 044002(2002)-(135 MeV). 
}
\end{itemize}
\end{frame}

%slide 24 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{5.0in}{!}{
\includegraphics{p2fig1_ting.pdf}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}


\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{5.0in}{!}{
\includegraphics{fig4_correct_rel_and_3NF.pdf}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}


\end{document}


\begin{frame}
\begin{center}
{\bf 
What have we learned
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item[{\color{red}$\bullet$}] Good model interactions between protons and neutrons. 
\vfill
\item[{\color{red}$\bullet$}] Good models of quark structure of nucleons. 
\vfill
\item[{\color{red}$\bullet$}] High and low energy limits of QCD understood and consistent with 
experiment.
}
\end{itemize}

\end{frame}

\begin{frame}

\begin{center}
{\bf 
What can we learn
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] How do we separate the effects due to relativity 
from the effects new relevant degrees of freedom?
\vfill
\item [{\color{red}$\bullet$}] What does is mean for scattering and 
confinement to coexist.
\vfill
\item [{\color{red}$\bullet$}] At what energies do we see signatures of 
individual quarks rather than protons and neutrons.
\vfill
\item [{\color{red}$\bullet$}] How is this distorted by confining interactions.
\vfill
\item [{\color{red}$\bullet$}] Can we understand QCD at the few-GeV scale.
}
\end{itemize}
\end{frame}


\begin{frame}
\vfill
\begin{itemize}
{\bf 
\item [] H. Wita{\l}a, J. Golak, R. Skib\'nski, W. Gl\"ockle, 
H. Kamada, and W. N. Polyzou,  Phys. Rev. C83,044001(2011),
Data from: (135)
K. Sekiguchi et al., Phys. Rev. C65, 034003(2002)-(250 MeV),
K. Hatanaka et al., Phys. Rev. C66, 044002(2002)-(135 MeV). 
}
\end{itemize}
\vfill
\end{frame}



\end{document}


\begin{frame}
\begin{center}
{\bf 
Collaborators
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Ch. Elster, T. Lin, M. Haziadeh (Ohio)
\vfill
\item [{\color{red}$\bullet$}] W. Gl\"ockle (Bochum)
\vfill
\item [{\color{red}$\bullet$}] J. Golak, R. Skib\'inski, H. Wita{\l}a (Jagiellonian U)
\vfill
\item [{\color{red}$\bullet$}] B. Keister (NSF)
\vfill
\item [{\color{red}$\bullet$}] F. Coester, P. Kopp (Iowa)
%\vfill
%\item [{\color{red}$\bullet$}] 
}
\end{itemize}
\end{frame}

%slide 3 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 

\begin{frame}
\begin{center}
{\bf 
Outline
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Relativistic invariance in quantum mechanics
\vfill
\item [{\color{red}$\bullet$}] Poincar\'e invariance
\vfill
\item [{\color{red}$\bullet$}] Cluster properties
\vfill
\item [{\color{red}$\bullet$}] Spin 
\vfill
\item [{\color{red}$\bullet$}] Position
\vfill
\item [{\color{red}$\bullet$}] Relativistic effects 
%\item [{\color{red}$\bullet$}] Wigner -Eckart theorem
\vfill
\item [{\color{red}$\bullet$}] Realizations and Relations
\vfill
\begin{itemize}
\item [{\color{red}$\bullet$}] Minkowski quantum field theory
\vfill
\item [{\color{red}$\bullet$}] Euclidean quantum field theory
\vfill
\item [{\color{red}$\bullet$}] {\color{blue} Covariant constraint dynamics} 
\vfill
\item [{\color{red}$\bullet$}] {\color{blue} Relativistic quantum mechanics of particles}
\vfill
\item [{\color{red}$\bullet$}] {\color{blue} Green function methods}
\vfill
\item [{\color{red}$\bullet$}] {\color{blue} Quasipotential methods}
\vfill
\item [{\color{red}$\bullet$}] {\color{blue} Euclidean relativistic quantum mechanics}
\vfill
\end{itemize}
%\item [{\color{red}$\bullet$}] Example: relativistic effects
%in the three nucleon system.
}
\end{itemize}
\end{frame}

%slide 4 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Special relativity - classical
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] {\color{blue}Inertial coordinate systems} 
(preferred coordinate systems). 
\vfill
\item [{\color{red}$\bullet$}]  Related by Lorentz 
transformations {\color{blue}and} space-time translations.
\vfill
\item [{\color{red}$\bullet$}] ``Dynamical equations have the same form 
in all inertial coordinate systems.''
}
\end{itemize} 
\end{frame}

%slide 5 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Special relativity - classical 
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [] Dynamical equations have the same form in all inertial coordinate 
systems. 
\[
{\color{blue}\Updownarrow}
\]
\item [] Solutions of dynamical equations have the same form in all 
inertial coordinate systems.
\[
{\color{blue}\Updownarrow}
\]
\item [] Theory predicts equivalent results in different inertial 
coordinate systems.
}
\end{itemize} 
\end{frame}

%slide 6 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Special relativity $+$ quantum mechanics \\
{\color{blue}wave functions are not observable!}
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] {\color{blue} ``Theory predicts equivalent results in all inertial coordinate systems.''}

\vfill

\item [{\color{red} X}] Solutions of dynamical equations have the same 
form in all inertial coordinate systems.

\vfill

\item [{\color{red} X}] Dynamical equations have the same form in 
all inertial coordinate systems. 

\vfill


\item [{\color{red}$\bullet$}] {\color{blue} Manifest covariance of 
dynamical equations is not required by special relativity in a quantum theory.}

}
\end{itemize} 
\end{frame}

%slide 7 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Classical relativity $=$ {\color{blue} manifest covariance of equations in ICS.}
}
\end{center}
\vfill
\[
\Downarrow
\]
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Classical field theories:
\vfill
\begin{itemize}
\item [{\color{red}$\bullet$}] Klein-Gordon equation.
\vfill
\item [{\color{red}$\bullet$}] Dirac equation.
\vfill
\item [{\color{red}$\bullet$}] Maxwell's equations.
\end{itemize} 
}
\end{itemize}
\vfill 
\begin{center}
{\bf
Quantum relativity $=$ {\color{blue}equivalence of experiments in ICS.} 
}
\end{center}
\vfill
\[
\Downarrow
\]
\vfill
\[
%\boxed{
\mathbf{U}(\pmb{\Lambda} ,\mathbf{a}) 
%}
\]
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Unitary representation of the Poincar\'e group,\\
{\color{blue} Wigner (Ann. Math. 40(1939)140)}.
\vfill
\item [{\color{red}$\bullet$}] Quantum field theories.
\vfill
\item [{\color{red}$\bullet$}] Poincar\'e invariant quantum mechanics.
}
\end{itemize} 

\end{frame}

%slide 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Special relativity in a quantum theory: 
}
\end{center}
\vfill
\[
\mathbf{X} \, \leftarrow\, (\Lambda ,a )\, \rightarrow \,  \mathbf{X}'
\]
\begin{center}
{\bf 
inertial
}
\end{center}
\vfill
\[
\vert \Psi \rangle \leftarrow\, (\Lambda ,a )\, 
\rightarrow \vert \Psi' \rangle
\]
\vfill
\[
\vert \Psi' \rangle = {\color{blue}U(\Lambda ,a)} \vert \Psi \rangle
\qquad 
A' = {\color{blue}U(\Lambda ,a)} A {\color{blue}U^{\dagger} (\Lambda ,a)}
\]
\vfill
\[
\rho' = {\color{blue}U(\Lambda ,a)} \rho {\color{blue}U^{\dagger} (\Lambda ,a)}
\]
\vfill
\begin{center}
{\bf
All experimental observables are independent of $\mathbf{X}$:
}
\end{center}
\vfill
\[
P = \vert \langle \Psi \vert \Phi \rangle\vert^2 =
\vert \langle \Psi' \vert \Phi' \rangle \vert^2 = P'
\]
\vfill
\[
\langle \Psi \vert A \vert \Psi \rangle
= \langle \Psi' \vert A' \vert \Psi' \rangle
\qquad 
\mbox{Tr}(\rho A) = \mbox{Tr}(\rho' A')
\]
\end{frame}

%slide 9 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Poincar\'e invariance\\
{\color{blue}(exact symmetry of theory)}
}
\end{center}
\vfill
\[
U(\Lambda ,a) \qquad \langle \Psi \vert \Phi \rangle = 
\langle \Psi' \vert \Phi' \rangle 
\]
\vfill
\[
\Updownarrow
\]
\vfill
\[
\{ H, \mathbf{P}, \mathbf{J}, \mathbf{K} \}
\]
\vfill
\[
{\color{blue}
[P^m,K^n] = i \delta_{mn} H
}
\]
\vfill
\begin{center}
{\bf 
{\color{blue}Hamiltonian on right !}
}
\end{center}

\vfill
\begin{center}
{\bf 
three equations $(n=x,y,z)$.
}
\end{center}
\vfill
\[
\cdots \, \vdots \, \cdots
\]
\end{frame}

%slide 10 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Consistent initial value problem
}
\end{center}


%\begin{center}
%{\bf 
%symmetric outgoing protons
%}
%\end{center}
%\begin{center}
\centering
\begin{figure} 
\rotatebox{00}{\resizebox{4.0in}{!}{
\includegraphics{pic.pdf}}
%\includegraphics[clip,viewport =000.0 100.0 600.0 350.0]{pic.pdf}}
%\includegraphics{p2fig1_ting.pdf}}
%\includegraphics*[0.0in,0.0in][8.0in,11.0in]{p2fig1_ting.pdf}}
%\includegraphics{p2fig1_ting.pdf}}
%\caption{breakup - symmetric outgoing protons}
%\label{fig:1}
}
\end{figure}
%\end{center}
\end{frame}
 
%slide 11 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
{\color{blue}Consistent initial value problem} means that\\
{\color{blue}at least three} generators have interactions,\\
{\color{blue}(Dirac 1949 RMP 21(1949)392).}
}
\end{center}
\begin{center}
{\bf 
Instant-form dynamics
}
\end{center}
\[
H = H_0 + {\color{blue}V}_I \qquad \mathbf{K} = \mathbf{K}_0 + {\color{blue}\mathbf{K}}_I  
\]
\begin{center}
{\bf 
Point-form dynamics
}
\end{center}
\[
H = H_0 + {\color{blue}V}_P \qquad \mathbf{P} = \mathbf{P}_0 + 
{\color{blue}\mathbf{P}}_I
\]
\begin{center}
{\bf 
Front-form dynamics
}
\end{center}
\[
P^- = H_0 - \hat{\mathbf{n}}\cdot \mathbf{P}_0 + {\color{blue}V}_F \qquad
\hat{\mathbf{n}} \times \mathbf{J} =  
\hat{\mathbf{n}} \times \mathbf{J}_0 +
\hat{\mathbf{n}} \times {\color{blue}\mathbf{J}}_I   
\]
\end{frame} 

%slide 12 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Cluster properties
}
\end{center}
\vfill
\[
U(\Lambda ,a) \to U_{s_1} (\Lambda ,a) \otimes  U_{s_2} (\Lambda ,a)
\otimes \cdots 
\]
\vfill
\[
\Downarrow
\]
\vfill
\[
\mathbf{K} = \sum_i \mathbf{K}_i + \sum_{ij} \mathbf{K}_{ij} + \sum_{ijk} \mathbf{K}_{ijk} + \cdots    
\]
\vfill
\[
\mathbf{J} = \sum_i \mathbf{J}_i     
\]
\vfill
\[
[K^x,K^y] = -i J^z
\]
\vfill
\[
\Downarrow
\]
\vfill
\[
[K_{12}^x,K_{23}^y] + [K_{23}^x,K_{12}^y] 
+[K_{12}^x,K_{31}^y] + [K_{23}^x,K_{32}^y] 
\]
\vfill
\[
+ [K_{31}^x,K_{23}^y] + [K_{31}^x,K_{12}^y] + {\color{blue}\cdots} =0
\]
\vfill
\begin{center}
{\bf
{\color{blue} 
Requires three-body $\mathbf{K}_{123} (\mathbf{K}_{12},\mathbf{K}_{23},
\mathbf{K}_{31}) \cdots $
}
}
\end{center}
\end{frame}

%slide 13 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Cluster properties
}
\end{center}
\vfill
\begin{center}
{\bf
{\color{blue}
What happens when they fail?
}
}
\end{center}
\vfill
\begin{center}
{\bf
Relativistic Jacobi momenta:
}
\end{center}
\vfill
\[
P := \sum p_i,\, {q}_i := B^{-1}(P) p_i,\,
k_{ij} := B^{-1} (q_i+q_j) q_i
\]
\vfill
\[
{\color{blue}
\mathbf{p}_3 = \mathbf{q}_3 + \mathbf{P} 
\Phi (\mathbf{P},\mathbf{q}_3, \mathbf{k}_{12})   
}
\]
\vfill
\[
\langle \mathbf{P} , \mathbf{q}_3, \mathbf{k}_{12} \vert V_{12} \vert
\mathbf{P}' , \mathbf{q}_3', \mathbf{k}_{12}' \rangle =
\]
\vfill
\[
\delta (\mathbf{P} - \mathbf{P}') 
\delta ( \mathbf{q}_3-\mathbf{q}_3') 
\langle \mathbf{k}_{12} \vert V_{12} \vert \mathbf{k}'_{12} \rangle
\]
\vfill
\[
\lim_{\vert \mathbf{a}_3\vert \to \infty} 
\Vert e^{i \mathbf{p}_3 \cdot \mathbf{a}_3} V_{12} 
e^{-i \mathbf{p}_3 \cdot \mathbf{a}_3} \vert \Psi \rangle \Vert  =
\]
\vfill
\[
{\color{blue}
\lim_{\vert \mathbf{a}_3\vert \to \infty} 
\Vert V_{12} 
e^{-i \mathbf{P} \cdot \mathbf{a}_3(\Phi - \Phi')  } \vert \Psi \rangle \Vert  =0 
\qquad
\mbox{\bf For $\mathbf{P}\not= 0$} 
}
\]
\end{frame} 

%slide 14 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Position in relativistic quantum mechanics:
}
\end{center}
\vfill
\begin{center}
{\bf 
{\color{blue} Particles}  cannot be localized !
}
\end{center}
\vfill
\[
\langle m, \mathbf{p} \vert x \rangle = 
\langle m, \mathbf{p} \vert U(I,x) \vert 0 \rangle = 
\]
\[
e^{i p \cdot x} \langle m, \mathbf{p} \vert 0 \rangle =
e^{i p \cdot x} \langle m, \mathbf{p} \vert 
U(\Lambda (p),0) \vert 0 \rangle =
\]
\[
e^{i p \cdot x} \sqrt{{m \over \omega_m (\mathbf{p})}} 
\langle m, \mathbf{0}  
\vert 0 \rangle 
\]
\vfill
\begin{center}
{\bf 
Calculate
}
\end{center}
\vfill 
\[
\langle \mathbf{x},0 \vert \mathbf{y},0 \rangle = 
\int e^{i \mathbf{p} \cdot (\mathbf{x}-\mathbf{y})}
{c \over \omega_m(\mathbf{p})} d\mathbf{p} =
2 \pi i c {K_1 (m \vert \mathbf{x} -\mathbf{y} \vert)
\over  m \vert \mathbf{x} -\mathbf{y} \vert}
\]
\vfill
\begin{center}
{\bf 
{\color{blue}
$\not= 0$ for $\mathbf{x} \not= \mathbf{y}$
}
}
\end{center}
\vfill
\begin{center}
{\bf 
Vanishes exponentially beyond a Compton wavelength.
}
\end{center}
\vfill
\begin{center}
{\bf 
Retardation meaningless for {\color{blue}particles}, 
(OK for fields). 
}
\end{center}

\end{frame}

%slide 15 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
``Position'' and spin
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [] To compare spins they should be transformed to a common frame.
\vfill
\[
[K_i, K_j] = - i \epsilon_{ijk} J_k
\]
\vfill
\item [] The choice of spin observables depends on the 
choice of ``standard'' boost to the common frame:
\vfill
\[
(0,\mathbf{j}_a) = {1 \over 2m} ({\color{blue} 
B_a^{-1}(p/m)})^{\mu}{}_{\nu}
\epsilon^{\nu \alpha \beta \gamma}p_{\alpha}J_{\beta \gamma}
\]  
\vfill
\item [] Different boosts are related by  {\color{blue}momentum-dependent} 
rotations
\vfill
\[
B_a^{-1} (p/m)B_b (p/m) = R_{ab}(p/m) \qquad 
\mathbf{j}_a = R_{ab}(\mathbf{p}) \mathbf{j}_b
\]
}
\end{itemize}
\end{frame}

%slide 16 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
``Position'' with spin
}
\end{center}
\vfill
\[
\mathbf{j}_h = R_{hc} (\mathbf{p}) \mathbf{j}_c  =
R_{hf} (\mathbf{p}) \mathbf{j}_{f}
\]
\vfill
\[
R_{ab}(\mathbf{p}) = B_a(\mathbf{p})^{-1} B_b (\mathbf{p}) 
\]
\vfill
\begin{center}
{\bf 
(Generalized Melosh rotations)
}
\end{center}
\vfill
\[
[\pmb{\nabla}_{p} , \mathbf{j}_h ]=0
\]
\vfill
\[
\Downarrow
\]
\vfill
\[
[\pmb{\nabla}_{p} , \mathbf{j}_c ] \not=0
\] 
\vfill
\[
[\pmb{\nabla}_{p} , \mathbf{j}_{lf} ] \not=0
\]
\vfill
\begin{center}
{\bf 
{\color{blue} Which spin is held constant when computing the 
partial derivative?
}
}
\end{center}
\end{frame}

%slide 17 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
2 and 4 component spinors/
Lorentz vs Poincar\'e invariance
}
\end{center}
\vfill
\begin{center}
{\bf 
Single-particle state
}
\end{center}
\vfill
\[
\vert (m,j) \mathbf{p}, \mu \rangle
\]
\vfill
\begin{center}
{\bf
Vector in mass $m$, spin $j$ irreducible representation space of the 
Poincar\'e group ($2j+1$ spin components)
}
\end{center}
\vfill
\[
U(\Lambda,0) \vert (m,j) \mathbf{p}, \mu \rangle =
\]
\vfill
\[
\sum \vert (m,j) \pmb{\Lambda}{p}, \mu' \rangle
\sqrt{{ \omega_m (\pmb{\Lambda}p) \over \omega_m (\pmb{p})}}
\underbrace{{\color{blue}D^j_{\mu' \mu}[B^{-1}(\pmb{\Lambda} p/m) \Lambda B(\mathbf{p}/m)]}}_{\mbox{\bf Wigner rotation}} 
\]
\end{frame}

%slide 18 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\[
\Downarrow 
\]
\vfill
\begin{center}
{\bf 
Define {\color{blue}equivalent} Lorentz covariant states\\
by breaking the Wigner rotation into products of LTs:
}
\end{center}
\vfill
%\[
%U(\Lambda,0) \sum_{\mu}\vert (m,j) \mathbf{p}, \mu \rangle
%\sqrt{\omega_m (\pmb{p})}
%D^j_{\mu \sigma}[ B^{-1}(\mathbf{p}/m)]  
%\]
%\vfill
%\[
%\sum \vert (m,j) \pmb{\Lambda}{p}, \mu \rangle
%\sqrt{ \omega_m (\pmb{\Lambda})}
%D^j_{\mu \sigma'}[B^{-1}(\pmb{\Lambda} p/m)] 
%{\color{blue}D^j_{\sigma'\sigma}[\Lambda]}  
%\]
%\vfill
\[
\vert {p},j, \sigma \rangle_c :=
\vert (m,j) \mathbf{p}, \mu \rangle \sqrt{\omega_m (\mathbf{p})}
D^j_{\mu \sigma}[ B^{-1}(\mathbf{p}/m)]
\]
\vfill
\[
U (\Lambda ,0) \vert p,j, \sigma \rangle_c  = \sum
\vert \Lambda p ,j, \sigma' \rangle_c
{\color{blue}D^j_{\sigma\sigma'}[\Lambda]}  
\]
\vfill
\begin{center}
{\bf 
$U(\Lambda )$ unitary with respect to inner product 
}
\end{center}
\vfill
\[
\langle \Psi \vert \Phi \rangle =
\]
\vfill
\[
\sumint \Psi^* (\mathbf{p}, \sigma)  
{\color{blue}{m d \mathbf{p}  \over \omega_{m}(\mathbf{p}) }
D^j_{\sigma \sigma'} [ B(\mathbf{p}/m)
B^{\dagger} (\mathbf{p}/m)]}
\Phi (\mathbf{p}, \sigma')
\]
\vfill
\[
\Phi (\mathbf{p}, \sigma) :=
_c\langle {p},j, \sigma \vert \Phi \rangle
\]

\end{frame}

%slide 19 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
What's the problem?
}
\end{center}
\vfill
\[ 
D[(R^{\dagger})^{-1}] = D[R] \qquad
{\color{blue}D[(\Lambda^{\dagger})^{-1}] \not\sim D[\Lambda]}    
\]
\vfill
\[
D[\Lambda]\, \underbrace{\longrightarrow}_{\mbox{\bf space reflection}}\, 
D[(\Lambda^{\dagger})^{-1}]
\]
\vfill
\begin{center}
{\bf 
Solution - double 
}
\end{center}
\vfill
\[
D^j_{\mu \mu'}[ R_w(\Lambda,p) ] \to 
D^j_{\mu \mu'}[ R_w(\Lambda,p) ] \oplus 
D^j_{\nu \nu'}[ (R_w^{\dagger})^{-1}(\Lambda,p) ]
\]
\vfill
\begin{center}
{\bf 
Redundant in Poincar\'e irreducible representations.
}
\end{center}
\end{frame}

