\documentclass{beamer}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amscd}
\usepackage{graphicx}
% \usepackage{graphics}
\usepackage{color}
\mode<presentation>
\def\p{{\sf p}}
\def\w{{\sf w}}
\def\x{{\sf x}}
\def\t{{\sf t}}
\def\k{{\sf k}}
\def\q{{\sf q}}
\def\s{{\sf s}}
\def\y{{\sf y}}
\def\z{{\sf z}}
\def\a{{\sf a}}
\def\b{{\sf b}}
\def\E{{\sf E}}
\def\O{{\sf O}}
\def\X{{\sf X}}
\def\Y{{\sf Y}}
\def\notto{\to\mspace{-20mu}/\mspace{20mu}}
\def\sumint{\sum\mspace{-25mu}\int}
\def\directsumint{\sum\mspace{-25mu}\int\,_{\oplus}}
\newcommand{\h}{\mathbf{h}}
\newcommand{\dg}{\mathbf{d}}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\setbeamertemplate{navigation symbols}{} %no nav symbolsy
%\hypersetup{pdfpagemode=FullScreen}
%\usepackage{beamerthemesplit}

\begin{document}
% slide 1
%slide 1 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
\begin{frame}
\begin{center}
{\bf 
Relativistic quantum mechanics  
}
\end{center}

\vfill

\begin{center}
{\bf 
W. N. Polyzou
}
\end{center}
\vfill

\begin{center}
{\bf
The University of Iowa 
}
\end{center}

\begin{figure}
\begin{minipage}[t]{.2\linewidth}
\centering
\includegraphics[angle=000,scale=.25]{ui.pdf}
%\caption{Full matrix, $\lambda$=0, mass=1}
%\label{fig:8}
\end{minipage}
\begin{minipage}[t]{.2\linewidth}
\centering
\includegraphics[angle=000,scale=.3]{doe.pdf}
%\caption{Full matrix, $\lambda$=0.2, mass=1}
%\label{fig:9}
\end{minipage}
\begin{minipage}[t]{.2\linewidth}
\centering
\includegraphics[angle=000,scale=.35            ]{u_logo.pdf}
%\caption{Full matrix, $\lambda$=0.2, mass=1}
%\label{fig:9}
\end{minipage}

\end{figure}

%slide 2 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\end{frame}

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

\begin{itemize}
{\bf 
\item[${\color{red}\bullet}$]Relativity in Quantum Mechanics
\vfill
\item[${\color{red}\bullet}$]Poincar\'e covariant theories
\vfill
\item[${\color{red}\bullet}$]Lorentz covariant theories  
\vfill
\item[${\color{red}\bullet}$]Euclidean covariant theories  
\vfill
\item[${\color{red}\bullet}$]Dynamics    
}  
\end{itemize}  

\end{frame}

%slide 3 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Relativity in quantum mechanics
}
\end{center}  

\begin{itemize}
{\bf 
\item[${\color{red}\bullet}$] Quantum observables:
\[
P=\vert \langle \psi \vert \phi \rangle \vert^2 \qquad  
\langle \psi \vert A \vert  \psi \rangle  \qquad
\mbox{Tr}(\rho A)
\]
\item[${\color{red}\bullet}$] Inertial coordinate systems:
\[
X \underbrace{\longrightarrow}_{(\Lambda ,a)} X'
\]
\item[${\color{red}\bullet}$] Relativistic invariance of quantum observables:
\[
P'=P \qquad   \langle \psi' \vert A' \vert  \psi' \rangle =
\langle \psi \vert A \vert  \psi \rangle  \qquad
\mbox{Tr}(\rho' A') =
\mbox{Tr}(\rho A)
\]
\item[${\color{red}\bullet}$] Wigner's theorem: $U(\Lambda ,a)$
\[
\vert \psi'\rangle = U(\Lambda ,a)\vert \psi \rangle
\qquad
A'= U(\Lambda ,a)A U^{\dagger}(\Lambda ,a)
\]
\[   
\rho'= U(\Lambda ,a)\rho U^{\dagger}(\Lambda ,a) 
\]
}  
\end{itemize}  