%slide 20 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{itemize}
{\bf 
\item[{\color{red}$\bullet$}] Poincar\'e covariant representation: 2 component spinors that Wigner rotate -
depend on boost choice and momentum.
\vfill
\item[{\color{red}$\bullet$}] Lorentz covariant representation: 4 component spinors that transform 
under 
\vfill
\[
S[\Lambda] =D[\Lambda ]\oplus D[(\Lambda^{\dagger})^{-1}],
\]
\vfill
\item[] independent of boost choice and momentum.
\vfill
\item[{\color{red}$\bullet$}] Lorentz covariant measure 
for mass $m$ spin $j$ irreducible representations
\vfill
\[
{\color{blue} \to \delta (p^2 + m^2 ) \theta(p^0) d^4p 
D^j[{\sigma\cdot p\over m}]\oplus  D^j[\sigma_2{\sigma^*\cdot p\over m}\sigma_2
]
}
\]
}\end{itemize}
\end{frame}

%slide 21 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Two and four-component spins are related by intertwiners (boosts) 
}
\end{center}
\vfill
\begin{center}
{\bf 
Example: Dirac spinor $u(p)$
}
\end{center}
\vfill
\[
S(\Lambda ) u(p) =  u(\Lambda p)D^{1/2} (R_w(\Lambda ,p)) \qquad u(p) = 
S(B(p) ) u(p_{\mbox{rest}}) 
\]
\vfill
\begin{itemize}
{\bf 
\item [] Matrix elements of fields in Poincar\'e irreducible eigenstates 
also act as intertwiners relating Lorentz covariant quantities to
Poincar\'e covariant quantities:
}
\end{itemize}
\vfill
\[
\langle (j,m) \mathbf{p}, \mu \vert \Psi^j_\sigma (x)  \vert 0 \rangle
\]
\vfill
\[
\langle (j,m) \mathbf{p}, \mu \vert U(\Lambda, 0) \vert \Psi^j_\sigma (x)  \vert 0 \rangle =
\]
\vfill
\[
D^j_{\mu \mu'} [R_w^{-1} (\Lambda,  \Lambda^{-1} p )] 
{\color{blue}
\langle (j,m) \pmb{\Lambda}^{-1}{p}, \mu' \vert \Psi^j_\sigma (x)  \vert 0 \rangle} =
\]
\vfill
\[
{\color{blue}\langle (j,m) {p}, \mu \vert \Psi^j_{\sigma'} (\Lambda x)  \vert 0 \rangle}
S^j_{\sigma'\sigma} [\Lambda ] 
\] 
\end{frame} 

%slide 22 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Relativistic corrections
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item[] Experimental data transformed to center of momentum frame by 
Lorentz transformations.
\vfill
\item[] For the strong interaction relativistic and non-relativistic interactions are 
fit to transformed data as a function of CM momentum,
$\mathbf{k}$.
\vfill
\item[] {\color{blue}No relativistic effects in the two-nucleon problem.}
\vfill
\item [] First appear in the three-nucleon problem - related to 
two-nucleon problem by cluster properties. 
\vfill
\item [] $\mathbf{v}/c$ or $\mathbf{k}/m$ 
expansions {\color{blue}make no sense for $N=2$.}
}
\end{itemize}
\end{frame}

%slide 23 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
{\color{blue}Realizations - Hilbert space inner products}
}
\end{center}
\vfill
\begin{center}
{\bf 
Single particles - free fields
}
\end{center}
\vfill
\begin{center}
{\bf 
Relativistic quantum mechanics
}
\end{center}
\vfill
\[
\int \Psi^* (\mathbf{p}) \Psi (\mathbf{p}) d \mathbf{p} = 
\]
\vfill
\begin{center}
{\bf 
Relativistic quantum field theory
}
\end{center}
\vfill
\[
\int f(x)^* W(x-y) f(y) d^4x d^4y =  
\]
\vfill
\begin{center}
{\bf 
Euclidean quantum field theory
}
\end{center}
\vfill
\[
\int g(\mathbf{x},{\color{blue}-\tau_x} )^* S(x-y) g(\mathbf{y},\tau_y) d^4x_e d^4y_e
\]
\vfill
\[
W(x-y) = \langle 0 \vert 
\phi(x) \phi(y) \vert 0 \rangle = 
c \int d^4p \theta (p^0) \delta (p^2 + m^2) e^{i p \cdot (x-y)}
%\qquad \tilde{W}(p) = c \delta(p^2 + m^2) 
\]
\vfill
\[
\tilde{S} (p_e) = c/(p_e^2 + m^2)
\]
\end{frame}

%slide 24 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


\begin{frame}
\begin{center}
{\bf 
Relations - wave functions
}
\end{center}
\vfill
\begin{center}
{\bf 
(free particles/free field of mass $m$) 
}
\end{center}

\vfill
\[
{\color{blue}\Psi (\mathbf{p})}  =  
\langle \mathbf{p} \vert \phi(f) \vert 0 \rangle 
\]
\vfill
\[
=\int {\color{blue}f(\mathbf{x},t)} e^{-i \omega_m (\mathbf{p})t - i \mathbf{p} \cdot x}
\sqrt{{m \over \omega_{m}(\mathbf{p})}}
% \langle m, \mathbf{p}=\mathbf{0} \vert \phi(x=0) \vert 0 \rangle
d^4 x 
\]
\vfill
\[
= \int {\color{blue}g(\mathbf{x},\tau_x)} e^{- \omega_m (\mathbf{p})\tau_x - i \mathbf{p} \cdot x}
\sqrt{{m \over \omega_{m}(\mathbf{p})}} d^4x_e
\]
\end{frame}

%slide 25 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Unitary representations of the Poincar\'e group
}
\end{center}
\vfill
\[
f(x) \to f_{\Lambda ,a}(x) = f(\Lambda^{-1}( x-a))
\]
\vfill
\begin{center}
{\bf 
Unitary $\Leftrightarrow$  $W$ invariant
}
\end{center}
\vfill
\[
W(x-y) = W(\Lambda (x - y))  
\]
\vfill
\[
\Downarrow
\]
\vfill
\begin{center}
{\bf 
Poincar\'e transformation properties of
$\Psi(\mathbf{p})$ and $g(\mathbf{y},\tau)$. 
}
\end{center}
\vfill
\[
\Downarrow
\]
\begin{center} 
{\bf 
Dynamical generators
}
\end{center}
\[
H g(x) = {\partial g \over \partial \tau} (x) \qquad
K^i g(x) = \tau {\partial g \over \partial x_i} (x)-
x_i {\partial g \over \partial \tau} (x)
\]
\vfill
\[
H \Psi (\mathbf{p}) = \sqrt{\mathbf{p}^2 + m^2} \Psi (\mathbf{p})
\qquad
K_i \Psi (\mathbf{p}) = {i \over 2}\{ H ,
{\partial  \over \partial p_i}  \} 
\Psi (\mathbf{p})
\]
\end{frame} 

%slide 26 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
General case
}
\end{center}
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] General case

\vfill

\[
\langle\Psi \vert\Psi \rangle =
\]
\vfill
\[
\sum_{kn}( Rg_k^{\dagger}, S^{k+n} g_n) = \sum_{kn}(f_k^{\dagger}, W^{k+n} f_n) = 
\]
\vfill
\[
\int \sum_{kn} 
\underbrace{
\langle 0 \vert \prod_{i=1}^k \phi (f^*_i) 
\vert \mathbf{p},m \rangle}_{\langle \Psi \vert \mathbf{p} ,\cdots \rangle}  d\mathbf{p} dm
\underbrace{\langle \mathbf{p},m \vert \prod_{j=1}^m \phi (f_j)
\vert 0 \rangle}_{\langle \mathbf{p}, \cdots \vert \Psi \rangle }    
\]
\vfill
\item [{\color{red}$\bullet$}] $R$ is a Euclidean time reflection.
\vfill
\item [{\color{red}$\bullet$}] The general correspondence is precise but 
it requires a choice of Poincar\'e irreducible basis and 
knowing the {\color{blue}fields} or the
{\color{blue}Euclidean Green functions}.
}
\end{itemize}
\end{frame}

%slide 27 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Field theory
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [] Minkowski field theory
\vfill
\item[] Euclidean field theory
}
\end{itemize}
\vfill
\begin{center}
{\bf 
Models
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [] Covariant constraint dynamics
\vfill
\item [] Direct interaction methods 
\vfill
\item [] Green function methods
\vfill
\item [] Quasipotential methods 
\vfill
\item [] Euclidean Methods
}
\end{itemize}
\end{frame}

%slide 28 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Minkowski quantum field theory
}
\end{center}
\vfill

\begin{center}
{\bf 
Input
}
\end{center}
\[
W_n (x_1, \cdots x_n ):=
\langle 0 \vert \prod \phi_i (x_i) \vert 0 \rangle
\]
\begin{center}
{\bf 
Hilbert space
}
\end{center}
\[
\langle f \vert g \rangle 
 = \sum_{mn} \int f^*_m(x)W_{m+n} (x,y )  g_n (y) d^{4m}x d^{4n}y
\]
\begin{center}
{\bf 
Unitary Poincar\'e transformations
}
\end{center}
\[
g_n(x) \to  D(\Lambda^{-1}) g_n (\Lambda x+a)
\]
\end{frame}


%slide 29 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Euclidean quantum field theory
}
\end{center}
\vfill

\begin{center}
{\bf 
Input reflection positive Euclidean Green functions
}
\end{center}
\vfill
\[
S_n (x_1, \cdots x_n ) = 
\int D[\phi]e^{-A[\phi]} \prod \phi_i (x_i)
\]
\vfill
\begin{center}
{\bf 
Hilbert space (positive relative time functions)
}
\end{center}
\vfill
\[
\langle f \vert g \rangle 
 = \sum_{mn} \int f^*_m(Rx)S_{m+n} (x,y )  g_n (y) d^{4m}x d^{4n}y
\]
\vfill
\begin{center}
{\bf 
Euclidean group $=$ 10 parameter subgroup of complex 
Poincar\'e group $\Rightarrow$ Poincar\'e generators
}
\end{center}
\end{frame}

%slide 30 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Few-body models: 1
}
\end{center}
\vfill
\begin{center}
{\bf 
{\color{blue}Covariant constraint dynamics}
}
\end{center}
\vfill
\begin{center}
{\bf 
Model Wightman functions
}
\end{center}
\vfill
\[
W = \langle 0 \vert \phi(x_1) \cdots \phi(x_n) \vert 0 \rangle
\]
\vfill
\begin{center}
{\bf 
Covariant constraints: 
}
\end{center}
\[
C_1 \psi = (p_1^2 + m_1^2 + V_1)\psi \thickapprox  0
\]
\[
\vdots
\]
\[
C_n \psi = (p_n^2 + m_n^2 + V_n)\psi  \thickapprox 0
\]
\vfill
\begin{center}
{\bf 
First class condition:
}
\end{center}
\vfill
\[
[C_i, C_j]  \thickapprox 0 
\]
\vfill
\begin{center}
{\bf
realization ($N=2, \,  V_1=V_2=V$)
}
\end{center}
\vfill
\[
[p_1^2 - p_2^2 , V ]  \thickapprox0
\]
\end{frame}


%slide 31 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\vfill
\begin{center}
{\bf
Model Wightman function
}
\end{center}
\vfill
\[
W = \prod_i \delta (C_i)  
\]
\vfill
\begin{center}
{\bf
Covariant scalar product ($N=2$)
}
\end{center}
\vfill
\[
\langle g \vert f \rangle =  
\]
\vfill
\[
\int g^*(x_1, x_2)
\langle x_1, x_2 \vert \delta (C_1)\delta (C_2) \vert y_2, y_1 \rangle f (y_1,y_2)
d^8x d^8y
\]
\vfill
\begin{center}
{\bf
Poincar\'e transformations
}
\end{center}
\vfill
\[
f (y_1,y_2) \to f (\Lambda y_1+a ,\Lambda y_2 + a )
\]
\vfill
\begin{center}
{\bf
Unitary for covariant constraints
}
\end{center}
\end{frame}

%slide 32 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
{
\tiny
{\bf Constraint dynamics equations for two spin $1/2$ particles
interacting by vector and scalar potentials:}%
\begin{equation*}
\mathcal{C}_{1}\Psi \psi \equiv (\gamma _{1}\cdot (p_{1}-\tilde{V}%
_{1})+m_{1}+\tilde{S}_{1})\Psi =0,~~\mathcal{C}_{2}\Psi \equiv (\gamma
_{2}\cdot (p_{2}-\tilde{V}_{2})+m_{2}+\tilde{S}_{2})\Psi =0.
\end{equation*}%
{\bf Compatibility :} $[\mathcal{C}_{1},\mathcal{C}_{2}]\Psi =0,$%
\vfill
\begin{table}[t] % [hbt]                                                        
%{\bf Table 1: } \\[1.0ex]                                                      
\caption{\bf Covariant constraint dynamics: results for positronium}
\centering
\begin{tabular}{lllllllll}
\hline
$l$&$s$&$j$&$n$&$N_c$& &perturbative&numerical&diff/${\mu\alpha^4}$\\
0&0&0&1&1&uu &-6.8033256279&-6.8032861579&\phantom{-}5.45E-02\\
0&0&0&1&2&uu \&  ll &-6.8033256279&-6.8033256719&-6.08E-05\\
0&1&1&1&1&uu&-6.8028426132&-6.8028074990&\phantom{-}4.84E-02\\
0&1&1&1&2&uu \& ll&-6.8028426132&-6.8028082195&\phantom{-}4.75E-02\\
0&1&1&1&2&uu \& tensor&-6.8028426132&-6.8028239499&\phantom{-}2.58E-02\\
0&1&1&1&4&uu, ll \& tensor&-6.8028426132&-6.8028426636&-6.97E-05\\
\hline
\end{tabular}
\end{table}
%}
\vfill
\begin{itemize}
{\bf 
\item [] H. W. Crater, R. Becker, C.
Y. Wong, and P. Van Alstine, Phys. Rev. D46 , 5117 (1992),
Horace Crater and Cheuk-Yin Wong, Phys. Rev. D 85, 
116005(2012).
}
\end{itemize}
%\end{frame}
\vfill
%\begin{frame}
\tiny
\begin{table}[t] % [hbt]
%{\bf Table 1: } \\[1.0ex]
\caption{\bf Covariant constraint dynamics: results for mesons}
\centering
\begin{tabular}{lllllll}
\hline
$ \pi (140)\text{/}0.144 $ &&&\\
$ \rho (767)\text{/}0.792$ &&&\\%
$ \phi (1.019)\text{/}1.033$ &&&\\
$K(494)\text{/}0.492 $&$ K^{\ast }(892)\text{/}0.910 $&\\
$B(5.279)\text{/}5.273 $&$ B^{\ast }(5.325)\text{/}5.321 $&$ B_{s}(5.369\text{)/}%
5.368 $
&$ B_{s}^{\ast }(5.416)\text{/}5.427$\\
$D(1.865)\text{/}1.866 $&$ D^{\ast }(2.007)\text{/}2.000$ &$ D_{s}(1.968)\text{/}%
1.976 $&$ D_{s}^{\ast }(2.112)\text{/}2.123~$ \\
$c\bar{c}(2.980)\text{/}2.978 $&$c\bar{c}(3.097)\text{/}3.129 $&$c\bar{c}(3.526)\text{/}3.520 $&$c\bar{c}(3.415)\text{/}3.407 $\\
&$c\bar{c}(3.510)\text{/}3.507 $&$c\bar{c}(3.556)\text{/}3.549 $&$c\bar{c}(3.686)\text{/} 3.688$\\%
$\Upsilon (9.460)\text{/}9.453$ & $\Upsilon (9.860)\text{/}9.842$ & $\Upsilon
\left( 9.892\right) \text{/}9.889$ & $\Upsilon \left( 9.913\right) \text{/}%
9.921$\\
& $\Upsilon \left( 10.023\right) \text{/}10.022$&&\\
\hline
\end{tabular}
\end{table}
\begin{itemize}
{\bf  
\item [] Horace Crater
and Peter Van Alstine, Phys. Rev. D 37 1982 (1988),
Horace W. Crater, James Schiermeyer 
Phys. Rev. D82,094020 (2010).
}
\end{itemize}
}
\end{frame}


%slide 33 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Few-body models: 2
}
\end{center}
\vfill
\begin{center}
{\bf
{\color{blue}
Relativistic quantum mechanics of particles}
}
\end{center}
\vfill
\[
{\cal H} = \oplus (\otimes {\cal H}_{jm})
\]
\vfill
\[
H = \sum_i H_i + \sum_{ij} H_{ij} + \sum_{ijk} H_{ijk} + \cdots 
\]
\vfill
\[
\mathbf{P} = 
\sum_i \mathbf{P}_i + \sum_{ij} \mathbf{P}_{ij} + \sum_{ijk} 
\mathbf{P}_{ijk} + \cdots 
\]
\vfill
\[
\mathbf{J} = 
\sum_i \mathbf{J}_i + \sum_{ij} \mathbf{J}_{ij} + \sum_{ijk} 
\mathbf{J}_{ijk} + \cdots 
\]
\vfill
\[
\mathbf{K} = 
\sum_i \mathbf{K}_i + \sum_{ij} \mathbf{K}_{ij} + \sum_{ijk} 
\mathbf{K}_{ijk} + \cdots 
\]
\vfill
\begin{itemize}
{\bf 
\item [] Input: interactions constrained to satisfy Poincar\'e 
commutation relations, cluster properties, and spectral condition.
}
\end{itemize}
\end{frame}

%slide 34 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Models satisfying commutation relations - {\color{blue}
B. Bakamjian and L. H. Thomas, Phys Rev 92,1300(1953).}

\vfill

\item [{\color{red}$\bullet$}] Models satisfying $S$-matrix cluster properties for 
$N=3$ particles  - {\color{blue}
F. Coester, Helv. Phys. Acta, 38,7(1965).}

\vfill

\item [{\color{red}$\bullet$}] Models satisfying cluster properties, all $N$
- {\color{blue}
S. N. Sokolov, Dokl. Akad, Nauk SSSR, 233, 575(1977).}

\vfill

\item [{\color{red}$\bullet$}] Models with different kinematic subgroups related by 
S-matrix preserving unitary transformations

\vfill

\item [{\color{red}$\bullet$}] Numerous applications to few-body  
models with nucleon or constituent quark degrees of freedom.
}
\end{itemize}


\end{frame}


%slide 36 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
%\begin{center}
%{\bf 
%symmetric outgoing protons
%}
%\end{center}
\begin{itemize}
{\bf 
\item [] T. Lin, Ch. Elster, W. N. Polyzou, W. Gl\"ockle,
Physics Letters B660(2008),345., Data from  
V. Punjabi et al., Phys. Rev. C38, 2728 (1988).
}
\end{itemize}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{5.0in}{!}{
\includegraphics[clip,viewport =00.0 00.0 600.0 350.0]{p2fig1_ting.pdf}}
%\includegraphics{p2fig1_ting.pdf}}
%\includegraphics*[0.0in,0.0in][8.0in,11.0in]{p2fig1_ting.pdf}}
%\includegraphics{p2fig1_ting.pdf}}
%\caption{breakup - symmetric outgoing protons}
%\label{fig:1}
}
\end{figure}
\end{center}
\end{frame}

%slide 35 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%\begin{frame}
%\begin{center}
%{\bf 
%Examples
%}
%\end{center}
%\begin{center}
%{\bf 
%Example 1; Relativistic quantum mechanics of particles
%}
%\end{center}
%\vfill
%\begin{center}
%{\bf 
%Selected spin-dependent pd breakup observables at 270 
%MeV with and without 3N forces.
%Selected polarization observables comparing relativistic and 
%non-relativistic pd scattering with and without 3NFs. 
% for different
% proton angle pairs $\theta_2-\theta_2$ around the beam axis as
% function of the laboratory kinetic energy of one of the outgoing
% protons.
%}
%\end{center}
%\vfill
%
%\begin{center}
%{\bf 
%Calculations: H. Wita{\l}a, et al,\\
%J. Golak, R. Skib\'inski, W. Gl\"ockle, H. Kamada, and W. 
%N. Polyzou
%Phys. Rev. C.83.044001(2011).
%}
%\end{center}
%
%\vfill
%\begin{center}
%{\bf 
%Data from: \\
%K. Sekiguchi et al., Phys. Rev. C79, 054008(2009).
%}
%\end{center}
%
%\vfill
%\begin{itemize}
%{\bf
%\item [{\color{red}$\bullet$}] {\color{red}red} - non-relativistic - CD Bonn 
%\item [{\color{red}$\bullet$}] {\color{blue}blue dots} - 
%non-relativistic - CD Bonn $+$ TM 3Nf 
%\item [{\color{red}$\bullet$}] {\color{blue}blue dashes}  - 
%relativistic - CD Bonn - no Wigner rotations
%\item [{\color{red}$\bullet$}] {\color{brown}brown dash-dot} - 
%relativistic - CD Bonn $+$ TM 3Nf - no Wigner rotations
%}
%\end{itemize}
%\end{frame}

%slide 37 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


\begin{frame}
\vfill
\begin{itemize}
{\bf 
\item [] H. Wita{\l}a, J. Golak, R. Skib\'nski, W. Gl\"ockle, 
H. Kamada, and W. N. Polyzou,  Phys. Rev. C83,044001(2011),
Data from: (135)
K. Sekiguchi et al., Phys. Rev. C65, 034003(2002)-(250 MeV),
K. Hatanaka et al., Phys. Rev. C66, 044002(2002)-(135 MeV). 
}
\end{itemize}
\vfill
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig4_correct_rel_and_3NF.pdf}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}

%slide 38 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%\begin{frame}
%\vfill
%\begin{itemize}
%{\bf 
%\item [] H. Wita{\l}a, J. Golak, R. Skib\'nski, W. Gl\"ockle, 
%H. Kamada, and W.N. Polyzou,  Phys. Rev. C.83.044001(2011),
%Data from: 
%W. Pairsuwan, et al. Phys. Rev. C52, 2552 (1995).
%}
%\end{itemize}
%\vfill
%\begin{center}
%\begin{figure} 
%\rotatebox{00}{\resizebox{2.5in}{!}{
%\includegraphics{fig16.pdf}}
%\includegraphics{fig18.pdf}}
%}
%\end{figure}
%\end{center}
%\end{frame}
%
%slide 39 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\vfill
\begin{itemize}
{\bf 
\item [] H. Wita{\l}a, J. Golak, R. Skib\'nski, W. Gl\"ockle, 
H. Kamada, and W.N. Polyzou,  Phys. Rev. C.83.044001(2011),
Data from: 
K. Sekiguchi et al., Phys. Rev. C79, 054008(2009).
}
\end{itemize}
\vfill
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig18.pdf}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}

%slide 40 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{4.0in}{!}{
\includegraphics{Plessas-SM-LC2013-28.pdf}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}

%slide 43 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Few-body models: 3
}
\end{center}
\vfill
\begin{center}
{\bf 
{\color{blue}
Green function methods}
}
\end{center}
\vfill
\begin{center}
{\bf 
{\color{blue}
Input - model covariant Bethe-Salpeter kernels, 
multipoint generalizations.
}
}
\end{center}
\vfill
\begin{itemize}
{\bf 

\item[] Solve Bethe-Salpeter (Schwinger-Dyson) equation 
for time-ordered Green functions given model input kernel.

\vfill

\item[] Assumes underlying Hilbert space structure and complete sets
of Poincar\'e irreducible eigenstates.

\vfill

\item[] Extract matrix elements of
operators from residues of poles of computed Green functions $G$
}
\end{itemize}

\end{frame}

%slide 44 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
{\footnotesize
\begin{center}
{\bf 
$N=2$ case
}
\end{center}
\vfill
\[
K = - (G^{-1})_c := G_0^{-1}- {\color{blue}G^{-1}} 
\qquad \mbox{\bf cluster properties} 
\]
\vfill
\[
G = G_0 + G_0 K G 
\]
\vfill
\[
\tilde{G}(P) = \tilde{G}_{0}(P) + \tilde{G}_{0}(P) \tilde{K}(P) 
\tilde{G}(P) 
\]
\vfill
\begin{center}
{\bf 
Time ordering (heaviside function)
}
\end{center}
\vfill
\[
\Downarrow
\]
\vfill
\[
\tilde{G}(P) = {\chi (P) \bar{\chi} (P) \over P^0-E} + \cdots  
\]
\vfill
\begin{center}
{\bf 
Bound states
}
\end{center}
\vfill
\[
\chi (P) = \tilde{G}_0(P)\tilde{K}(P) \chi (P) 
\qquad P^2 = - {\color{blue}M}^2 
\]
\vfill
\begin{center}   
{\bf
They satisfy the {\color{blue}normalization condition} 
}
\end{center} 
\vfill
\[
1= {i \over 2\pi} \chi(P) {\partial \tilde{G}^{-1}(P) \over \partial P^0} 
\bar{\chi}(P)
\]
}
\end{frame}

%slide 45 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\vfill
\begin{center}   
{\bf
Formal relation to wave functions
}
\end{center} 
\vfill
\[
{\color{blue}{\chi} (P) = \langle 0  \vert  T
(\phi (x) \phi (y) )\vert (m,j)\mathbf{p},\mu  \rangle} 
\]
%\vfill
%\[
%{\color{blue}\{} \langle 0 \vert \phi (x_1) 
%\phi (x_2 ) \cdots \phi (x_n )\vert 
%\mathbf{p},m \rangle {\color{blue}\}}_{n= 0}^\infty 
%\]
\vfill
\begin{center}
{\bf 
Matrix elements expressed as quadratic forms
}
\end{center}
\vfill
\[
\langle (m,j)\mathbf{p}, \mu \vert I^{\mu}(0) \vert (m',j')\mathbf{p}', \mu' \rangle =
\]
\vfill
\[
- \lim {(P^0-E) (P^{0\prime}-E')\over 4 \pi^2} 
\times 
\]
\[
\bar{\chi} (P) 
{\partial \tilde{G}^{-1}(P) \over \partial P^0}
\tilde{R}^{\mu}  (P,P')
{\partial \tilde{G}^{-1\prime }(P') \over \partial P^{0\prime}}
{\chi}'(P')
\] 
\vfill
\begin{center}
{\bf 
in terms of new input current vertex 
}
\end{center}
\vfill
\[
R^{\mu} = \langle 0 \vert T(\phi(x_1) \phi(x_2) I^{\mu} (0) 
\phi(x_4) \phi(x_5) )\vert 0 \rangle
\]
\end{frame}

%slide 47 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Few-body models: 4
}
\end{center}
\vfill
\begin{center}
{\bf 
{\color{blue}
Quasipotential methods
}
}
\end{center}
\vfill
\begin{center}
{\bf 
Based on Green function methods - reduce 4(n-1) variable 
equations to 3(n-1) variable equations.
}
\end{center}
\vfill
\[
\tilde{G}_0(P) = \tilde{g}_0 (P) + \tilde{\Delta} (P)
\]
\vfill
\begin{center}
{\bf 
$\tilde{\Delta} (P)$ non-singular 
}
\end{center}
\vfill
\begin{center}
{\bf 
Quasipotential
}
\end{center}
\vfill
\[
\tilde{U} (P) = \tilde{K}(P) + \tilde{K}(P) \tilde{\Delta} (P) \tilde{U}(P) 
\]
\vfill
\[
\tilde{g}(P) = \tilde{g}_0(P) + \tilde{g}_0(P) \tilde{U}(P) \tilde{g}(P)
\]

\begin{center}
{\bf 
$\tilde{g}(P)$ has same poles as $\tilde{G}(P)$
}
\end{center}

\end{frame}

%slide 47 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Bound state equation
}
\end{center}
\vfill
\[
\xi (P) = \tilde{g}_0 (P) \tilde{U}(P) \xi (P)   
\]
\vfill
\begin{center}
{\bf 
Relation to BS wave function
}
\end{center}
\vfill
\[
\tilde{K}(P) \chi (P)  =  \tilde{U} (P) \xi (P) 
\]
\vfill
\[
\chi (P)  =  \tilde{G}_0^{-1}(P) \tilde{U} (P) \xi (P) 
\]
\vfill
\begin{center}
{\bf 
Normalization condition 
}
\end{center}
\vfill
\[
1= {i \over 2\pi} \xi(P) {\partial g^{-1}(P) \over \partial P^0} \bar{\xi}(P)
\]
\vfill
\begin{center}
{\bf 
Can be used to calculate matrix elements as before
}
\end{center}
\end{frame}

%slide 48 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\vfill
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{3.5in}{!}{
\includegraphics{CSTslide.pdf}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}

%slide 41 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Few-body models: 5
}
\end{center}
\vfill
\begin{center}
{\bf 
{\color{blue}Model: Euclidean relativistic quantum mechanics}
}
\end{center}
\vfill

\begin{center}
{\bf 
Input: model {\color{blue}reflection positive} Euclidean Green functions
}
\end{center}
\vfill
\[
S_n (x_1, \cdots x_n ) 
\]
\vfill
\begin{center}
{\bf 
Hilbert space (positive relative time functions)
}
\end{center}
\vfill
\[
\langle f \vert g \rangle 
 = \sum_{mn} \int f^*_m(Rx)S_{m+n} (x,y )  g_n (y) d^{4m}x d^{4n}y
\]
\vfill
\begin{center}
{\bf 
Euclidean group $=$ 10 parameter subgroup of complex 
Poincar\'e group $\Rightarrow$ Poincar\'e generators
}
\end{center}
\end{frame}

%slide 42 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
{\footnotesize
\begin{center}
\begin{center}
{\bf
GeV-scale elastic scattering using matrix elements of $e^{-\beta H}$ 
}
\end{center}
\vfill
\[
\langle \psi \vert \Omega^{\dagger}_+ \Omega_- \vert \phi \rangle 
\qquad \Omega_{\pm} = 
\lim_{n \to \pm \infty} e^{-ine^{-\beta H}}e^{+ine^{-\beta H_0}}
%\qquad S = \Omega_+\Omega_-
\]
%
%{\bf 
%Convergence with respect time ``$n$''
%}
\end{center}
\vbox{
\begin{center}
%{\bf Table 2: $k_0=2.0$[GeV], $\alpha=135$[GeV$^{-2}$]  }
{\bf Table 2: $k_0=2.0$[GeV] }
\end{center}
\begin{center}
\begin{tabular}{lll}
\hline
$n$ & Re $\langle \phi \vert (S_n-I) \vert \phi \rangle$ & Im $\langle \phi \vert (S_n-I) \vert \phi \rangle$ \\
\hline
50  & {\color{red}-2.}60094316473225e-6 & 1.94120750171791e-3 \\ 
100 & {\color{red}-2.8}2916859895010e-6 & {\color{red}2.3}5553585404449e-3 \\
150 & {\color{red}-2.831}71624670953e-6 & {\color{red}2.374}71383801820e-3 \\ 
200 & {\color{red}-2.831659}46257657e-6 & {\color{red}2.3749}2460997990e-3 \\ 
250 & {\color{red}-2.83165905}312632e-6 & {\color{red}2.37492527}186858e-3 \\ 
300 & {\color{red}-2.831659052}57121e-6 & {\color{red}2.374925272}62432e-3 \\ 
350 & {\color{red}-2.83165905}190508e-6 & {\color{red}2.374925272}62493e-3 \\
400 & {\color{red}-2.831659052}34917e-6 & {\color{red}2.374925272}62540e-3 \\
\hline                                             
ex  & -2.83165905227843e-6 & 2.37492527259701e-3 \\
\hline
\end{tabular}
\end{center}
}
\begin{itemize}
{\bf 
\item [] P. Kopp and W. N. Polyzou, Phys. Rev. D85,016004(2012).
}
\end{itemize}
}
\end{frame}

%slide 49 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Many formulations of relativistic few-body quantum mechanics . 
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] All have underlying unitary representation of 
the Poincar\'e group.
\vfill
\item [{\color{red}$\bullet$}] All representations of the Poincar\'e 
group can be decomposed in direct integrals of irreducible representations. 
\vfill
\item [{\color{red}$\bullet$}] These representations are the key to 
understanding the relation between different formalisms.
\vfill
\item [{\color{red}$\bullet$}] Relativistic few-body methods provide a means 
to perform realistic calculations of few-GeV scale observables that are not
easily calculated using other means. 
}
\end{itemize}
\end{frame}


\end{document}

\end{document}

\begin{frame}

\begin{center}
{\bf 
Examples
}
\end{center}
\begin{center}
{\bf 
Example 1; Relativistic quantum mechanics of particles
}
\end{center}
\vfill
\begin{center}
{\bf 
%Selected spin-dependent pd breakup observables at 270 
%MeV with and without 3N forces.
%Selected polarization observables comparing relativistic and 
%non-relativistic pd scattering with and without 3NFs. 
% for different
% proton angle pairs $\theta_2-\theta_2$ around the beam axis as
% function of the laboratory kinetic energy of one of the outgoing
% protons.
}
\end{center}
\vfill

\begin{center}
{\bf 
Calculations: H. Wita{\l}a, et al,\\
%J. Golak, R. Skib\'inski, W. Gl\"ockle, H. Kamada, and W. 
%N. Polyzou
Phys. Rev. C.83.044001(2011).
}
\end{center}

\vfill
\begin{center}
{\bf 
Data from: \\
K. Sekiguchi et al., Phys. Rev. C79, 054008(2009).
}
\end{center}

 

\vfill
\begin{itemize}
{\bf
\item [{\color{red}$\bullet$}] {\color{red}red} - non-relativistic - CD Bonn 
\item [{\color{red}$\bullet$}] {\color{blue}blue dots} - non-relativistic - CD Bonn $+$ TM 3Nf 
\item [{\color{red}$\bullet$}] {\color{blue}blue dashes}  - relativistic - CD Bonn - no Wigner rotations
\item [{\color{red}$\bullet$}] {\color{brown}brown dash-dot} - relativistic - CD Bonn $+$ TM 3Nf - no Wigner rotations
}
\end{itemize}

\end{frame}


\begin{frame}
\begin{center}
{\bf 
symmetric outgoing protons
}
\end{center}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{6.0in}{!}{
\includegraphics[clip,viewport =00.0 00.0 600.0 350.0]{p2fig1_ting.pdf}}
%\includegraphics{p2fig1_ting.pdf}}
%\includegraphics*[0.0in,0.0in][8.0in,11.0in]{p2fig1_ting.pdf}}
%\includegraphics{p2fig1_ting.pdf}}
%\caption{breakup - symmetric outgoing protons}
%\label{fig:1}
}
\end{figure}
\end{center}
\end{frame}


\begin{frame}
\vfill
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig4_correct_rel_and_3NF.pdf}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\vfill
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{3.5in}{!}{
\includegraphics{CSTslide.pdf}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
{\bf 
Many formulations of relativistic few-body quantum mechanics . 
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] All have underlying unitary representation of 
the Poincar\'e group.
\vfill
\item [{\color{red}$\bullet$}] All representations of the Poincar\'e 
group can be decomposed in direct integrals of irreducible representations. 
\vfill
\item [{\color{red}$\bullet$}] These representations are the key to 
understanding the relation between different formalisms.
\vfill
\item [{\color{red}$\bullet$}] Relativistic few-body methods provide a means 
to perform realistic calculations few-GeV scale observables that are not
easily calculated using other means. 
}
\end{itemize}
\end{frame}

\end{document}


\begin{frame}
\vfill
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
%\includegraphics{fig6.pdf}}
\includegraphics{fig5.pdf}}
}
\end{figure}
\end{center}
\end{frame}


\begin{frame}
\vfill
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig7.pdf}}
%\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}



\begin{frame}
\vfill
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
%\includegraphics{fig6.pdf}}
\includegraphics{fig8.pdf}}
}
\end{figure}
\end{center}
\end{frame}


\begin{frame}
\vfill
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
%\includegraphics{fig6.pdf}}
\includegraphics{fig9.pdf}}
}
\end{figure}
\end{center}
\end{frame}



%slide 11 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
%\includegraphics{fig7.pdf}}
\includegraphics{fig6.pdf}}
}
\end{figure}
\end{center}
\end{frame}


\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig7.pdf}}
%\includegraphics{fig6.pdf}}
}
\end{figure}
\end{center}
\end{frame}




\begin{frame}
\begin{center}
{\bf 
Direct interaction methods
}
\end{center}
\vfill
\[
{\cal H} = \oplus (\otimes {\cal H}_{j_im_i} ) 
\]
\vfill
\begin{center}
{\bf 
${\cal H}_{jm}$ mass $m$ spin $j$ irreducible representation spaces 
}
\end{center}
\vfill
\[
U_0 (\Lambda ,a) = \sum  \otimes U_{0i} (\Lambda ,a) 
\]
\vfill
\[
\{ H_0, \mathbf{P}_0, \mathbf{J}_0 , \mathbf{K}_0 \} \to 
\{ H, \mathbf{P}, \mathbf{J} , \mathbf{K} \} 
\]
\end{frame}

\begin{center}
{\bf 
$pd$ breakup reactions at $508 MeV$.
% for different
% proton angle pairs $\theta_2-\theta_2$ around the beam axis as
% function of the laboratory kinetic energy of one of the outgoing
% protons.
}
\end{center}
\vfill

\begin{center}
{\bf 
Calculations from T. Lin, et. al. \\
% Ch. Elster, W.N. Polyzou, W. Gl\"ockle,\\
Phys. Lett. B660,345,2008.
}
\end{center}

\vfill
\begin{center}
{\bf 
Data from V. Punjabi et al., \\
Phys. Rev. C 38, 2728 (1988).
}
\end{center}

\vfill
\begin{itemize}
{\bf
\item [{\color{red}$\bullet$}] solid {\color{red}red} lines: 
relativistic calculation. 
\vfill
\item [{\color{red}$\bullet$}] dashed {\color{green}green} lines: 
non-relativistic calculation 
}
\end{itemize}


\begin{frame} 
\begin{center}
{\bf 
Find simultaneous eigenstates of mass, spin, linear 
momentum and magnetic quantum numbers. 
}
\end{center}
\vfill
\[
M=M_0+V,  [V,\mathbf{j}_0] = [V,\mathbf{P}_0] = [V,
\pmb{\nabla}_{{P_0}{\vert_{\mathbf{j}_0}}}]=0
\]
\vfill
\[
\Uparrow
\]
\vfill
\begin{center}
{\bf 
Instant form
}
\end{center}
\vfill
\begin{center}
{\bf 
All forms of dynamics scattering equivalent
}
\end{center}
\vfill
\[
\langle (j_0,m_0) \mathbf{P}_0 , \mu_0 , X 
\vert (j,m) \mathbf{P} , \mu \rangle
\]
\end{frame}

\begin{frame}

\vfill
\begin{center}
{\bf 
Like BS - and infinite number of components are replaced by 
a few components
}
\end{center}
\vfill
\begin{center}
{\bf 
Choosing $\mathbf{j}=\mathbf{j}_0$ always possible; leads to 
violations of cluster properties in generators
}
\end{center}
\vfill
{\bf
Cluster properties can be restored by introducing 
appropriate momentum dependent many-body forces.
}
\end{frame}


\begin{frame}
\begin{itemize}
\item [] Irreps of the Lorentz group
\vfill
\item [] Irreps of the Poincar\'e group
\vfill
\item [] Types of spin 
\vfill
\item [] Intertwiners
\vfill
\item [] Dirac spinors 
\end{itemize}
\end{frame}


\begin{frame}
\begin{center}
{\bf Position }
\end{center}
\vfill
\begin{itemize}
\item [] Compton \vfill
\item [] Spin \vfill
\item [] Intertwiners \vfill
\item [] Dirac spinors
\end{itemize}
\end{frame}


\begin{frame}
\begin{center}
{\bf 
Cluster properties
}
\end{center}
\vfill
\begin{itemize}
\item [] Observables
\vfill
\item [] Generators
\vfill
\item [] Interactions
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
{\bf 
Relativistic effects
}
\end{center}
\vfill
\begin{itemize}
\item [] Gross V18
\vfill
\item [] embedding 
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
{\bf 
Applications
}
\end{center}
\vfill
\begin{itemize}
\item [] Large angle 
\vfill
\item [] Spin effects 
\end{itemize}
\end{frame}



\begin{frame}
spin
\end{frame}

%\end{document}


\begin{frame}
\begin{center}
{\bf 
Finding interactions consistent with experiment\\ 
{\color{blue}and} the commutation relations is 
one of the fundamental problems of relativistic quantum mechanics.
}
\end{center}

\vfill

\begin{itemize}
{\bf 
\item [] The Poincar\'e group has two independent invariant functions of 
the generators related to {\color{blue}mass} and 
{\color{blue}spin} of particles.

\vfill

\item [] All unitary representations of the Poincar\'e group can
  transformed to direct integrals or direct sums of known irreducible
  representations.