\end{frame}

%slide 4 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
The Lorentz group and $SL(2,\mathbb{C})$
}
\end{center}
\begin{center}
{\bf
$2 \times 2$ matrix representation of coordinates
}
\end{center}  
\vfill
\[
X = \sigma_{\mu} x^{\mu} =
\left (
\begin{array}{cc}
x^0+x^3 & x^1-i x^2 \\
x^1+i x^2 & x^0-x^3 \\
\end{array}  
\right ) = X^{\dagger}
\]
\vfill
\[
\mbox{det} (X) = (x^0)^2 -\mathbf{x}^2
\]
\begin{center}
{\bf
Poincar\'e transformations
}
\end{center}  

\[
X \to X' = \Lambda X \Lambda^{\dagger}+A \qquad
\mbox{det}(\Lambda)=1 \qquad A=A^{\dagger}
\]
\vfill
\[
\Lambda^{\mu}{}_{\nu}={1 \over 2} \mbox{tr}(\sigma_{\mu}\Lambda \sigma_{\nu}\Lambda^{\dagger})
\qquad
a^{\mu} = {1 \over 2} \mbox{tr}(\sigma_{\mu}A)
\]
\end{frame}

%slide 5 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Polar decomposition - Boosts
}
\end{center} 
\[
\Lambda = \underbrace{((\Lambda \Lambda)^{\dagger})^{1/2}}_{P}
\underbrace{((\Lambda \Lambda)^{\dagger})^{-1/2}\Lambda}_{U}
\]
\begin{center}
{\bf
Rotationless (canonical) boost ($\pmb{\rho}=$ rapidity)
}
\end{center} 
\[
P = e^{{1 \over 2} \pmb{\rho}\cdot \pmb{\sigma}}= \Lambda_c(p) 
\]
\begin{center}
{\bf
Rotation (generalized Melosh rotation)
}
\end{center} 
\[
U = e^{{i \over 2} \pmb{\theta}\cdot \pmb{\sigma}} = R(p)
\]
\vfill
\[
\Lambda (p) = \Lambda_c(p)R(p)  \qquad R(p) = \Lambda_c^{-1}(p) \Lambda (p) 
\]  

\end{frame}  


%slide 6 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Relativistic dynamics
}
\end{center}  
\[
U(\Lambda ,a) :{\cal H} \to {\cal H}
\]
\begin{center}
{\bf
Relativistic analog of diagonailzing Hamiltonian
}
\end{center}  

\[
W U(\Lambda,a) W^{\dagger} = \int_{\oplus_{m,s,d}}  U_{m,s}(\Lambda ,a) d\mu(m,s,d)
\]
\[
m=\mbox{\bf mass}\qquad s=\mbox{\bf spin} \qquad d=\mbox{\bf degeneracy}
\]

\begin{itemize}
{\bf 
\item[${\color{red}\bullet}$] $U_{m,s}(\Lambda ,a)$ fixed by group theory,
basis choice and representation of ${\cal H}$
}  
\end{itemize}  
\end{frame}