\vfill 

\item [] Connection with interactions
}
\end{itemize}

\end{frame}

\begin{frame}
\[
\vert \Psi \rangle = \Phi(f) \vert 0 \rangle
\qquad
\Phi(f) := \int f(x) \Phi(x) d^4x
\]
\[
\int d^4x d^4y f^*(y) 
\langle 0 \vert  \Phi(x) \Phi(y)\vert 0 \rangle
f(x) 
\]
\[
\int d^4x d^4y d\mathbf{p} f^*(y) 
\langle 0 \vert  \Phi(x)\vert \mathbf{p} \rangle 
d \mathbf{p} \langle \mathbf{p} \vert  \Phi(y)\vert 0 \rangle
f(x) 
\]
\end{frame}



\begin{frame}
\[
T(P) = K(P) + K(P) G_0(P) T(P) =
\]
\vfill
\[
 = U(P) + U(P) g_0(P) T(P)
\]
\[
T(P) = {\Gamma(P) \bar{\Gamma}(P) \over P^0-E} + R(P) 
\]
\[
\Gamma (P) = U(P) g_0(P) = K(P) G_0 (P) 
\]
\[
\Gamma (P) \bar{\Gamma}(P) = U(P) g_0(P) \Gamma(P) 
\bar{\Gamma}(P) = K(P) G_0(P) \Gamma(P) \bar{\Gamma}(P)
\]
\vfill
\[ 
1= {i \over 2\pi} \chi(P) {\partial G^{-1}(P) \over \partial P^0} \bar{\chi}(P)
\]
\vfill
\[
1= {i \over 2\pi} \chi(P) {\partial g^{-1}(P) \over \partial P^0} \bar{\chi}(P)
\]
\vfill
\[
\chi {\partial G^{-1}(P) \over \partial P^0} (P-E)(P'-E') G {\partial G^{-1}(P) \over \partial P^0}\bar \chi = <O> 
\]


\end{frame}

\begin{frame}
\[
G(P) = G_0(P) + G_0(P) K(P) G(P) 
\]
\vfill
\[
G_0(P) = g_0(P) + \Delta(P)
\]
\vfill
\[
T(P) = K(P) + K(P) G_0(P) T(P) =
\]
\vfill
\[
= K(P) + K(P)(g_0(P)+\Delta(P)) T(P)
\]
\vfill
\[
 = U(P) + U(P) g_0(P) T(P)
\]
\end{frame}

\begin{frame}
\[ 
U(P) = K(P) + K(P) \Delta (P) U(P)
\]
\vfill
\[
g(P) = g_0(P) + g_0(P) U(P) g(P) 
\]
\vfill
\[
T(P) = {\Gamma(P) \bar{\Gamma}(P) \over P^0-E} + R(P) 
\]
\vfill
\[
\Gamma (P) \bar{\Gamma}(P) = U(P) g_0(P) \Gamma(P) 
\bar{\Gamma}(P) = K(P) G_0(P) \Gamma(P) \bar{\Gamma}(P)
\]
\vfill
\[ 
1= {i \over 2\pi} \chi(P) {\partial G^{-1}(P) \over \partial P^0} \bar{\chi}(P)
\]
\vfill
\[
1= {i \over 2\pi} \chi(P) {\partial g^{-1}(P) \over \partial P^0} \bar{\chi}(P)
\]
\vfill
\[
\chi {\partial G^{-1}(P) \over \partial P^0} (P-E)(P'-E') G {\partial G^{-1}(P) \over \partial P^0}\bar \chi = <O> 
\]
\end{frame}



%\end{document}


%slide 4 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
%Commented outline
%
% 1 Why relativity
% Relativitiy and quantum mechanics
%   principle of special relativity in qantum mechanics
%   Wigner - 1939
% General Axioms
%   Cluster
%   QM 
%   Poincare symm
%   Spectral cond
%   Microscopic locality
% Critique of standard methods
% Problems 
%   Commutation relations
%   Cluster properties
%   Spinor, vector and tensor operators
% Dirac - 1949 
%   Kinematic subgroups - forms of dynamics
%   Sokolov equivalence
%   Bakamjian Thomas
%   Coester 
%   Sokolov
% Groups - Lorentz Poincare 
%   Covariance - irreducible representations
% Realizations
%   Inner products - abstractions  int psi*[cl] k[cl][cl']psi[cl'])dcl dcl'
%   Minkowski QFT
%   Euclidean QFT
%   Constraint dynamics
%   Quantum mechanics of particles
%   Quasipotenital equations
%   Abstract HS - Gauge theories 
% Issues
%   Spin
%   Position on RQM
%   Cluster
%   Currents - Wigner Eckart theorem 
% Relativistic effects in strong interactions
%   Gross
%   CPS
%   Kamada
% Physics
%   relativistic effects
%   models and results questions 
% Outlook 
%   problems to do
%
% 3 relativistic 3N as a computational problem
% 4 relativistic in QM
% 5 particles vs fields
% 6 NR + rel + cluster
%   abstract structure of equations
%   actual structure of equations
% 7 realistic forces - justification 
% 8 computational issues
%   partial waves
%   invariants
%   permutation operators
%   relativistic kernel
%   resolvent equations
%   matrix equations.  
%   singulartities   
%   gram schmidt
%   outlook for few GeV
%   future - production 
%   future - interactions
%   future - other degrees of freedom
%   future - cluster properties
%

\begin{frame}
\begin{center}
{\bf 
Outline
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Motivation.
%: study 
%few-GeV scale few-hadron physics.

\vfill
\item [{\color{red}$\bullet$}] Relativistic effects - examples.

\vfill
\item [{\color{red}$\bullet$}] The structure of relativistic models.
%: relativity,  
%quantum mechanics, cluster properties, realistic interactions.
\vfill
\item [{\color{red}$\bullet$}] Computational issues in the relativistic
3N problem.
}
\end{itemize}
\end{frame}

%slide 5 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Motivation
}
\end{center}
\vfill
\begin{itemize}
{\bf
\item [{\color{red}$\bullet$}] Understand few-GeV-scale hadronic
physics using few-body methods.

\vfill

\item [{\color{red}$\bullet$}] At this scale:

\vfill

\begin{itemize}

\item [{\color{black}$\bullet$}] Sub-nucleon degrees
of freedom should be relevant.

\vfill

\item [{\color{black}$\bullet$}] QCD is non-perturbative.

\vfill

\item [{\color{black}$\bullet$}] Relativistic effects are important.

\end{itemize}

\vfill

\item [{\color{red}$\bullet$}] Assumption: dynamics dominated by a
small number of degrees of freedom.

%\vfill
%\begin{itemize}
%\item [{\color{blue}$\bullet$}] Extend the success of low-energy few-body 
%physics to the few-GeV scale. 
%\vfill
%\item [{\color{blue}$\bullet$}] Sub-nucleon
%degrees of freedom may be relevant at this scale.
%\vfill 
%\item [{\color{blue}$\bullet$}] QCD non-perturbative at this scale.
%
%\vfill
%\item [{\color{blue}$\bullet$}] Understand structure of GeV-scale 
%effective interactions that might be understood from QCD.  
%
%\end{itemize}
%\vfill 
%
%\item [{\color{red}$\bullet$}] Understand the role of relativistic 
%effects on few-GeV scale scattering observables.


%\item [{\color{red}$\bullet$}] 
%
%Understand the role of sub-nucleon 
%degrees of freedom at the few-GeV scale.  Relevant scale: .2 fm 
%$\leftrightarrow$ 1 GeV.
%\vfill
%
%\item [{\color{red}$\bullet$}] Understand the structure of effective
%few-GeV few-body models that might eventually be derivable from QCD.
}
\end{itemize}
\end{frame}

%slide 6 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%\begin{frame}
%\begin{center}
%{\bf 
%Requirements
%}
%\end{center}
%\vfill
%\begin{iemize}
%{\bf 
%\item [{\color{red}$\bullet$}] 
%Extend the success of few-body quantum mechanics to the few-GeV scale.
%\vfill
%\item [{\color{red}$\bullet$}] Relativistic treatment required.
%\vfill
%\item [{\color{red}$\bullet$}] Cluster properties required to relate 
%few and many-body problems.
%\vfill
%\item [{\color{red}$\bullet$}] Must be able to separate relativistic effects
%from effects of new degrees of freedom (sub-nuclear).
%\vfill
%\item [{\color{red}$\bullet$}] Realistic interactions are needed to identify
%spin-dependent effects. 
%\vfill
%\item [{\color{red}$\bullet$}] Numerically stable treatment GeV scale
%dynamics. 
%QCD $\to$ few-Gev scale effective theory 
%of nucleons and mesons - nature of interactions? 
%}
%\end{itemize}
%\end{frame}

%
%\begin{frame}
%\begin{center}
%{\bf 
%Why few-body quantum theory?
%}
%\end{center}
%\begin{itemize}
%{\bf 
%\item [{\color{red}$\bullet$}] Small enough number of degrees
%of freedom so almost complete experiments are possible.
%understanding nuclear 
%ineraction at the few Gev scale  
%\vfill
%\item [{\color{red}$\bullet$}] Small enough number of degrees
%of freedom so ``numerically exact'' calculations are possible. 
%\vfill
%\item [{\color{red}$\bullet$}] Cluster properties relate the 
%dynamics of larger systems to the dynamics of few-body systems.
%\vfill
%\item [{\color{red}$\bullet$}] Few-body methods have proved to be 
%quantitatively successful at describing low-energy nuclear physics.
%\vfill
%\item [{\color{red}$\bullet$}] Motivation for this research is the
%assumption that nuclear physics at the few GeV scale is dominated by
%a small number of degrees of freedom.
%}
%\end{itemize}
%\end{frame} 

\begin{frame}
\begin{center}
{\bf
Examples of relativistic effects in 
{\color{red}scattering reactions} below the few-GeV scale.
}
\end{center}

%\vfill
%\begin{center}
%{\bf
%Two illustrative examples
%}
%\end{center}
\vfill

\begin{center}
{\bf
Models are constrained so (1) two-body 
}
\end{center}
\vfill
\[
S_{2b\,r}(\mathbf{p}=0)=S_{2b\,nr}(\mathbf{p}=0)
\]
\vfill
\begin{center}
{\bf
and (2) three-nucleon $S$-matrices satisfy cluster properties
}
\end{center}
\vfill
\[
S_{3b}\to S_{2b\,ij} \otimes I_k
\]
%\vfill
%\begin{center}
%{\bf
%Correct for lab vs cm energy mismatch
%}
%\end{center}

\end{frame}

%slide 7 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Example 1
}
\end{center}

\begin{center}
{\bf 
$pd$ breakup reactions at $508 MeV$.
% for different
% proton angle pairs $\theta_2-\theta_2$ around the beam axis as
% function of the laboratory kinetic energy of one of the outgoing
% protons.
}
\end{center}
\vfill

\begin{center}
{\bf 
Calculations from T. Lin, et. al. \\
% Ch. Elster, W.N. Polyzou, W. Gl\"ockle,\\
Phys. Lett. B660,345,2008.
}
\end{center}

\vfill
\begin{center}
{\bf 
Data from V. Punjabi et al., \\
Phys. Rev. C 38, 2728 (1988).
}
\end{center}

\vfill
\begin{itemize}
{\bf
\item [{\color{red}$\bullet$}] solid {\color{red}red} lines: 
relativistic calculation. 
\vfill
\item [{\color{red}$\bullet$}] dashed {\color{green}green} lines: 
non-relativistic calculation 
}
\end{itemize}

%{\bf 
%\noindent  The solid {\color{red}red} lines represent the solution of the
%relativistic Faddeev equation
%\vfill
%The corresponding solution of the
%nonrelativistic the Faddeev equation is given by the {\color{green}green} 
%short-dashed
%curves.  The data are from V. Punjabi et al., Phys. Rev. C 38,
%2728 (1988).  The model uses a Malfliet-Tjon interaction.
%}
%
\end{frame}

%\end{document}

%slide 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
symmetric outgoing protons
}
\end{center}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{6.0in}{!}{
\includegraphics[clip,viewport =00.0 00.0 600.0 350.0]{p2fig1_ting.pdf}}
%\includegraphics{p2fig1_ting.pdf}}
%\includegraphics*[0.0in,0.0in][8.0in,11.0in]{p2fig1_ting.pdf}}
%\includegraphics{p2fig1_ting.pdf}}
%\caption{breakup - symmetric outgoing protons}
%\label{fig:1}
}
\end{figure}
\end{center}
\end{frame}

%slide  xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%\begin{frame}
%\begin{center}
%{\bf 
%asymmetric outgoing protons
%}
%\end{center}
%\begin{center}
%\begin{figure} 
%\rotatebox{00}{\resizebox{6.0in}{!}{
%\includegraphics{p2fig2_ting.pdf}}
%\includegraphics[clip,viewport =00.0 00.0 600.0 350.0]{p2fig2_ting.pdf}}
%}
%\end{figure}
%\end{center}
%\end{frame}

%slide 9 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%\begin{frame}

%\begin{center}
%{\bf 
%Examples
%}
%\end{center}

%\begin{center}
%{\bf 
%Elastic pd scattering with and without 3N forces.
%
%Selected polarization observables comparing relativistic and 
%non-relativistic pd scattering with and without 3NFs. 
% for different
% proton angle pairs $\theta_2-\theta_2$ around the beam axis as
% function of the laboratory kinetic energy of one of the outgoing
% protons.
%}
%\end{center}
%\vfill

%\begin{center}
%{\bf 
%Calculations: H. Wita{\l}a, et al,\\
%J. Golak, R. Skib\'inski, W. Gl\"ockle, H. Kamada, and W. 
%N. Polyzou
%Phys. Rev. C.83.044001(2011).
%}
%\end{center}
%
%\vfill
%\begin{center}
%{\bf 
%Data from: \\
%K. Sekiguchi et al., Phys. Rev. C65, 034003 (2002),\\
%Y. Maeda et al., Phys. Rev. C76, 014004 (2007),\\
%K. Hatanaka et al., Phys. Rev. C66, 044002 (2002).\\
%Phys. Rev. C65, 034003 (2002) (slide 1)\\
%and\\
%B.V. Przevoski et al., \\
%Phys. Rev. C74, 064003 (2006) (slide 2).
%}
%\end{center}

 

%\vfill
%\begin{itemize}
%{\bf
%\item [{\color{red}$\bullet$}] {\color{red}red} - non-relativistic - CD Bonn 
%\item [{\color{red}$\bullet$}] {\color{blue}blue dots} - non-relativistic - CD Bonn $+$ TM 3Nf 
%\item [{\color{red}$\bullet$}] {\color{blue}blue dashes}  - relativistic - CD Bonn - no Wigner rotations
%\item [{\color{red}$\bullet$}] {\color{brown}brown dash-dot} - relativistic - CD Bonn $+$ TM 3Nf - no Wigner rotations
%}
%\end{itemize}
%
%\begin{center}
%{\bf 
%Example 3,4: next two slides\\
%Calculations: H. Wita{\l}a, et al,\\
%J. Golak, R. Skib\'inski, W. Gl\"ockle, H. Kamada, and W. 
%N. Polyzou
%Phys. Rev. C.83.044001(2011).
%}
%\end{center}
%\vfill
%
%{\bf 
%\noindent
%The vector (deuteron) $A_y(d)$ and tensor analyzing powers $A_{xx}$,
%$A_{yy}$, and $A_{xz}$ in elastic $nd$ scattering at the incoming
%neutron lab. energy $E = 135 MeV$.  
%}
%\vfill
%{\bf
%\noindent The tensor analyzing powers $A_{xx} − A_{yy}$ and $A_{zz}$ in elastic
%$nd$ scattering at the incoming neutron lab. energy $E = 135$ and 
%$E = 200 MeV$.  The $pd$ data are from
%}
%\vfill
%{
%\parindent=0pt
%\obeylines
%\bf
%{\color{red}red} - non-relativistic - CD Bonn \\
%{\color{blue}blue dots} - non-relativistic - CD Bonn $+$ TM 3bf \\
%{\color{blue}blue dashes}  - relativistic - CD Bonn - no Wigner rotations\\
%{\color{brown}brown dash-dot} - relativistic - CD Bonn $+$ TM 3bf - no Wigner rotations
%}
%\end{frame}

%slide 1 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

%\begin{center}
%{\bf 
%Examples
%}
%\end{center}
\begin{center}
{\bf 
Example 2
}
\end{center}
\vfill
\begin{center}
{\bf 
Selected spin-dependent pd breakup observables at 270 
MeV with and without 3N forces.
%
%Selected polarization observables comparing relativistic and 
%non-relativistic pd scattering with and without 3NFs. 
% for different
% proton angle pairs $\theta_2-\theta_2$ around the beam axis as
% function of the laboratory kinetic energy of one of the outgoing
% protons.
}
\end{center}
\vfill

\begin{center}
{\bf 
Calculations: H. Wita{\l}a, et al,\\
%J. Golak, R. Skib\'inski, W. Gl\"ockle, H. Kamada, and W. 
%N. Polyzou
Phys. Rev. C.83.044001(2011).
}
\end{center}

\vfill
\begin{center}
{\bf 
Data from: \\
K. Sekiguchi et al., Phys. Rev. C79, 054008(2009).
}
\end{center}

 

\vfill
\begin{itemize}
{\bf
\item [{\color{red}$\bullet$}] {\color{red}red} - non-relativistic - CD Bonn 
\item [{\color{red}$\bullet$}] {\color{blue}blue dots} - non-relativistic - CD Bonn $+$ TM 3Nf 
\item [{\color{red}$\bullet$}] {\color{blue}blue dashes}  - relativistic - CD Bonn - no Wigner rotations
\item [{\color{red}$\bullet$}] {\color{brown}brown dash-dot} - relativistic - CD Bonn $+$ TM 3Nf - no Wigner rotations
}
\end{itemize}

\end{frame}


\begin{frame}

\vfill
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
%\includegraphics{fig6.pdf}}
\includegraphics{fig18.pdf}}
}
\end{figure}
\end{center}
\end{frame}


%slide 11 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%\begin{frame}
%\begin{center}
%\begin{figure} 
%\rotatebox{00}{\resizebox{2.5in}{!}{
%\includegraphics{fig7.pdf}}
%\includegraphics{fig6.pdf}}
%}
%\end{figure}
%\end{center}
%\end{frame}

%slide 13 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%\begin{frame}
%\begin{center}
%{\bf 
%Computational issues
%}
%\end{center}
%
%\vfill
%
%\begin{itemize}
%{\bf 

%\item [{\color{red}$\bullet$}] Formulation?
%\vfill

%\item [{\color{red}$\bullet$}] Few-nucleon physics has traditionally 
%been associated with large-scale computing.

%\vfill

%\item [{\color{red}$\bullet$}] A relativistic treatment presents 
%additional challenges. 

%\vfill

%\item [{\color{red}$\bullet$}] This talk will discuss the resolution
%of the additional computational challenges that arise in the 
%relativistic three-nucleon problem.
%}
%\end{itemize}
%\end{frame}

%slide 13 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Model structure 
}
\end{center}
\vfill 
\begin{center}
{\bf 
Relativistic quantum theory (Wigner) 
}
\end{center}
\vfill
\[
{\cal H} = \oplus (\otimes {\cal H}_{m_i\, j_i})
\]
\vfill
\[
U(\Lambda ,a):{\cal H} \to {\cal H}
\]
\vfill

\begin{center}
{\bf 
Cluster properties
}
\end{center}
\vfill
\[
\lim_{\vert r_{ij} - r_k\vert \to \infty} 
[U(\Lambda ,a) - U_{ij}(\Lambda ,a) \otimes U_k(\Lambda ,a)] =0
\]
%\vfill
%\begin{center}
%{\bf 
%Challenges
%}
%\end{center}
\end{frame} 

%slide 13 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Computational issues
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Few-nucleon physics has traditionally 
been associated with large-scale computing.
\vfill
%\vfill
\item [{\color{red}$\bullet$}] A relativistic treatment presents 
additional challenges. 
}
\end{itemize}
%\vfill
\end{frame}

%slide 14 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
New computational issues (relativistic n$=$3)
}
\end{center}
\vfill
\begin{itemize}
{\bf
\item [{\color{red}$\bullet$}] The Poincar\'e commutation relations put 
non-linear dynamical constraints on the interactions.

\vfill

\item [{\color{red}$\bullet$}] Cluster properties require the addition
of three (many)-body interactions to the cluster expansions of generators
to preserve Poincar\'e CRs.

\vfill
\item [{\color{red}$\bullet$}] Permutation operators involve momentum-dependent
spin rotations.

%
%\item [{\color{red}$\bullet$}] Large singular matrix equations.
%

\vfill

%\item [{\color{red}$\bullet$}] Electromagnetic probes require a consistent
%treatment of cluster properties, current covariance and conservation.
%
%\item [{\color{red}$\bullet$}] Few GeV interactions can produce particles; 
%NN interactions only fit to about 300 MeV.
}
\end{itemize}
\end{frame}

%slide 15 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Structure of dynamical 3N models (Coester) 
}
\end{center}
\[
\bar{M} := \bar{M}_{12,3} + \bar{M}_{23,1}+ \bar{M}_{31,2}- 2M_0
\]
\vfill
\[
\bar{M}_{ij,k} = M_0 + \bar{V}_{ij} 
\]
\vfill
\[
M_0 = \sqrt{\mathbf{q}_k^2 + (\sqrt{\mathbf{k}_{ij}^2 + m_i^2 }
+\sqrt{\mathbf{k}_{ij}^2 + m_j^2 })^2} +
\sqrt{\mathbf{q}_k^2 + m_k^2} 
\]
\vfill
\[
\mathbf{q}_i := \pmb{\Lambda}(P/M_0)^{-1} p_i  \qquad 
\mathbf{k}_{ij} := \pmb{\Lambda}({q_i + q_j \over m_{0ij}})^{-1} q_i 
\]
\vfill
\[
[\bar{V}_{ij},\mathbf{j}^2_0]=0
\]
\vfill
\[
\downarrow
\]
\end{frame}

%slide 16 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{itemize}
{\bf 

\item [{\color{red}$\bullet$}] One Casimir operator, $\bar{M}$, has 
interactions.

\vfill

\item [{\color{red}$\bullet$}] Diagonalize $\bar{M}$ in 
invariant subspaces of $\mathbf{j}^2_0$.\\
(Bakamjian-Thomas)
\vfill

\[
\bar{U}(\Lambda ,a) = \directsumint   U_{\mathbf{j}_0,m}(\Lambda ,a)
\]

\vfill

\item [{\color{red}$\bullet$}] Problem - $\bar{U}(\Lambda,a)$  violates
cluster properties.
\vfill

\item [{\color{red}$\bullet$}] No loss of generality - $\exists \, 
S$-matrix-preserving unitary transformations $A$: $A\mathbf{j}^2 A^{\dagger}=
\mathbf{j}_0^2$.
}
\end{itemize}
\end{frame}

%\end{document}

%slide 17 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\[
\bar{U}(\Lambda ,a) \rightarrow  \bar{U}_{ij,k}(\Lambda ,a)
{\color{red} \not=}
\bar{U}_{ij} \otimes {U}_{k}(\Lambda ,a)
\]
\vfill
\begin{center}
{\bf 
however 
}
\end{center}
\vfill
\[
\bar{S}_{ij,k} = \bar{S}_{ij} \otimes I_k 
\]
\vfill
\[
\Downarrow
\]
\vfill
\[
\exists \, A_{ij,k} \qquad (\mbox{\bf Ekstein})
\]
\vfill
\[
A_{ij,k} \bar{U}_{ij,k}(\Lambda ,a) 
A_{ij,k}^{\dagger} = \bar{U}_{ij} \otimes {U}_{k}(\Lambda ,a)  
\]
\vfill
\[
A_{ij,k} \bar{M}_{ij,k} 
A_{ij,k}^{\dagger} = {M}_{ij \otimes k} 
\]
\vfill
\[
A_{ij,k} \mathbf{j}^2_0 
A_{ij,k}^{\dagger} = 
\mathbf{j}^2_{ij\otimes k} 
{\color{red} \not=} \mathbf{j}^2_0 
\]

%\begin{center}
%{\bf 
%and 
%}
%\end{center}
%\vfill
%\[
%\langle \mathbf{p}, \mathbf{q}_r, \mathbf{k}_r \vert S_{ij,k\, r} \vert
%\mathbf{p}', \mathbf{q}_r', \mathbf{k}_r' \rangle = 
%\]
%\[
%\delta (\mathbf{p}- \mathbf{p}' )\delta (\mathbf{q}_r-\mathbf{q}_r' )
%{\color{red}\langle \mathbf{k}_r \vert s_{ij} \vert \mathbf{k}_r' \rangle}
%\]
%\vfill
%\[
%\langle \mathbf{p}, \mathbf{q}_{nr}, \mathbf{k}_{nr} \vert S_{ij,k\, nr}
%\vert 
%\mathbf{p}', \mathbf{q}_{nr}', \mathbf{k}_{nr}' \rangle =
%\]
%\[ 
%\delta (\mathbf{p}- \mathbf{p}' )\delta (\mathbf{q}_{nr}-\mathbf{q}_{nr}' )
%{\color{red}\langle \mathbf{k}_{nr} 
%\vert s_{ij} \vert \mathbf{k}_{nr}' \rangle}
%\]

\end{frame}


%\end{document}

%slide 18 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
%\[
%\bar{S}_{ij,k} = \bar{S}_{ij} \otimes I_k \, \Rightarrow \, 
%\exists \, A_{ij,k}\, \mbox{unitary}:  
%\]
%\vfill
\[
C_{ij,k} := i (A_{ij,k}-I)(A_{ij,k}+I)^{-1} 
\]
\vfill
\[
C := C_{12,3}+C_{23,1}+C_{31,2} 
\]
\vfill
\[
A:=(I - i C)(I+i C)^{-1} \qquad A \to A_{ij,k} \to I
\]
\vfill
%\[
%A_{ij,k} \bar{U}_{ij,k} A^{\dagger}_{ij,k}
%= \bar{U}_{ij} ( \Lambda ,a) \otimes U_k (\Lambda ,a)   
%\]
%\vfill
%\vfill
\[
U(\Lambda ,a) := A^{\dagger} \bar{U}(\Lambda ,a) A
\]
\vfill
\[
U(\Lambda ,a) \to \bar{U}_{ij}(\Lambda ,a)
\otimes {U}_{k}(\Lambda ,a)
\]
\vfill
\[
{\color{red}
M = A (\sum A^{\dagger}_{ij,k} {M}_{ij \otimes k} A_{ij,k}- 2M_0 )A^{\dagger}
= A \bar{M} A^{\dagger}
}
\]
\vfill
\begin{center}
{\bf 
(Sokolov)
}
\end{center}


\end{frame}

%slide 19 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
%\end{document}


\begin{frame}

\begin{center}
{\bf 
Simplifications:
}
\end{center}

\vfill

\begin{itemize}
{\bf
\item [{\color{red}$\bullet$}] $A$, $A_{ij,k}^{\dagger}$ generate 
many-body forces that restore 
Poincar\'e CR to cluster expansions of generators.
\vfill
\[
{\color{red}A(\mathbf{p}=0) = I}
\qquad 
S(\mathbf{p}) = S(\mathbf{p}=0) = \bar{S}(\mathbf{p}=0) = \bar{S}(\mathbf{p})
\]
\vfill
\item [{\color{red}$\bullet$}] To calculate 3N $S$ no need to 
calculate 3NFs generated by $A$, $A_{ij,k}^{\dagger}$!
}
\end{itemize}
\vfill
\[
\Downarrow
\]
\vfill
\begin{center}
{\bf 
To calculate on-shell $S$ set $A\to I$,
{\color{red}diagonalize  $\bar{M}$} 
}
\end{center}
\end{frame} 
%\end{document}
%slide 20 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Three-Particle Scattering (Operator Equations)
}
\end{center}
\vfill
\[
\bar{M}=M_0 +\bar{V} \qquad \bar{V}= \sum_{\alpha} \bar{V}_{\alpha}  
\qquad \alpha \in \{ (12,3),(23,1),(31,2) \}
\]
\vfill
\[
\bar{V}_\alpha = \bar{M}_\alpha -M_0    \qquad \bar{V}^{\alpha} = \bar{M}-
\bar{M}_{\alpha}
\]
\vfill
\[
\bar{T}^{\alpha \beta}(m) := 
\bar{V}^\beta + \bar{V}^{\alpha}(m-\bar{M}+i 0^+)^{-1} \bar{V}^{\beta} 
\]

\vfill
\[
\langle a_0 \vert S^{\alpha \beta}\vert  b_0 \rangle  = 
\langle a_0 \vert b_0\rangle  - 2 \pi i \delta (m_a - m_b)
\langle a_0 \vert { \bar{T}^{\alpha\beta} (m_a+i0^+)}\vert b_0 \rangle
\] 
\end{frame}

%slide 21 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
{\color{black}\bf
\begin{center}
Faddeev Equations
\end{center}
\vfill
\[
{\color{red}
\boxed{
{\color{black}
\bar{T}^{\alpha\beta}(z) = \bar{V}^\beta + \sum_{\gamma \not= \alpha} 
\bar{T}_\gamma (z-M_0)^{-1} \bar{T}^{\gamma\beta}(z) 
}
}
}
\]
\vfill
\[
{\color{red}
\bar{T}_\gamma (z) = \bar{V}_\gamma +  
\bar{V}_\gamma (z-M_0)^{-1} \bar{T}_\gamma (z) }
\]
\vfill
\begin{center}
{\bf 
Iterated kernel of coupled equations compact
}
\end{center}
\vfill
\[
\bar{T}(z) = \bar{D}(z) + \bar{K}(z)\bar{T}(z)  
\qquad \bar{K}(z)^2 \quad \mbox{\bf compact}
\]
\vfill
\[
\bar{T}(z) = (I- \bar{K}(z)^2)^{-1} (\bar{D}(z) + \bar{K}(z)\bar{T}(z))
\]
}
\end{frame}

%slide 21 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Interactions I\\
(Coester, Pieper \& Serduke)
}
\end{center}
 
\vfill
\[
\bar{M} := M_0 + \bar{V}_{12} + \bar{V}_{23}+ \bar{V}_{31}
\]
\vfill
\[
\bar{V}_{ij} :=
\]
\[
\sqrt{\mathbf{q}_k^2 + (\sqrt{{\color{red}\mathbf{k}_{ij}^2} + m_i^2 + {\color{red}2 \mu_{ij}
v_{nr\,ij}}}
+\sqrt{{\color{red}\mathbf{k}_{ij}^2} + m_j^2 + 
{\color{red}2 \mu_{ij}v_{nr\,ij}}})^2} -
\]
\[
\sqrt{\mathbf{q}_k^2 + (\sqrt{\mathbf{k}_{ij}^2 + m_i^2}
+\sqrt{\mathbf{k}_{ij}^2 + m_j^2 })^2} 
\]
\vfill
\[
\bar{M}_{ij,k} = M_{ij,k}(h_{nr\,ij})
\]

\vfill
\[
\langle \mathbf{p}, \mathbf{q}_r, \mathbf{k}_r \vert S_{ij,k\, r} \vert
\mathbf{p}', \mathbf{q}_r', \mathbf{k}_r' \rangle = 
\delta (\mathbf{p}- \mathbf{p}' )\delta (\mathbf{q}_r-\mathbf{q}_r' )
{\color{red}\langle \mathbf{k}_r \vert s_{ij} \vert \mathbf{k}_r' \rangle}
\]
\vfill
\[
\langle \mathbf{p}, \mathbf{q}_{nr}, \mathbf{k}_{nr} \vert S_{ij,k\, nr}
\vert 
\mathbf{p}', \mathbf{q}_{nr}', \mathbf{k}_{nr}' \rangle =
\delta (\mathbf{p}- \mathbf{p}' )\delta (\mathbf{q}_{nr}-\mathbf{q}_{nr}' )
{\color{red}\langle \mathbf{k}_{nr} \vert s_{ij} \vert \mathbf{k}_{nr}' \rangle}
\]
\end{frame}

%slide 22 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Interactions II
}
\end{center}

\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] $\bar{V}_{\alpha}$ is a
  complicated operator involving square roots of the non-relativistic
  interaction.

\vfill

\[
\langle \Phi_0 \vert
\bar{T}_{\alpha} \vert \Psi_0 \rangle =
\langle \Phi_0 \vert
\bar{V}_{\alpha} \vert \Psi^- \rangle =  
\langle \Phi_0 \vert
(\bar{M}_{\alpha}-M_0) \vert \Psi^- \rangle 
\]

\vfill

\item [{\color{red}$\bullet$}] Using the identity of the NN
wave functions, the {\color{red}half-shell} relativistic Faddeev 
kernel can be expressed {\color{red}exactly} in terms of the non-relativistic half-shell 
two-body transition matrix elements!

\vfill

\[
\langle \mathbf{k}  \vert \Psi^-_{nr} \rangle =
\langle \mathbf{k}  \vert \Psi^-_r \rangle 
\]

\vfill

\item [{\color{red}$\bullet$}] The two-nucleon subsystem is not at rest
in the three-nucleon rest frame.

}
\end{itemize}
\end{frame}

%slide 23 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%\vfill
%\[
%\langle \mathbf{q}_\alpha, \mathbf{k}_\alpha \vert  
%\bar{V}_{\alpha} (z-\bar{M}_{\alpha})^{-1} \vert 
%\mathbf{q}'_\alpha, \mathbf{k}'_\alpha \rangle =
%\langle \mathbf{q}_\alpha, \mathbf{k}_\alpha \vert  
%\bar{T}_{\alpha}(z) (z-\bar{M}_{0})^{-1} \vert 
%\mathbf{q}'_\alpha ,\mathbf{k}'_\alpha \rangle =
%\]
%\vfill
%\[
%\langle \mathbf{q}_\alpha (\mathbf{k}_\alpha)^+ \vert  
%\bar{V}_{\alpha} (z-\bar{M}_{0})^{-1} 
%\vert \mathbf{q}'_\alpha ,\mathbf{k}'_\alpha \rangle =
%\]
%\vfill
%\[
%\mbox{for} \qquad z= M_0 (\mathbf{q}_\alpha \mathbf{k}_\alpha)+ i0^+
%\]
%\item [] and the identity of $\langle \mathbf{k}_{\alpha}' 
%\vert (\mathbf{k}_\alpha)^+
%\rangle$ for the relativistic and non relativistic system can be 
%use to derive the 
%following half shell relation:
%}
%\end{itemize}
%\end{frame}

%slide 18 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx



\begin{frame}

\[
\langle \mathbf{q}_\alpha, \mathbf{k}_{\alpha} \vert
T_{\alpha}(z) (z-\bar{M}_{0})^{-1}  \vert \mathbf{q}_\alpha', \mathbf{k}_{\alpha}' \rangle = 
\]
\vfill
\[
\delta(\mathbf{q}_{\alpha}-\mathbf{q}'_{\alpha}) 
{m_{0\alpha}(\mathbf{k}) + m_{0\alpha}(\mathbf{k}') \over 
( \sqrt{\mathbf{q}_{\alpha}^2 + m_{0\alpha}^2 (\mathbf{k}_{\alpha})} +
\sqrt{\mathbf{q}_{\alpha}^2 + m_{0\alpha}^2(\mathbf{k}_{\alpha}')}
)} \times
\]
\vfill
%\[
%\times
%({2 \mu \over \sqrt{\mathbf{k}^2_{\alpha} + m_i^2}+ 
%\sqrt{\mathbf{k}^{\prime 2}_{\alpha} + m_j^2}}  + 
%{2 \mu \over \sqrt{\mathbf{k}_{\alpha}^2 + m_i^2}+ 
%\sqrt{\mathbf{k}_{\alpha}^{\prime 2} + m_j^2}}) 
%\times 
%\]
%\vfill
\[
{\color{red}\langle \mathbf{k}_{\alpha} 
\vert t_{r}(z) 
\vert \mathbf{k}'_{\alpha} \rangle} 
%{\color{red}\langle \mathbf{k}_{\alpha} 
%\vert t_{nr}(\mathbf{k}_{\alpha}^2/2\mu+i0^+)
%\vert \mathbf{k}'_{\alpha} \rangle} 
{1 \over M_{0}(\mathbf{q}_{\alpha}, \mathbf{k}_{\alpha} )- 
M_{0}(\mathbf{q}_{\alpha}, \mathbf{k}'_{\alpha} )+i0^+}
\]
{\bf where }
\[
m_{0\alpha}(\mathbf{k}_{\alpha}) :=
\sqrt{\mathbf{k}_{\alpha}^2 + m_i^2} + \sqrt{\mathbf{k}_{\alpha}^2 + m_j^2} 
\]
{\bf and}
\[
z= M_0 (\mathbf{q}_\alpha \mathbf{k}_\alpha)+ i0^+
\]
\end{frame}

%slide 24 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\[
\langle \mathbf{k}_{\alpha} 
\vert t_{r}(z) 
\vert \mathbf{k}'_{\alpha} \rangle =
\]
\vfill
\[
({2 \mu \over \sqrt{\mathbf{k}^2_{\alpha} + m_i^2}+ 
\sqrt{\mathbf{k}^{\prime 2}_{\alpha} + m_j^2}}  + 
{2 \mu \over \sqrt{\mathbf{k}_{\alpha}^2 + m_i^2}+ 
\sqrt{\mathbf{k}_{\alpha}^{\prime 2} + m_j^2}}) 
\times 
\]
\vfill
\[
{\color{red}\langle \mathbf{k}_{\alpha} 
\vert t_{nr}(\mathbf{k}_{\alpha}^2/2\mu+i0^+)
\vert \mathbf{k}'_{\alpha} \rangle} 
\]
\end{frame} 

%slide 25 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Permutation operators - different NN rest frames
}
\end{center}
\[
\langle \mathbf{q}_i, \mu_i, \mathbf{q}_j, \mu_j \vert t_{r}(z) 
\vert \mathbf{q}_i', 
\mu_i', \mathbf{q}_j' , \mu_j' \rangle = 
\]
\[
\delta (\mathbf{q}_k-\mathbf{q}_k')
%\left ( {\omega_i (\mathbf{k}_{ij})+\omega_j(\mathbf{k}_{ij}) 
%\over \omega_i (\mathbf{q}_i)+\omega_j(\mathbf{q}_j) }
%{\omega_i (\mathbf{q}_i) \over  
%\omega_i (\mathbf{k}_{ij})} 
%{\omega_j(\mathbf{q}_j)\over 
%\omega_j(\mathbf{k}_{ij}) }
%\right )^{1/2}
%\[
%\[
\left ( {\omega_i (\mathbf{q}_{i})+\omega_j(\mathbf{q}_{j}) 
\over \omega_i (\mathbf{k}_{ij})+\omega_j(\mathbf{k}_{ji}) }
{\omega_i (\mathbf{k}_{ij}) \over  
\omega_i (\mathbf{q}_{i})} 
{\omega_j(\mathbf{k}_{ji})\over 
\omega_j(\mathbf{q}_{j}) }
\right )^{1/2} \times
\]
\vfill
\[
\sum 
D^{j_i}_{\mu_i \nu_i}
[R_{wc}(B_c(q_{ij}),k_{ij})]
D^{j_j}_{\mu_j \nu_j}
[R_{wc}(B_c(q_{ij}),k_{ji})] \times
\]
\vfill
\[
{\color{red}\langle \mathbf{k}_{ij}, \nu_i, \nu_j 
\vert t_{r}(z) \vert \mathbf{k}_{ij}', 
\nu_i', \nu_j' \rangle }\times
\]
\vfill
\[
D^{j_i}_{\nu_i' \mu_i'}
[R_{wc}(B^{-1}_c(q_{ij}),q_{i})]
D^{j_j}_{\nu_j' \mu_j'}
[R_{wc}(B^{-1}_c(q_{ij}),q_{j})] \times
\]
\vfill
\[
\left ( {\omega_i (\mathbf{q}'_{i})+\omega_j(\mathbf{q}'_{j}) 
\over \omega_i (\mathbf{k}'_{ij})+\omega_j(\mathbf{k}'_{ji}) }
{\omega_i (\mathbf{k}'_{ij}) \over  
\omega_i (\mathbf{q}'_{i})} 
{\omega_j(\mathbf{k}'_{ji})\over 
\omega_j(\mathbf{q}'_{j}) }
\right )^{1/2}
\]
\end{frame}


%slide 26 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{itemize}
{\bf 

\item [{\color{red}$\bullet$}] The relation to the half-shell
non-relativistic transition operator avoids the problem of 
explicitly computing square roots of non-commuting operators.

\vfill

\item [{\color{red}$\bullet$}] The fully {\color{red}off-shell} 
transition operator is needed
in the Faddeev kernel.  It can be obtained by solving the first 
resolvent equation in the form
}
\end{itemize}
\vfill
\[
\bar{T}_{\alpha}(z)= \bar{T}_{\alpha}(z') + \bar{T}_{\alpha}(z) {z'-z \over (z-M_0)(z'-M_0)} 
\bar{T}_{\alpha}(z')  
\]

\end{frame}

%slide 27 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Interactions III
}
\end{center}
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Partial-wave expansion numerically 
difficult above 250 MeV.
\vfill
\item [{\color{red}$\bullet$}] Use direct three-dimensional integration.
\begin{itemize}
\vfill
\item [{\color{red}$\bullet$}] Adds continuous variables (larger 
matrices). 
\vfill
\item [{\color{red}$\bullet$}] One channel (Malfliet-Tjon) model
convergent at  2 GeV. 
\vfill
\item [{\color{red}$\bullet$}] Requires realistic momentum-space 
interaction in operator form.  
\vfill
\item [{\color{red}$\bullet$}] Fourier transform of Argonne V18
\end{itemize}
}
\end{itemize}
\end{frame}

%slide 28 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\[
\langle \mathbf{k} \vert v_{nr} \vert
\mathbf{k}' \rangle = \sum V_n  
W_n 
\]
\vfill
\[
W_1 := I
\]
\vfill
\[
W_2 :=\mathbf{j}_1 \cdot \mathbf{j}_2
\]
\vfill
\[
W_3 := (\mathbf{j}_1 \cdot \hat{\mathbf{K}}) \otimes
(\mathbf{j}_2 \cdot \hat{\mathbf{K}}) 
\]
\vfill
\[
W_4 := (\mathbf{j}_1 \cdot \hat{\mathbf{Q}})\otimes
(\mathbf{j}_2 \cdot \hat{\mathbf{Q}}) 
\]
\vfill
\[
W_5 :=  (\mathbf{j}_1 \cdot \hat{\mathbf{N}})
\otimes I_2 + I_1 \otimes 
(\mathbf{j}_2 \cdot \hat{\mathbf{N}}) 
\]
\vfill
\[
W_6 := (\mathbf{j}_1 \cdot \hat{\mathbf{K}})\otimes
(\mathbf{j}_2 \cdot \hat{\mathbf{Q}}) +
(\mathbf{j}_1 \cdot \hat{\mathbf{Q}})\otimes
(\mathbf{j}_2 \cdot \hat{\mathbf{K}})
\]
\vfill
\[
\mathbf{K} =\mathbf{k}'-\mathbf{k}
\qquad
\mathbf{Q} =\mathbf{k}'+\mathbf{k}
\qquad
\mathbf{N} =\mathbf{k}'\times \mathbf{k}
\]
\end{frame}

%slide 29 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Spins and continuous variables must be 
chosen to Wigner rotate together.

\vfill

\item [{\color{red}$\bullet$}] Six independent spin operators off shell,
five on shell - can lead to numerical instabilities.

\vfill

\item [{\color{red}$\bullet$}] Choose the 6 off-shell operators 
to match smoothly 
with the 5 on-shell operators.

\vfill

\item [{\color{red}$\bullet$}] Generate three-nucleon
 spin operators by iteration.

\vfill

\item [{\color{red}$\bullet$}] Faddeev equation has $24^3$ traces - 
developed dedicated symbolic package (using GiNaC) to perform traces.

\vfill

\item [{\color{red}$\bullet$}] $NN$ scattering tested against PW calculations
and GW database.

}
\end{itemize}
\end{frame}


%slide 30 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
V18 results \qquad pp \qquad 350 MeV
}
\end{center}
\vfill
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{Wolfenstein_350_ppall2.pdf}}
}
\end{figure}
\end{center}
\end{frame}

%slide 31 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
%
%\begin{frame}
%
%\vfill
%\begin{center}
%{\bf 
%V18 results \qquad np \qquad 350 MeV
%}
%\end{center}
%\begin{center}
%\begin{figure} 
%\rotatebox{00}{\resizebox{2.5in}{!}{
%\includegraphics{Wolfenstein_350_npall2.pdf}}
%}
%\end{figure}
%\end{center}
%\end{frame}
%
%slide 33 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
%
%\begin{frame}
%\begin{center}
%{\bf 
%V18 results \qquad pp \qquad 100 MeV
%}
%\end{center}
%\vfill
%\begin{center}
%\begin{figure} 
%\rotatebox{00}{\resizebox{2.5in}{!}{
%\includegraphics{Wolfenstein_100_ppall2.pdf}}
%}
%\end{figure}
%\end{center}
%\end{frame}
%
%slide 34 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
%\begin{frame}
%\begin{center}
%{\bf 
%V18 results \qquad np \qquad 100 MeV
%}
%\end{center}
%\vfill
%\begin{center}
%\begin{figure} 
%\rotatebox{00}{\resizebox{2.5in}{!}{
%\includegraphics{Wolfenstein_100_npall2.pdf}}
%}
%\end{figure}
%\end{center}
%\end{frame}

%slide 35 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Established computational methods can 
still be applied.
\vfill
\item [{\color{red}$\bullet$}] Use powers of kernel on driving term
plus Gram-Schmidt orthogonalization to generate a small basis (Krylov 
method). 
\vfill
\item [{\color{red}$\bullet$}] Use rotational covariance simplify 
the computation of the kernel (Balian - Brezin method).
}
\end{itemize}
\end{frame}

%slide 32 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Summary
} 
\end{center}
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Poincar\'e invariance - choose to 
add interactions to only {\color{red}one} Casimir operator.

\item [{\color{red}$\bullet$}] Cluster properties in the 
{\color{red}rest frame}
is sufficient to get cluster properties for {\color{red} $S$}.  

\item [{\color{red}$\bullet$}] Realistic NN interactions parameterize 2-body 
data - can be used to construct {\color{red} equivalent relativistic} 
interactions.

\item [{\color{red}$\bullet$}] Identity of $nr$ and $r$ wave functions 
gives half-shell relativistic Faddeev kernel in terms of nr 
half-shell transition matrix. 

\item [{\color{red}$\bullet$}] First resolvent equation must be 
solved to get relativistic off-shell kernel. 
}
\end{itemize}
\end{frame}

%slide 33 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{itemize}
{\bf 

\item [{\color{red}$\bullet$}] Permutation operators involve Jacobians and 
Wigner rotations. Permutations can be made trivial by including these 
factors in the relativistic two-nucleon transition matrices.

\item [{\color{red}$\bullet$}] Partial-wave expansions unstable - need
a realistic interaction.  Fourier transform AV18.

\item [{\color{red}$\bullet$}] Expand in terms of Wolfenstein parameters
to easily extract cross sections, extra operator needed.  Must be 
chosen carefully for stability.

\item [{\color{red}$\bullet$}] Many traces needed.  Can be automated.

\item [{\color{red}$\bullet$}] Standard computational methods can still
be applied.
 
\item [{\color{red}$\bullet$}] Non-relativistic Faddeev equation arises 
by setting relativistic corrections to $1$.
}
\end{itemize}

\end{frame}

\begin{frame}
\begin{center}
{\bf 
Future
}
\end{center}

\vfill

\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] GeV-scale 3N calculations with V18 
in progress (Mohammadreza). 

\vfill

\item [{\color{red}$\bullet$}]  Cluster expansions for generators and
consistent covariant conserved currents.

\vfill

\item [{\color{red}$\bullet$}]  Inclusion of production channels 
in a manner consistent with cluster properties.
}
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
{\bf 
Thanks!\\ 
Andrey, Bruce, Pieter \\
and the Organizing Committee.
}
\end{center}
\end{frame}

\end{document}
%slide 36 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Consistent calculations involving electromagnetic probes of 
3N systems require $U(\Lambda ,a)$ and currents that cluster
}
\end{center}
\vfill
\[
\langle \Psi_f \vert J^{\mu} (0) \vert \Psi_i \rangle =
\langle \bar{\Psi}_f \vert 
A^{\dagger}J^{\mu} (0)A \vert \bar{\Psi}_i \rangle \approx 
\langle \bar{\Psi}_f \vert 
J^{\mu} (0) \vert \bar{\Psi}_i \rangle    
\]
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] If $A-I$ small.
\vfill
\item [{\color{red}$\bullet$}] Solutions of Faddeev equation can be 
transformed to equivalent light-front wave functions. 
}
\end{itemize}
\end{frame}

%slide 37 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Size of $A_{ij,k}$
}
\end{center}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{6.0in}{!}{
\includegraphics[clip,viewport =00.0 -100.0 600.0 250.0]{instant_3d_perp.pdf}}
%\includegraphics{p2fig1_ting.pdf}}
%\includegraphics*[0.0in,0.0in][8.0in,11.0in]{p2fig1_ting.pdf}}
%\includegraphics{p2fig1_ting.pdf}}
%\caption{breakup - symmetric outgoing protons}
%\label{fig:1}
}
\end{figure}
\end{center}
\end{frame}

%slide 38 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Particle production
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Cluster properties $+$ few-body problems.
\vfill
\item [{\color{red}$\bullet$}] Bare particles replaced by physical particles.
\vfill
\item [{\color{red}$\bullet$}] Vertex replaced by 
$2\leftrightarrow 3$ interactions.
\vfill
\item [{\color{red}$\bullet$}] Subsystem replaced by sub-energy.
\vfill
\item [{\color{red}$\bullet$}] General development in progress.
}
\end{itemize}
\end{frame}



\end{document}
\begin{frame}

\begin{itemize}
{\bf 

\item [{\color{red}$\bullet$}] Fourier transform V18 - use functions with analytic FT

\item [{\color{red}$\bullet$}] Express potential in terms of a minimal set of 
invariant spin operators.

\item [{\color{red}$\bullet$}] 20 on shell 24 off shell

\item [{\color{red}$\bullet$}] On and off shell must match smoothly 

\item [{\color{red}$\bullet$}] Many traces

}
\end{itemize}
\end{frame}


\begin{frame}

\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Fourier transform computed for V18 
as expansion on 24 spin-isospin operators.

\item [{\color{red}$\bullet$}] Six independent 2N spin operators off
shell - Five on shell.  Numerical instabilities result unless 5 of 
the six off shell operators approach the on shell operators smoothly.

\item [{\color{red}$\bullet$}] Three-body spins operators generated 
from two body spin operators by iterating the Faddeev equations. 

\item [{\color{red}$\bullet$}] Need expansions for $t_r$ including 
Wigner rotations.
}
\end{itemize}
\end{frame}


\begin{frame}
\begin{itemize}
{\bf
\item [{\color{red}$\bullet$}] Many-spin isospin degrees of freedom.
\vfill
\item [{\color{red}$\bullet$}] Need realistic effective interactions
\vfill
\begin{itemize}
\vfill
\item [{\color{red}$\bullet$}] Use fact the nr NN interactions fit to data to 
construct phase-equivalent relativistic NN interaction.
\vfill
\item [{\color{red}$\bullet$}] Half shell matrix elements can be constructed
directly from NR matrix elements
\vfill
\item [{\color{red}$\bullet$}] Additional first resolvent equations need to be solved.
\vfill
\item [{\color{red}$\bullet$}] Need momentum space interaction in operator form
\vfill
\end{itemize}
\vfill
\item [{\color{red}$\bullet$}] Need to satisfy cluster properties
\vfill
\begin{itemize}
\vfill
\item [{\color{red}$\bullet$}] Generating 3 body interactions necessary for 
cluster properties requires solving three body intergral equation.
\vfill
\item [{\color{red}$\bullet$}] Model can be reformulated to 3NF vanished in
rest frame.  Gives on shell equivalent NN model. 
\vfill
\item [{\color{red}$\bullet$}] Addiional first resolvent equations need to be solved.
\vfill
\item [{\color{red}$\bullet$}] Need momentum space interaction in operator form
\vfill
\end{itemize}
\vfill
\item [{\color{red}$\bullet$}] Standard numerical problems 
\vfill
\begin{itemize}
\vfill
\item [{\color{red}$\bullet$}] Krylov methods for large matrices
\vfill
\item [{\color{red}$\bullet$}] Balian Brezin method utilize rotational 
covariance to compute kernel
\end{itemize}
}
\end{itemize}
\end{frame}

% 

\begin{frame}
\begin{center}
{\bf 
Quantum mechanics - formal observables
}
\end{center}
\vfill
\[
P_{AB} = {\langle A \vert B \rangle \langle B \vert A \rangle
\over \langle A \vert A \rangle \langle B \vert B \rangle} 
= \vert \langle \hat{A} \vert \hat{B} \rangle \vert^2
\]
\vfill
\[
\langle A \vert O \vert A \rangle  = \sum_n  o_n {\color{red}P_{n,A}} 
\qquad
O= \sum_n \vert \hat{n} \rangle o_n \langle \hat{n} \vert
\qquad
\]
\vfill
\[
\rho= \sum_n c_m \vert \hat{A}_m \rangle \langle \hat{A}_m \vert
 \qquad \sum_m c_m =1 
\]
\vfill
\[
\mbox{Tr} (\rho O) = \sum_{mn} c_{m} o_n {\color{red}P_{n,A_m}} 
\]
\end{frame}

%slide 6 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
%
%\begin{frame}
%\begin{center}
%{\bf 
%Experimental observables
%}
%\end{center}
%\vfill
%\begin{itemize}
%{\bf 
%\item [{\color{red}$\bullet$}] Mass, linear momentum,
%spin and magnetic quantum number of each asymptotically 
%separated initial and final particle.
%\vfill
%\item [{\color{red}$\bullet$}] A complete experiment measures
%all of these quantities for a large ensemble of reactions.
%\vfill
%\item [{\color{red}$\bullet$}] Additional non-observable degrees
%of freedom may simplify the description of the dynamics,
%however they are not required to describe the dynamics. 
%}
%\end{itemize}
%\end{frame}

%slide 7 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Special relativity in quantum mechanics
}
\end{center}
\vfill

\begin{center}
{\bf 
Inertial frames are related by Poincar\'e transformations
}
\end{center}
\vfill

\begin{center}
{\bf 
Quantum observables have same values in all inertial frames
}
\end{center}
\vfill
\end{frame}



%slide 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Wigner - Ann. Math 40(149)1939 
}
\end{center}
\vfill
\[
X \underbrace{\to}_{(\Lambda ,a )} X'  
\]
\vfill
\[
\vert A \rangle  \to \vert A' \rangle , \vert B \rangle \to \vert B'\rangle
\]
\vfill
\[
P_{AB} = P_{A' B'}
\]
\vfill
\[
\updownarrow
\]
\vfill
\[
\vert A' \rangle = U(\Lambda ,a) \vert A \rangle  \qquad
\vert B' \rangle = U(\Lambda ,a) \vert B \rangle  
\]
\vfill
\[
U(\Lambda_2 ,a_2) U(\Lambda_1 ,a_1) =
U(\Lambda_2 \Lambda_1,\Lambda _2 a_1 + a_2)e^{i \phi_{21}}
\]
\vfill
\[
\langle O \rangle_A = \langle O' \rangle_{A'}
\qquad \mbox{Tr}(\rho O) = \mbox{Tr}(\rho' O') 
\]
\end{frame} 

%\end{document}

%slide 9 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\[
M = M_0 +V 
\]
\vfill
\[
[V,\mathbf{P}]= [V, \mathbf{j}] = [\pmb{\nabla}_{p_j} , V]=0
\]
\vfill
\[
\langle m_0, \mathbf{p} , j, \mu_j  \vert 
M \vert m_0', \mathbf{p}' , j', \mu_j' \rangle
= 
\delta_{jj'}\delta (\mathbf{p}'-\mathbf{p})\delta_{\mu \mu'} 
\langle m_0 \Vert M^j \Vert m_0' \rangle 
\]
\vfill
\[
M^j \vert \lambda, \mathbf{p} , j, \mu  \rangle = 
\lambda \vert \lambda, \mathbf{p} , j, \mu  \rangle
\]
\vfill
\[
U(\Lambda ) \vert \lambda, \mathbf{p} , j, \mu  \rangle =
\vert \lambda, \pmb{\Lambda} {p} , j, \nu  \rangle
\sqrt{{\omega_\lambda (\pmb{\Lambda}p) \over
\omega_\lambda (\mathbf{p})}}
D^j_{\nu \mu}[R_w (\Lambda ,p) ] 
\]
\end{frame}



\begin{frame}
\begin{center}
{\bf 
Relation to field theory - Lorentz vs Poincar\'e invariance
}
\end{center}
\vfill
\[
\langle f \vert g \rangle = 
\int f^*(x) \langle 0 \vert \phi^{\dagger}(x) \phi(y) \vert 0 \rangle g(y) d^4x d^4y =
\]
\vfill
\[
\int f^*(x) d^4x \langle 0 \vert \phi^{\dagger}(x) \vert m, j, \mathbf{p},\mu, d \rangle 
{d \mathbf{p} \over \omega_m (\mathbf{p}^2) }
\langle m, j, \mathbf{p},\mu, d \vert \phi(y) \vert 0 \rangle 
d^4 y g(y) 
\]
\vfill
\[
\Psi_g (\mathbf{p},\mu) : = 
\int \langle m, j, \mathbf{p},\mu, d \vert \phi(y) \vert 0 \rangle g(y) d^4y
\]
\vfill
\[
\langle f \vert g \rangle = 
\sum_{\mu=-j}^j \int \Psi_f^* (\mathbf{p},\mu) 
{d \mathbf{p} \over \omega_m (\mathbf{p}^2) } \Psi_g (\mathbf{p},\mu) 
\]
\end{frame}


\begin{frame}
Challenge: Need realistic relativistic effective interactions
\end{frame}

\begin{frame}
\begin{center}
{\bf 
Relativistic interactions have to be
refit (to the same experimental data as the non-relativistic
interactions) 
}
\end{center}
\vfill
\[
H_{nr}= {\mathbf{P}^2 \over 2(m_1+m_2)} + {\mathbf{k}^2\over 2 \mu_{12}} + V_{nr}
\]
\vfill
\[
\mathbf{P}=0 \to H \to h_{nr} := {\mathbf{k}^2\over 2 \mu_{12}} + V_{nr}
\]
\vfill
\[
H_r= \sqrt{\mathbf{P}^2 + M^2} 
\]
\vfill
\[
\mathbf{P}=0 \to H \to M := M_0 + V_r
\]
\vfill
\[
M: =
\sqrt{\mathbf{k}^2 +m_1^2+ 2 \mu V_{nr}} +
\sqrt{\mathbf{k}^2 +m_2^2+ 2 \mu V_{nr}} 
\]
\vfill
\[
M_0=
\sqrt{\mathbf{k}^2 +m_1^2} +
\sqrt{\mathbf{k}^2 +m_2^2} 
\]
\end{frame}






\begin{frame}

Challenge: Partial wave expansion difficult above 250 MeV

Replace partial waves by 3-variable integral equation.  One variable 
only integrates over kernel.   The integrand is smooth.

Converges at 2GeV with Malfliet 


 

\end{frame}




\begin{frame}
\[
f(x) \to S(\Lambda^{-1}) f(\Lambda x+a)  
\]
\vfill
\[
\Updownarrow
\]
\vfill
\[
\Psi_g (\mathbf{p},\mu) \to 
\sum_{\nu=-j}^j D^j_{\mu \nu} [R_w^{-1} (\Lambda, p)] 
\sqrt{{\omega_m ((\pmb{\Lambda}{p})^2)\over \omega_m (\mathbf{p}^2)}}
e^{i \Lambda p \cdot a} 
\Psi_g (\pmb{\Lambda}{p},\nu)
\]
\vfill
\begin{itemize}
{\bf
\item [{\color{red}$\bullet$}] Generalizations of 
$\langle m, j, \mathbf{p},\mu, d \vert \phi(y) \vert 0 \rangle$
intertwine finite dimensional representations of the Lorentz group 
with unitary representations of the Poincar\'e group.
}
\end{itemize}
\end{frame}

\begin{frame}

\begin{itemize}
{\bf 
%\item [{\color{red}$\bullet$}] This generalizes to more complex
%  systems because the vacuum expectation values represent kernels of
%  the Hilbert space scalar product.
% 
%\vfill

\item [{\color{red}$\bullet$}] The Poincar\'e covariant representation
is more closely related to standard representations of non-relativistic 
quantum mechanics.

\vfill
\item [{\color{red}$\bullet$}] These relations can be utilized to motivate
the structure and properties of more complex few-body models.
}
\end{itemize} 
 
\end{frame}

%slide ?? xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
%{\footnotesize

\begin{center}
{\bf 
Strategy
}
\end{center}

\vfill

\begin{itemize}
{\bf  
\item [{\color{red}$\bullet$}] Scattering data transforms relativistically; 
it is possible to construct both relativistic and non-relativistic 
two-nucleon interactions that produce identical two-body 
$S$-matrix elements in one chosen frame.

\vfill

\item [{\color{red}$\bullet$}] Cluster properties determine how the
  nucleon-nucleon interactions are embedded in the three-nucleon
  problem.  This is different in the non-relativistic and relativistic
  three-nucleon models.

\vfill 

\item [{\color{red}$\bullet$}] Relativistic effects $=$ difference in
  three-nucleon observables in relativistic and non-relativistic
  models with ``equivalent'' two-nucleon input.

}
\end{itemize}
%}
\end{frame}


%slide 10  xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Computational challenges - Gev scale three-nucleon scattering 
}
\end{center}
\vfill
\begin{itemize}
{\bf  
\item [{\color{red}$\bullet$}] Relativistic interactions have to be
refit (to the same experimental data as the non-relativistic
interactions) 
\vfill
\item [{\color{red}$\bullet$}] Cluster properties requires solving 
an auxillary three-nucleon problem just to compute the kernel of the 
Faddeev equation.   Niave addition of interacting violates 
Poincar\'e symmetry.
\vfill
\item [{\color{red}$\bullet$}] The kernel of the Faddeev equations 
involve square roots of sums of non-commuting operators.
\vfill
\item [{\color{red}$\bullet$}] Permutations operators are expensive.
\vfill
\item [{\color{red}$\bullet$}] Partial waves are not efficient 
or stable at the few GeV scale.
\vfill
\item [{\color{red}$\bullet$}] Matrices are large, dense, and have 
moving singularities.
\vfill

\item [{\color{red}$\bullet$}] Particle production needs to be included.
}
\end{itemize}

\end{frame}

%slide 11 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Computational tricks
}
\end{center}
\vfill
\begin{itemize}
{\bf  
\item [{\color{red}$\bullet$}] A number of tricks can be used to deal with
each of these problems.  Some only apply to the three nucleon problem and
sum apply more generally.  
}
\end{itemize}
\end{frame}

%slide 12 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Background - structure of models
}
\end{center}
\vfill
\begin{itemize}
{\bf  
\item [{\color{red}$\bullet$}] Add interactions to non-interacting 
mass Casimir operator that commute with the free spin Casimir operator.
Standard construction gives $\bar{U}(\Lambda ,a)$.

\vfill

\item [{\color{red}$\bullet$}] Note: Since there exists $S$-matrix
  preserving unitary transformations that transform the spin Casimir
  operator to the free spin Casimir operator, there is no loss of
  generality in using this representation.

\vfill

\item [{\color{red}$\bullet$}] In this representation the exact
  unitary representations of the Poincar\'e group requires that (1)
  interactions are rotationally invariant in the kinematic rest frame
  and (2) have a positive mass spectrum.  }

\end{itemize}
\end{frame}

%slide 13 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


%slide 14 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{itemize}
{\bf  
\item [{\color{red}$\bullet$}] Choose $V_{nr}$ as realistic NN interaction.
It  has the required rotational invariance.
\vfill
\item [{\color{red}$\bullet$}] Since $M=M(h_{nr})$, the rest scattering 
wave functions,
$\psi^{\pm}(\mathbf{k},\mu_1,\mu_2)$,
are identical to the non-relativistic rest wave functions.
\vfill
\item [{\color{red}$\bullet$}] Since 
$S = \langle \psi^+ \vert \psi^- \rangle$
both $M$ and $h_{nr}$ produce the experimental $S$ matrix elements in the
rest frame. 
\vfill
\item [{\color{red}$\bullet$}] Poincar\'e symmetry determines 
relativistic $S$ in other frames.
\vfill
\item [{\color{red}$\bullet$}] Non-relativistic limits are not involved.
\vfill
\item [{\color{red}$\bullet$}] Eliminates the need to refit interactions.
\vfill
\item [{\color{red}$\bullet$}] Quantum mechanical models with the same
  degrees of freedom that give the same 2-body $S$-matrix are
  experimentally indistinguishible (interactions are not $S$-matrix
  observables). $\therefore$ no loss of generality in this specific
  choice of realistic relativistic interactions.
\vfill 


}
\end{itemize}

\end{frame}

%slide 15 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{itemize}

{\bf 
\item [{\color{red}$\bullet$}] The addition of these
interactions as required by cluster properties violates Poincar\'e
symmetry without additional special three-nucleon interactions.  

\vfill

\item [{\color{red}$\bullet$}] 
An auxillary ``three-nucleon'' problem must be solved 
to compute the three-nucleon forces required by 
cluster properties. 

\vfill

\item [{\color{red}$\bullet$}] 
The three-nucleon system is special case were this can be avoided 
using a related $S$-matrix equivalent model.

}
\end{itemize}

\end{frame}

%slide 16 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


\begin{frame}
\[
\bar{M} := \bar{M}_{12,3} + \bar{M}_{23,1}+ \bar{M}_{31,2}- 2M_0
\]
\vfill
\[
\bar{M}_{ij,k} := 
\]
\vfill
\[
\sqrt{\mathbf{q}_k^2 + (\sqrt{\mathbf{k}_{ij}^2 + m_i^2 + 2 \mu_{ij}v_{nr\,ij}}
+\sqrt{\mathbf{k}_{ij}^2 + m_j^2 + 2 \mu_{ij}v_{nr\,ij}})^2} +
\]
\vfill
\[
\sqrt{\mathbf{q}_k^2 + m_k^2} 
\]
\vfill
\[
[\bar{M}_{ij,k},\mathbf{j}^2_0]=0
\]
\vfill
\[
\downarrow
\]
\vfill
\[
\bar{U}(\Lambda ,a),\, \bar{U}_{ij,k}(\Lambda ,a)\not =
\bar{U}_{ij}(\Lambda ,a) \otimes U_k (\Lambda ,a)
 \]
\[
\bar{U}(\Lambda ,a) \nrightarrow  \bar{U}_{ij}(\Lambda ,a)
\otimes {U}_{k}(\Lambda ,a)
\]
\end{frame}

\begin{frame}
\[
\exists \, A_{ij,k}\, \mbox{unitary} \qquad  
A_{ij,k} \bar{U}_{ij,k} A^{\dagger}_{ij,k}
= \bar{U}_{ij} ( \Lambda ,a) \otimes U_k (\Lambda ,a)   
\]
\vfill
\[
A:=(I - i C)(I+i C)^{-1}
\qquad
C := C_{12,3}+C_{23,1}+C_{31,2} 
\]
\vfill
\[
C_{ij,k} := i (A_{ij,k}-I)(A_{ij,k}+I)^{-1} 
\]
\vfill
\[
U(\Lambda ,a) := A^{\dagger} \bar{U}(\Lambda ,a) A
\]
\vfill
\[
U(\Lambda ,a) \to \bar{U}_{ij}(\Lambda ,a)
\otimes {U}_{k}(\Lambda ,a)
\]
\vfill
\[
M = A (\sum A_{ij,k} \bar{M}_{ij,k} A^{\dagger}_{ij,k}- 2M_0 )A^{\dagger}
\]

\end{frame}

%slide 17 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{itemize}
{\bf  
\item [{\color{red}$\bullet$}] $A \to I$ in rest frame.
\vfill
\item [{\color{red}$\bullet$}] $A$ preserves $S$
\vfill
\item [{\color{red}$\bullet$}] $A$ generates three body forces
that restore Poincar\'e symmetry to cluster expansion.
\vfill
\item [{\color{red}$\bullet$}] $[S,U_f(\Lambda ,a)]=0$ 
\vfill
\item [{\color{red}$\bullet$}] For the three nucleon $S$-matrix 
it is sufficient to diagonalize $\bar{M} :=\sum \bar{M}_{ij,k} - 2M_0$ 
\vfill
\item [{\color{red}$\bullet$}] This avoids the calculation of $A$.
}
\end{itemize}

\end{frame}

%slide 18 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


\begin{frame}
\begin{center}
{\bf 
Permutation operators $P$
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [] The Faddeev kernel has the form $P\bar{T}_\gamma (z-M_0)^{-1}$, where 
$P$ is a permutation operator, which is the identity in a mixed basis
where the spins are defined differently. It has the form
}
\end{itemize}

\[
\langle \mathbf{q}_i, \mathbf{k}_{ij}, \mu_1, \mu_2, \mu_3
\vert \mathbf{q}_l, \mathbf{k}_{mn}, \mu_1, \mu_2, \mu_3
\rangle =
\]
\vfill
\[
\sum' D^{j_1}_{\mu_1 \mu_1'}[R^{-1}] 
D^{j_2}_{\mu_2 \mu_2'}[R^{-1}] 
D^{j_3}_{\mu_3 \mu_3'}[R^{-1}] 
\times
\]
\vfill
\[
\langle R\mathbf{q}_i, R\mathbf{k}_{ij}, \mu_1, \mu_2, \mu_3
R\mathbf{q}_l, R\mathbf{k}_{nm}, \nu_1', \nu_2', \nu_3'
\rangle \times
\]
\vfill
\[
D^{j_1}_{\nu_1' \nu_1}[R] 
D^{j_2}_{\nu_2' \nu_2}[R] 
D^{j_3}_{\nu_3' \nu_3}[R] =  
\]
\vfill
\[
\int_{SU(2)} dR \sum' D^{j_1}_{\mu_1 \mu_1'}[R^{-1}] 
D^{j_2}_{\mu_2 \mu_2'}[R^{-1}] 
D^{j_3}_{\mu_3 \mu_3'}[R^{-1}] \times
\]
\vfill
\[
\langle R\mathbf{q}_i, R\mathbf{k}_{ij}, \mu_1, \mu_2, \mu_3
R\mathbf{q}_l, R\mathbf{k}_{nm}, \nu_1', \nu_2', \nu_3' \rangle \times
\]
\vfill 
\[
D^{j_1}_{\nu_1' \nu_1}[R] 
D^{j_2}_{\nu_2' \nu_2}[R] 
D^{j_3}_{\nu_3' \nu_3}[R] 
\]
\end{frame}


\begin{frame}
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Can evaluated for $\mathbf{q}_k$ along
  the $z$ axis and $\mathbf{q}_l$ in the positive $z$ half of the $xz$
  plane.
\vfill
\item [{\color{red}$\bullet$}] It can also be expressed as a linear
combinations of scalar coefficients multiplies by 
the 16 independent rotationally invariant spin operators consisting 
of products of the form
\vfill
\[
I_j , \mathbf{q}_1 \cdot \mathbf{j}_j,
\mathbf{q}_2 \cdot \mathbf{j}_j,
(\mathbf{q}_1\times  \mathbf{q}_2) \cdot \mathbf{j}_j
\qquad 1 \leq j \leq 3
\]
\item [{\color{red}$\bullet$}] An important requirement is the 
choice of spins and momentum variables must be chosen so they all 
undergo identical Wigner rotations. 
}
\end{itemize}

\end{frame}


%slide 17 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{itemize} 
{\bf 
\item [{\color{red}$\bullet$}] Conventional few-body calculations are
  done using partial wave expansions.

\vfill
\item [{\color{red}$\bullet$}] At relativistic momentum transfers the
  number of partial waves needed becomes large.  Above about 250 MeV
  bookkeeping becomes complicated and there are convergence issues.

\vfill
\item [{\color{red}$\bullet$}] Direct integration methods are simpler
  to code, but require more evaluations of the kernel and lead to
  larger matrices.

\vfill
\item [{\color{red}$\bullet$}] Convergence has been tested up to
2 GeV.  

}
\end{itemize}
\end{frame}


\begin{frame}
\begin{itemize} 
{\bf 
\item [{\color{red}$\bullet$}] Expanding the kernel, transition operators
and permutation operators in a basis of rotationally invariant 
spin operators provides some simplifications,  In general there are
$4^3$ such operators;  To get equations for the coefficients requires 
evaluating traces of $(4^3)^3$ spin operators.  
\vfill
\item [{\color{red}$\bullet$}] This requires a realistic non-relativistic 
momentum-space potential in operator form as input.  This has been done
for the AV18 potential.
}
\end{itemize}
\end{frame}
  
\begin{frame}

\begin{itemize} 
{\bf 

\item [{\color{red}$\bullet$}] At relativistic momentum transfers the
  number of partial waves needed becomes large.  Above about 250 MeV
  bookkeeping becomes complicated and there are convergence issues.

\item [{\color{red}$\bullet$}] Direct integration methods are simpler
  to code, but require more evaluations of the kernel and lead to
  larger matrices.

\item [{\color{red}$\bullet$}] Convergence has been tested up to
2 GeV.  

}
\end{itemize}


\end{frame}

\begin{frame}
\begin{center}
{\bf 
Measurements that determine probabilities must be performed a statistically 
significant number of times.
}
\end{center}
\vfill
\begin{center}
{\bf 
It is difficult to produce a large number of identical measurements (with 
identical initial and final states).
}
\end{center}
\vfill
\[
\Downarrow
\]
\vfill

\begin{center}
{\bf 
Expectation values
}
\end{center}
\vfill

\[
O = O^{\dagger} : \qquad 
O = \sum_n  \lambda_n \hat{\phi}_n \hat{\phi}_n^{\dagger} 
\qquad
( \hat{\psi}, O \, \hat{\psi} ) = 
\sum_n \lambda_n {\color{red} \vert ( \hat{\phi}_n , 
\hat{\psi} )\vert^2}
\]
\vfill
\begin{center}
{\bf 
Ensemble averages
}
\end{center}
\vfill
\[
\rho = \sum_k p_k \hat{\psi}_k \hat{\psi}_k^{\dagger}
\qquad
E_\rho  [O ] = \mbox{Tr} ( O \rho ) =\sum_{nk} p_k \lambda_n 
{\color{red} \vert ( \hat{\phi}_n , \hat{\psi}_k )\vert^2}
\]
 
\end{frame}

%slide 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Spacetime symmetries in quantum mechanics 
}
\end{center}
\vfill
\begin{itemize}
{\bf
\item [{\color{red}$\bullet$}] Provide operator algebras that:
\vfill
\begin{itemize}
\item [{\color{red}$\bullet$}] Determine the structure of the 
dynamical theory.

\vfill

\item [{\color{red}$\bullet$}] Determine measurable properties of particles.

\vfill

\item [{\color{red}$\bullet$}] Determine the representation of the 
Hilbert space.
\end{itemize}
}

\end{itemize}
\end{frame}

%slide 9 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\color{black}\bf 
Symmetries and theory 
}
\end{center}

\vfill
\begin{center}
{\color{black}\bf 
Symmetries in quantum theories are ray 
correspondences  
}
\end{center}
\vfill
\[
\hat{\phi}, \hat{\psi} \to \hat{\phi}', \hat{\psi}'
\]
\vfill
\begin{center}
{\color{black}\bf 
that preserve 
}
\end{center}
\vfill
\[
P = \vert (\hat{\phi}, \hat{\psi} )\vert^2 =
\vert (\hat{\phi}', \hat{\psi}' )\vert^2 = P'
\]
\vfill
\begin{center}
{\color{black}\bf 
Wigner's Theorem
}
\end{center}
\vfill
\[
\hat{\phi}' = U [\hat{\phi}] \qquad \hat{\psi}'=U[ \hat{\psi}]
\]
\vfill
\begin{center}
{\color{black}\bf 
$U$ Unitary or antiunitary.
}
\end{center}
\end{frame}

%slide 10 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\color{black}\bf 
Poincar\'e Symmetry
}
\end{center}

\vfill

\begin{center}
{\bf (Principle of special relativity)
}
\end{center}

\vfill

\begin{itemize}

\item [{\color{red}$\bullet$}]  {\bf Inertial coordinate systems}
(free particles move with constant velocity - no mass-dependent 
forces).

\vfill

\item [{\color{red}$\bullet$}] {\color{blue}\bf Equivalent experiments done 
in different inertial coordinate systems give identical results.}

\vfill

\item [{\color{red}$\bullet$}]  {\bf Different inertial coordinate
systems are related by the group of Poincar\'e transformations, 
${\cal P}$} (experimental result). 

\vfill

\item [{\color{red}$\bullet$}]  {\bf The Poincar\'e group 
is a symmetry group of the quantum theory}

\end{itemize}
\end{frame}

%slide 11 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
{Poincar\'e group}
}
\end{center}

\begin{center}
{\color{black}\bf 
Space-time event coordinates
}
\end{center}
\vfill
\[
x_a = (ct_a , \mathbf{x}_a)
\]
\vfill
\begin{center}
{\bf 
Minkowski ``metric''
}
\end{center}
\vfill
\[
x_a^2 = \mathbf{x}_a \cdot \mathbf{x}_a - (ct_a)^2 
\]
\begin{center}
{\color{black}\bf 
Poincar\'e group ${\cal P}$: $x_a,x_b \to x_a',x_b'$
}
\end{center}
\vfill
\[
(x_a - x_b)^2 = (x_a'-x_b')^2 
\]
\vfill
\[
x_a \to x_a' = \Lambda x_a + a_a
\qquad 
(\Lambda x)^2 = x^2   
\]
\end{frame}

%slide 12 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Relativity $\pmb{+}$ quantum theory
}
\end{center}

\begin{center}
{\bf 
${\cal P}=$ Poincar\'e group $=$ symmetry group of the quantum theory. 
}
\end{center}

\vfill
\begin{center}
{\bf 
Wigner's theorem 
}
\end{center}
\vfill
\[
g:X\to X' \qquad g \in {\cal P} \qquad X \quad \mbox{\bf inertial}
\]
\vfill

\begin{center}
{\bf 
Poincar\'e invariance 
}
\end{center}

\vfill
\[
P_{{\phi}, {\psi}}=P_{{\phi}', {\psi}'} \qquad \Leftrightarrow \qquad 
\hat{\phi}' = U(g)  \hat{\phi} \quad
\hat{\psi}' = U(g) \hat{\psi}
\]
\vfill
\[
U(g) U^{\dagger} (g) = I \qquad U(g) U(g') = U(g\cdot g')
e^{i \phi (g,g')}
\]
\end{frame}

%slide 13 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{
\bf Structure of ${\cal P}$
}
\end{center}
\vfill

\begin{itemize}

\item [{\color{red}$\bullet$}]  {\bf Ten parameter group.}

\vfill

\item [{\color{red}$\bullet$}]  {\bf One-parameter subgroups:
space translations (3), time translations (1), rotations (3), 
rotationless Lorentz transformations (3).}

\vfill

\item [{\color{red}$\bullet$}]  {\bf Infinitesimal generators of 
unitary representations of one-parameter subgroups on ${\cal H}$:}

\end{itemize}

\vfill
\[
\{\mathbf{P},H,\mathbf{J},\mathbf{K}\}
\qquad U(t) = e^{-i H t }, \cdots 
\]

\end{frame}

%slide 14 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{
\bf Quantum mechanics + Poincar\'e symmetry 
}
\end{center}
\vfill
\[
\Downarrow
\]
\vfill
\begin{itemize} 
{\bf 
\item [{\color{red}$\bullet$}] Degrees of freedom
\vfill
\item [{\color{red}$\bullet$}] Hilbert space
\vfill
\item [{\color{red}$\bullet$}] Structure of dynamics
\vfill
}
\end{itemize}

\end{frame} 

%slide 15 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Particles - complete measurement of the state of a particle of
mass $m$ and spin $\mathbf{j}^2$
}
\end{center}

\vfill

\begin{itemize}
{\bf
\item  [{\color{red}$\bullet$}] Particle's momentum, $\mathbf{p}$.

\vfill

\item  [{\color{red}$\bullet$}] Particle's spin projection, $\hat{\mathbf{n}} 
\cdot \mathbf{j}$.

\vfill

\item  [{\color{red}$\bullet$}] $m$, $\mathbf{j}^2$, $\mathbf{p}$, 
$\hat{\mathbf{n}} \cdot \mathbf{j}$ maximal set of independent commuting 
self-adjoint functions of the Poincar\'e generators.
}
\end{itemize}

\end{frame} 

%slide 16 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Particles - model Hilbert space
}
\end{center}

\vfill
\begin{itemize}
 
\item [{\color{red}$\bullet$}]{\bf Single-particles have known 
transformation properties with respect to the Poincar\'e group.}

\vfill

\item [{\color{red}$\bullet$}]{\bf The mass $m$, spin $\mathbf{j}^2$,
momentum $\mathbf{p}$ and spin projection $\hat{\mathbf{n}} 
\cdot \mathbf{j}$ for a particle are functions of one-body
Poincar\'e generators.}

\vfill

\item [{\color{red}$\bullet$}]{\bf The spectrum of the commuting
  operators, ($\mathbf{p},\hat{\mathbf{n}} \cdot \mathbf{j}$), is
  determined by the Poincar\'e group and particle's mass and spin
  $(m,\mathbf{j}^2)$.}

\end{itemize} 
\end{frame}

%slide 17 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Particles - model Hilbert space
}
\end{center}

\vfill
\begin{itemize}

\item [{\color{red}$\bullet$}] {\bf The single-particle Hilbert space
can be represented by the space of square integrable functions $\psi
(\mathbf{p},\mu)$:}
\vfill
\[
{\cal H}_{m,j} = \{
\psi (\mathbf{p},\mu) \quad \vert \quad 
\int d\mathbf{p} \sum_{\mu=-j}^j \vert \psi (\mathbf{p},\mu) \vert < \infty 
\}
\]
\vfill
\[
(U(\Lambda ,a)\psi)(\mathbf{p},\mu) = 
\int d\mathbf{p}' \sum_{\mu'=-j}^j 
{\cal D}^{mj}_{\mathbf{p},\mu; \mathbf{p}',\mu'} [\Lambda ,a] 
\psi(\mathbf{p}',\mu')
\]

\end{itemize} 
\end{frame}

%slide 17 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


\begin{frame}
\begin{center}
{\bf 
Three-particle systems
}
\end{center}

\vfill

\[
{\cal H} = {\cal H}_{m_1j_1} \otimes {\cal H}_{m_2j_2} \otimes
{\cal H}_{m_3j_3} 
\]

\vfill

\begin{itemize} 

\item [{\color{red}$\bullet$}]{\bf The particle content of the system
plus the single-particle symmetries determine the representation of 
the Hilbert space.}

\vfill

\item [{\color{red}$\bullet$}]{\bf The representation of ${\cal H}$
has enough vectors to describe any measurement on the three-body 
system.}

\vfill

\item [{\color{red}$\bullet$}] {\bf The three-body system is the smallest
non-trivial interacting system; it is also are simple enough to perform 
complete experimental measurements.
}
\end{itemize}
\end{frame} 

%slide 18 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Poincar\'e symmetry and dynamics 
}
\end{center}
 
\vfill

\begin{itemize}

\vfill

\item [{\color{red}$\bullet$}] {\bf Denote the natural unitary representation
of the Poincar\'e group on each ${\cal H}_{mj}$ by $U_{mj}(\Lambda ,a)$.} 

\vfill

\item [{\color{red}$\bullet$}] {\bf The non-interacting unitary representation
of the Poincar\'e group on ${\cal H}$ is given by:} 

\vfill

\[
U_0 (\Lambda ,a) =  U_{m_1j_1}(\Lambda ,a) \otimes
U_{m_2j_2}(\Lambda ,a) \otimes
U_{m_3j_4}(\Lambda ,a)
\]

\vfill

\item [{\color{red}$\bullet$}] {\bf There is another physical unitary
  representation, $U (\Lambda ,a):{\cal H} \to {\cal H}$, of the
  Poincar\'e group.}

\end{itemize} 
\end{frame} 

%slide 19 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Poincar\'e symmetry and dynamics 
}
\end{center}
\vfill

\begin{center}
{\bf 
The one-parameter time translation subgroup implies
}
\end{center} 
\vfill
\[
U(t_2 )U(t_1) = U(t_2 + t_1) \qquad U(0) = I \qquad 
U(t) U^{\dagger}(t) = I  
\]
\vfill
\[
\Downarrow
\]
\vfill
\[
U(t) = e^{-i H t /\hbar} \qquad H = H^{\dagger} 
\]

\vfill
\[
\Psi (t+t_0)  = U(t) \Psi (t_0) 
\qquad 
i \hbar {d \Psi \over dt} = H \Psi \qquad
\mbox{\bf Schr\"odinger equation} 
\]
\vfill
\begin{center}
{\bf 
Goal - find $H$ (Hamiltonian)!
}
\end{center}
\end{frame}

%slide 20 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Poincar\'e symmetry
}
\end{center} 
\vfill
\[
\Downarrow
\]
\vfill
\[
H_0 = H_1 + H_2 + H_3
\]
\vfill
\begin{center}
{\bf 
Cluster properties $+$ two-body experiment $+$ theory
}
\end{center} 
\vfill
\[
\Downarrow
\]
\vfill
\[
H_{N=2}= H_0 + V_{ij}
\]
\vfill
\[
\Downarrow
\]
\vfill
\[
H_{N=3}= H_0 + V_{12} + V_{23} + V_{31} 
\]
\vfill
\[
\Downarrow
\]
\begin{center}
{\bf 
Predictions
}
\end{center} 
\end{frame} 

%slide 21 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Scattering theory
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [] What is the relation between $V$ ($H=H_0+V$) and measurements of
asymptotically free particles?
\vfill
\item [] Is it possible to determine $V$ uniquely given a complete
set of asymptotic measurements?
}
\end{itemize}
\end{frame} 

%slide 22 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
Scattering theory
}
\end{center}
\vfill
\[
H = \underbrace{H_0}_{\mbox{free particles}} + \underbrace{V}_{\mbox{interactions}}
\] 
\[
U_0 (t) = e^{-i H_0 t /\hbar} \qquad U (t) = e^{-i H t /\hbar}
\]
\[
\lim_{t \to \pm \infty} \Vert \Psi_{\pm}(t) -
\Phi_{0\pm}(t) \Vert = 0 
\]

\[
\lim_{t \to \pm \infty} \Vert U(t) \Psi_{\pm}(0) -
U_0(t) \Phi_{0\pm}(0) \Vert = 0 
\]

\[
\lim_{t \to \pm \infty} \Vert \Psi_{\pm}(0) -
U(-t) U_0(t) \Phi_{0\pm}(0) \Vert = 0 
\]

\[
{\color{red} P} = \vert ( \Psi_{+}(0) , \Psi_{-} (0) ) \vert^2= 
\vert S_{fi} \vert^2 
\]

\end{frame} 

\begin{frame}

\[
S_{fi} = \sumint \hat{\phi}_f^{\dagger} (v_f) [
\delta_{fi} - 2 \pi i \delta (E_f -E_i) T^{fi}(E_i) ](v_f,v_i) 
\hat{\phi}_i^{\dagger} (v_i) dv_f dv_i \approx   
\]
\vfill
\[
\delta_{fi} - [2 \pi i \hat{\phi}_f^{\dagger} (v_f) 
\delta (E_f -E_i) 
\hat{\phi}_i^{\dagger} (v_i) dv_f dv_i] T^{fi}(E_i)(\bar{v}_f,\bar{v}_i)
\]
\vfill
\begin{center} 
{\bf 
Dependence on $\hat{\phi}_{fi}$ separates from dependence on $V$,
}
\end{center}
\vfill
\[
T^{fi}(E_i) (\bar{v}_f,\bar{v}_i) = 
T^{fi}(E_i) (\mathbf{p}_{1f}, \mu_{1f} \cdots
\mathbf{p}_{nf}, \mu_{nf};
\mathbf{p}_{1i}, \mu_{1i} \cdots
\mathbf{p}_{mi}, \mu_{mi} )
\]
\vfill
\[
T^{fi}(E) = V^f + V^i (E-H + i \epsilon)^{-1} V^f
\]
\end{frame}


%slide 23 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center} 
{\bf 
Actual measurements - spin degrees of freedom  
}
\end{center} 
\vfill
\[
\rho_f = \sum p_{fm} \hat{\chi}_{fm} \hat{\chi}_{fm}^{\dagger}
\]
\vfill
\[
\rho_i = \sum p_{in} \hat{\chi}_{im} \hat{\chi}_{im}^{\dagger}
\]
\vfill
\[
E_{fi} = {\mbox{Tr} ( \rho_f T^{fi} \rho_i T^{fi \dagger}  )\over 
\mbox{Tr} ( T^{fi}T^{fi\dagger} ) }
\]
  
\end{frame}

%slide 24 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center} 
{\bf 
Does $\vert S_{fi} \vert^2$ determine $V$?  
}
\end{center} 
\vfill
\[
H_0 +V' = H' = AHA^{\dagger} = A(H_0 +V) A^{\dagger} 
\]
\vfill
\[
AA^{\dagger} =I \qquad \lim_{t \to \pm \infty} 
\Vert (I -A) U_0 (t) \Phi(0) \Vert =0
\]
\vfill
\[
\Updownarrow 
\]
\vfill
\[
S_{fi} = S_{fi}'
\]
\vfill
\begin{center} 
{\bf 
Determine $V$ up to scattering equivalence $A$!  
}
\end{center} 

\vfill

\begin{center} 
{\bf 
Impossible to find a unique $H$ - possible to find equivalence 
class of $H$'s  
}
\end{center} 
 
\end{frame}

%slide 25 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center} 
{\bf 
Structure of  $H \to M= \sqrt{H^2 -\mathbf{P}^2}$  (Mass = rest Hamiltonain) 
}
\end{center} 
\vfill
\[
H_0 \to M_0 = \sqrt{4m^2 + 4k^2 + q^2 } + \sqrt{m^2 + q^2}
\]
\vfill
\[
V_{12} = \sqrt{4m^2 + 4k^2 + q^2 + 4mv_{12}} - \sqrt{4m^2 + 4k^2 + q^2 } 
\]
\begin{itemize}
{\bf 
\item [] $q=$ momentum of particle 3 in the three-body rest frame 
\vfill
\item [] $k=$ relative momentum of particles 1 and 2 in the two-body rest frame
}
\end{itemize}
\end{frame} 

%slide 25 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


\begin{frame}
\[
v_{12} = \sum_{n=1}^{18} V_n (r) O_n 
\]
\[
O_1 = I, 
\qquad 
O_2 = (\btau_1 \cdot \btau_2), 
\qquad 
O_3 = (\pmb{\sigma}_1 \cdot \pmb{\sigma}_2), 
\]
\[  
O_4 = (\pmb{\sigma}_1 \cdot \pmb{\sigma}_2)(\btau_1 \cdot \btau_2), 
\qquad 
O_5 = S_{12}=3(\pmb{\sigma}_1 \cdot \hat{\mathbf{r}})( \pmb{\sigma}_2 \cdot
\hat{\mathbf{r}}) - \pmb{\sigma}_1 \cdot \pmb{\sigma}_2
\]
\[  
O_6 =S_{12} (\btau_1 \cdot \btau_2), 
\qquad  
O_7 = (\mathbf{L}\cdot \mathbf{S}), 
\qquad 
O_8 = (\mathbf{L}\cdot \mathbf{S})(\btau_1 \cdot \btau_2) 
\]
\[
O_9 = (\mathbf{L}\cdot \mathbf{L}) 
\qquad
O_{10}
=(\mathbf{L}\cdot \mathbf{L})(\btau_1 \cdot \btau_2), \qquad
O_{11}=(\mathbf{L}\cdot \mathbf{L})(\pmb{\sigma}_1 \cdot \pmb{\sigma}_2),
\]
\[
O_{12}= (\mathbf{L}\cdot \mathbf{L})(\pmb{\sigma}_1 \cdot
\pmb{\sigma}_2) (\btau_1 \cdot \btau_2), 
\qquad
O_{13}=
(\mathbf{L}\cdot \mathbf{S})^2, 
\]
\[ 
O_{14}= (\mathbf{L}\cdot
\mathbf{S})^2 (\btau_1 \cdot \btau_2)
\qquad 
O_{15}= T_{12} = ( 3
\tau_{1z}\tau_{2z} - \btau \cdot \btau), 
\]
\[
O_{16}= 
(\pmb{\sigma}_1 \cdot \pmb{\sigma}_2)T_{12}, 
\qquad 
O_{17} = S_{12} 
T_{12}, 
\qquad 
O_{18} = (\tau_{1z} + \tau_{2z}).  
\]
\[
\mathbf{L} = \mathbf{r} \times (-i \pmb{\nabla}_t ) 
\qquad
\mathbf{S}= {1 \over 2} (\pmb{\sigma}_1 \otimes I_2 +
I_1 \otimes \pmb{\sigma}_2) 
\]
\[
\pmb{\sigma}_i = 
\left ( 
\left (
\begin{array}{cc}
0 & 1 \\
1 & 0 
\end{array} 
\right ), 
\left (
\begin{array}{cc}
0 & -i \\
i & 0 
\end{array} 
\right ), 
\left (
\begin{array}{cc}
1 & 0 \\
0 & -1 
\end{array} 
\right ) 
\right )
\qquad 
\pmb{\tau}_i = {1 \over 2} \pmb{\sigma}
\]
\end{frame}


%\begin{frame}
%{\color{black}
%\begin{center}
%{\color{black}\bf 
%Two-body scattering 
%}
%\end{center}
%\vfill
%\[
%H = {\mathbf{p}^2 \over 4m} + h_{nr} \qquad M^2 = 4 m {\color{red} h_{nr}} + 4m^2
%\]
%\vfill
%\[
%\lim_{t \to \pm \infty} \Vert e^{-iH t } \vert \psi_{\pm} (0) \rangle -
%e^{-iH_0 t } \vert \psi_0 (0) \rangle \Vert =0 
%\]
%\vfill
%\[
%\Updownarrow 
%\]
%\vfill
%\[
%\lim_{t \to \pm \infty} \Vert  \vert \psi_{\pm} (0) \rangle -
%{\color{red}e^{iHt } e^{-iH_0 t }} \vert \psi_0 (0) \rangle \Vert =0 
%\]
%\vfill
%\[
%{\color{red}e^{iH_{nr}t } e^{-iH_{nr0} t }} =
%e^{ih_{nr}t } e^{-ih_0 t } = 
%e^{iM^2t' } e^{-iM^2_0 t' } \qquad t'=t/4m
%\]
%\vfill
%\[
%\lim_{t \to \pm \infty} e^{iM^2t' } e^{-iM^2_0 t' } =
%\lim_{t \to \pm \infty} e^{iH_r^2t' } e^{-iH^2_{r0} t' } =
%\lim_{t \to \pm \infty} {\color{red}e^{iH_rt' } e^{-iH_{r0} t' }}  
%\]
%\vfill
%
%\[
%S_{fi(nr)}({\color{red}\mathbf{k}^2})  = \langle \psi_+ \vert \psi_- \rangle = 
%S_{fi(r)}({\color{red}\mathbf{k}^2})  
%\]
%}
%\end{frame}

\begin{frame}
\begin{center}
{\bf 
Three-Particle Scattering ($\bar{M}$ Operator Equations)
}
\end{center}
\vfill
\[
M=M_0 +V \qquad V=\sum_{\alpha} V_{\alpha}  \qquad \alpha \in \{ (12,3),(23,1),(31,2) \}
\]
\vfill
\[
M_\alpha =M_0 +V_\alpha   \qquad V^{\alpha} = M-M_{\alpha}
\]
\vfill
\[
T^{\alpha \beta}(m) := 
V^\beta + V^{\alpha}(m-M+i 0^+)^{-1} V^{\beta} 
\]

\vfill
\[
(a_0,S^{\alpha \beta} b_0) = (a_0,b_0) - 2 \pi i \delta (m_a - m_b)
(a_0,{ T^{\alpha\beta} (m_a+i0^+)} b_0)
\] 
\end{frame}

\begin{frame}
\[
T^{\alpha\beta}(z) = V^\beta + V^\alpha (z-M)^{-1} V^\beta
\]
\vfill
\[
(z-M)^{-1} = (z-M_\alpha)^{-1} + (z-M_\alpha)^{-1}V^\alpha (z-M)^{-1}
\]
\vfill
\[
\boxed{
T^{\alpha\beta}(z) = V^\beta + V^\alpha (z-M_\alpha)^{-1} T^{\alpha\beta}(z)
}
\]
\vfill
\begin{center}
{\bf 
Kernel not compact; solutions not unique.
}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
{\bf 
Faddeev equations
}
\end{center}
\vfill

\[
V^\alpha (z-M)^{-1}=  \sum_{\gamma \not= \alpha} V_\gamma (z-M_\gamma)^{-1} 
[I +V^\gamma (z-M)^{-1}]
\]
\vfill
\[ 
T^{\alpha\beta}(z) = V^\beta + V^\alpha (z-M)^{-1} V^{\beta}
\]
\vfill
\[
{\color{black}
T^{\alpha\beta}(z) = V^\beta + \sum_{\gamma \not= \alpha} 
V_\gamma (z-M_\gamma)^{-1} T^{\gamma\beta}(z) 
}
\]
\vfill
\[
{\color{red}
\boxed{
{\color{black}
T^{\alpha\beta}(z) = V^\beta + \sum_{\gamma \not= \alpha} 
T_\gamma (z-M_0)^{-1} T^{\gamma\beta}(z) 
}
}
}
\]
\vfill
\[
T_\gamma (z) = V_\gamma +  
V_\gamma (z-M_0)^{-1} T_\gamma (z) 
\]
\vfill
\begin{center}
{\bf
Iterated kernel of coupled equations compact
}
\end{center}
\vfill
\[
T(z) = D(z) + K(z)T(z)  \qquad K(z)^2 \quad \mbox{\bf compact}
\]
\end{frame}


\begin{frame}
\begin{center}
{\bf 
Computation
}
\end{center}
\vfill
\[
M \to \oplus M_j 
\]
\vfill
\begin{center}
{\bf
Independent equations for each $j$
}
\end{center}
\vfill
\begin{center}
{\bf
Up to 143 coupled singular integral equations in 2 continuous variables
}
\end{center}
\vfill
\[
T\Psi_0  = D\Psi_0  + KD\Psi_0  + K^2 T \Psi_0
\]
\vfill 
\[
D\Psi_0,  K D \Psi_0 \cdots K^n D \Psi_0  
\]
\vfill
\begin{center}
{\bf 
Generates an orthonormal basis - solve in this basis
}
\end{center}  
\end{frame}

\begin{frame}
\begin{center}
{\bf 
Spin observables $O_{ij}$
}
\end{center}
\vfill
\[
\{A_i= A_i^{\dagger} \}_{i=1}^{64}  \qquad  \mbox{Tr} (A_i A_j) = \delta_{ij}  
\]
\vfill
\[
O_{ij} ={\mbox{Tr} ( A_i T A_j T^{\dagger}) 
\over \mbox{Tr} ( T T^{\dagger})}
\]
\end{frame}

\begin{frame}

\begin{center}
{\bf 
What do we learn ?
}
\end{center}

\vfill
\begin{itemize}
\item [{\color{red}$\bullet$}] {\bf 
Comparison of theory with experiment constrains $V$ to be an 
element of the equivalence class of correct interactions.
}
\vfill
\item [{\color{red}$\bullet$}] {\bf Cluster 
properties can be used to relate interactions in few particle 
systems to interactions for larger systems.
}
\vfill
\item [{\color{red}$\bullet$}] {\bf Properties of 
interactions in the correct equivalence class 
provides information about microscopic dynamics.
}
\vfill
\item [{\color{red}$\bullet$}] {\bf Approximation  
methods can be tested against exact solutions.}
\end{itemize}

\end{frame}

\begin{frame}
{\color{black}\bf
\begin{center}
Numerical issues
\end{center}
\vfill
\begin{itemize}
{\color{black}
\item [{\color{red}$\bullet$}] Matrices range from 
$10^6 \times 10^6$ to $10^7 \times 10^7$ ($N=3$) 
\vfill
\item [{\color{red}$\bullet$}] The kernel is dense and  
has moving singularities.
\vfill
\item [{\color{red}$\bullet$}] The kernel is expressed in terms
of solutions of subsystem singular integral equations.
\vfill
\item [{\color{red}$\bullet$}] For larger problems the kernel 
cannot be stored.
\vfill
\item [{\color{red}$\bullet$}] Solution methods - use iterations 
of kernel coupled with orthogonalization (Lanczos, Gram Schmidt) 
to reduce size of the linear system.  
}
\end{itemize}
}                          
\end{frame}

\end{document}


\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{6.0in}{!}{
%\includegraphics{p2fig3_ting.pdf}}
\includegraphics[clip,viewport =00.0 00.0 600.0 350.0]{p2fig3_ting.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{6.0in}{!}{
%\includegraphics{p2fig4_ting.pdf}}
\includegraphics[clip,viewport =00.0 00.0 600.0 350.0]{p2fig4_ting.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig1.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig2.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig3.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig4.pdf}}
}
\end{figure}
\end{center}
\end{frame}

%\end{document}


\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig5.pdf}}
}
\end{figure}
\end{center}

\end{frame}


\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig8.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig9.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig10.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
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\includegraphics{fig11.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{fig12.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
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\includegraphics{fig13.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
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\includegraphics{fig14.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
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\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
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\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
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\begin{frame}
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\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
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\includegraphics{fig1n.pdf}}
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\end{figure}
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\end{frame}

\begin{frame}
\begin{center}
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\begin{frame}
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\begin{frame}
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\begin{frame}
\begin{center}
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\includegraphics{fig21n.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
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\includegraphics{fig22n.pdf}}
}
\end{figure}
\end{center}
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\begin{frame}
\begin{center}
\begin{figure} 
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\includegraphics{fig23n.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
\begin{figure} 
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\includegraphics{fig24n.pdf}}
}
\end{figure}
\end{center}
\end{frame}

\begin{frame}
\begin{center}
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}
\end{figure}
\end{center}
\end{frame}




\begin{frame}
{\color{black}\bf
\begin{center}
Applications
\end{center}
\vfill
\begin{itemize}
{\color{black}
\item [{\color{red}$\bullet$}] Test the role of special relativity and
three body interactions in three-body neutron deuteron 
scattering for energies up to 250 $MeV$.
}
\end{itemize}
}                          
\end{frame}


%slide 14 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{3.0in}{!}{
\includegraphics %[50bp,50bp][410bp,302bp]
%[bb= 10 00 400 300]%
{fig4.pdf}}
}
\end{figure}
\end{center}
\vfill
\end{frame}

%slide 14 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
\begin{figure} 
\rotatebox{00}{\resizebox{3.0in}{!}{
\includegraphics %[50bp,50bp][410bp,302bp]
%[bb= 10 00 400 300]%
{fig5.pdf}}
}
\end{figure}
\end{center}
\vfill
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\begin{frame}

\begin{center}
{\bf 
Conclusions and outlook 
}
\end{center} 
\vfill
{\bf
\begin{itemize}
\item [{\color{red}$\bullet$}] Calculations show that there are 
interesting interference effects involving three body forces
and corrections due to relativity.

\vfill

\item [{\color{red}$\bullet$}] Many open problems remain. 
\vfill

\begin{itemize}

\item [{\color{red}$\bullet$}] Particle production.
\item [{\color{red}$\bullet$}] Electromagnetic interactions.
\item [{\color{red}$\bullet$}] Cluster properties.
\item [{\color{red}$\bullet$}] Direct solution.

\end{itemize}


\vfill

\item [{\color{red}$\bullet$}] Both pure and applied mathematics 
were essential for progress in this field.

\end{itemize} 
}
\end{frame} 


\begin{frame}
\begin{center}
{\bf 
Conclusions/Outlook
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] 
\vfill
\item [{\color{red}$\bullet$}] 
\vfill
\item [{\color{red}$\bullet$}] 
\vfill
\item [{\color{red}$\bullet$}] 
\vfill
\item [{\color{red}$\bullet$}] 
\vfill
\item [{\color{red}$\bullet$}] 
}
\end{itemize}
\end{frame}


\end{document} 


\begin{frame}
\begin{center}
{\bf Current:}
\end{center}
\begin{center}
\begin{figure}
\rotatebox{00}{\resizebox{2.5in}{!}{
\includegraphics{current1_2.pdf}} 
}
\end{figure}               
\end{center}                                                                   
\end{frame}

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\begin{frame}
\begin{center}
{\color{black}\bf
Collaborators
}
\end{center} 
\vfill
\begin{itemize}
{\color{black}\bf

\item [{\color{red}$\bullet$}] Iowa:
Fritz Coester, Gerald Payne, Bill Klink,  
Yunfei Huang, Saravanan Veerasamy, Ping Lin Chung, 
Fatih Buluth, Hsien Chi Jean 
 
\vfill

\item [{\color{red}$\bullet$}] NSF: Brad Keister

\vfill

\item [{\color{red}$\bullet$}] Ohio U: Ting Lin, Charlotte Elster

\vfill

\item [{\color{red}$\bullet$}] Bochum: Walter Gl\"ockle

\vfill

\item [{\color{red}$\bullet$}] Krakow: Henryk Wita{\l}a,
Jacek Golak, Roman Skibi\'nski

\vfill 
\item [{\color{red}$\bullet$}] Kyushu: Hiroyuki Kamada

}
\end{itemize}
\end{frame}

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%slide 5 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame} 
\begin{center}
{\bf 
What are the relevant degrees of freedom?
}
\end{center}

\vfill

\begin{itemize}
{\bf 
\item [{\color{red}$\bullet$}] Observable degrees of freedom 
are properties of particles far away from the reaction site.

\vfill

\item [{\color{red}$\bullet$}] All possible measurement outcomes 
can be expressed in terms of particle degrees of freedom.

\vfill

\item [{\color{red}$\bullet$}] Additional degrees of freedom are not 
required, but they may provide a more elegant understanding of the dynamics.  
}
\end{itemize}

\end{frame}

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\begin{frame}

\begin{center}
{\bf Theory: quantum mechanics
}
\end{center} 

\bigskip

\begin{itemize}
{\bf  
\item [{\color{red}$\bullet$}] States of a physical system are represented 
by one-dimensional subspaces of a complex Hilbert space, ${\cal H}$.  
These subspaces can be labeled by unit vectors, $\hat{\phi} \in {\cal H}$.

\bigskip

\item [{\color{red}$\bullet$}] Predictions of a quantum theory are probabilities.

\bigskip

\item [{\color{red}$\bullet$}] The probability of a measuring a system prepared
in a state labeled by the unit vector $\hat{\psi}$ to be found in a
state labeled by unit vector $\hat{\phi}$ is expressed in terms of the Hilbert 
space scalar product as
\[
P_{\hat{\phi},\hat{\psi}} = \vert ( \hat{\phi} , \hat{\psi}) \vert ^2 
\] 
}
\end{itemize}
\end{frame} 
 \begin{frame}
\begin{center}
{\bf
GeV-scale model calculation using 
}
\end{center}
\vfill
\[
\Omega_{\pm} = 
\lim_{n \to \pm \infty} e^{-ine^{-\beta H}}e^{+ine^{-\beta H_0}}
\qquad S = \Omega_+\Omega_-
\]
\vfill
\vbox{
\begin{center}
\begin{tabular}{llll}
\hline
$k_0$ & Real T & Im T & \% error \\
\hline
%0.05  &   2.18499e-1    &  -1.03160e+0	          &     0.0982   \\
0.1   &  -2.30337e-1    &  -4.09325e-1	          &     {\color{red}0.0956}   \\
%0.2   &  -1.01512e-1    &  -4.61420e-2	          &     0.0981   \\
0.3   &  -3.46973e-2    &  -6.97209e-3	          &     {\color{red}0.0966}   \\
%0.4   &  -1.39007e-2    &  -1.44974e-3	          &     0.0997   \\
0.5   &  -6.44255e-3    &  -3.86459e-4	          &     {\color{red}0.0986}   \\
%0.6   &  -3.34091e-3    &  -1.24434e-4	          &     0.0952   \\
0.7   &  -1.88847e-3    &  -4.63489e-5           &     {\color{red}0.0977}   \\
%0.8   &  -1.14188e-3    &  -1.93605e-5           &     0.0965   \\
0.9   &  -7.28609e-4    &  -8.86653e-6           &     {\color{red}0.0982}   \\
%1.0   &  -4.85708e-4    &  -4.37769e-6           &     0.0967   \\
1.1   &  -3.35731e-4    &  -2.30067e-6           &     {\color{red}0.0987}   \\
%1.2   &  -2.39235e-4    &  -1.27439e-6           &     0.0968   \\
1.3   &  -1.74947e-4    &  -7.38285e-7           &     {\color{red}0.0985}   \\
%1.4   &  -1.30818e-4    &  -4.44560e-7           &     0.0955   \\
1.5   &  -9.97346e-5    &  -2.76849e-7           &     {\color{red}0.0956}   \\
%1.6   &  -7.73390e-5    &  -1.77573e-7           &     0.0992   \\
1.7   &  -6.08794e-5    &  -1.16909e-7           &     {\color{red}0.0964}   \\
%1.8   &  -4.85672e-5    &  -7.87802e-8           &     0.0956   \\
1.9   &  -3.92110e-5    &  -5.42037e-8           &     {\color{red}0.0967}   \\
%2.0   &  -3.20000e-5    &  -3.80004e-8           &     0.0966   \\
\hline                                                                           \end{tabular}
\end{center}
}
\end{frame}