%slide 7 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Poincar\'e covariant RQM - basis choice:
}
\end{center}
\[
U(\Lambda ,a) \to P^{\mu}, J^{\mu \nu} \to
\mbox{\bf commuting observables}
\]  
\[
\{ \vert (m,s;d)b\rangle \}
\]
\begin{center}
{\bf   
Invariants:
}
\end{center}
\[
{\color{red}M^2} = -P^{\mu}P_{\mu} \qquad
M^2 {\color{red}S^2} = W^{\mu}W_{\mu} \qquad
W^{\mu} = {1 \over 2} \epsilon^{\alpha \beta \gamma \mu }
P_{\alpha}J_{\beta \gamma}
\]
\begin{center}
{\bf
Momentum observables:
}
\end{center}
\[
\mathbf{P} \qquad \mathbf {V}= \mathbf{P}/M
\qquad P^{+} = P^0+\hat{\mathbf{n}}\cdot \mathbf{P},\mathbf{P}_{\perp}
\]  
\begin{center}
{\bf
Spin observables (depend on choice of $\Lambda_x (p)$):
}
\end{center}
\[
\mathbf{s}_x^i = \Lambda_x^{-1}({\color{blue}P})^i{}_{\mu}\Lambda_x^{-1}({\color{blue}P})^i{}_{\nu}
J^{\mu\nu} 
\]
\[
{\color{blue}P} \qquad \mbox{\bf operator}  
\]
  
\end{frame}

%slide 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Hilbert space representations: ${\cal H}= L^2(m,s,d,{\color{red}b})$     
}
\end{center}

\begin{itemize}
{\bf 
\item[${\color{red}\bullet}$] $\{ (m,s;d) {\color{red}\mathbf{p},\hat{s}_c \cdot \hat{\mathbf{n}} } \}$ \qquad (Instant form )
\vfill  
\item[${\color{red}\bullet}$] $\{ (m,s;d) {\color{red} \mathbf{v},\hat{s}_c \cdot
  \hat{\mathbf{n}}}\}$ \qquad (Point form)  
\vfill  
\item[${\color{red}\bullet}$] $\{ (m,s;d) {\color{red}\mathbf{p},\hat{s}_h \cdot 
\hat{\mathbf{z}}}\}$ \qquad (Jacob-Wick helicity )
\vfill
\item[${\color{red}\bullet}$] $\{ (m,s;d) {\color{red}p^+,\mathbf{p}_{\perp},\hat{s}_f
\cdot \hat{\mathbf{n}}}\}$ \qquad (Front form)

\item[${\color{red}\bullet}$] $\vdots$
}
\end{itemize}

\begin{center}
{\bf
Spectrum of ${\color{red}b}$ in irreducible subspaces fixed by group theory.
}
\end{center}

\begin{center}
{\bf
Different basis choices are related by unitary transformations.
}
\end{center}

\end{frame}

%slide 9 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Irreducible representations
}
\end{center}
\vfill
\[
U(\Lambda ,a)\vert (m,s,d)b\rangle = \sum_{b'}\int db' \vert (m,s,d)b'\rangle
{\cal D}^{m,s}_{b'b}(\Lambda ,a)
\]
\vfill
\[
{\cal D}^{m,s}_{b'b}(\Lambda ,a) =
\langle (m,s,d)b \vert U(\Lambda ,a) \vert (m,s,d)b' \rangle
\]
\vfill
\[
\sum\int db' {\cal D}^{m,s}_{bb''}(\Lambda_2 ,a_2) {\cal D}^{m,s}_{b''b'}(\Lambda_1 ,a_1) =
{\cal D}^{m,s}_{bb'}(\Lambda_2\Lambda_1  ,\Lambda_2 a_1+ a_2) 
\]

\end{frame}

%slide 10 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


\begin{frame}
\begin{center}
{\bf
Structure of
${\cal D}^{m,s}_{b'b}(\Lambda  ,a)$
} 
\end{center}
\vfill
\begin{center}
{\bf
Depends on basis choice
} 
\end{center}

\vfill
\[
\delta (\mathbf{b}' -\pmb{\Lambda}( b))
\vert {\partial (\pmb{\Lambda} (b)) \over
\partial (\mathbf{b})}\vert^{1\over 2}
e^{i (\Lambda p) \cdot a}
D^s_{\mu \nu} [\Lambda_x^{-1} (\Lambda p)\Lambda \Lambda_x(p)]
\]
\vfill
\begin{center}
{\bf
Example $b=\{\mathbf{p},\mathbf{s}_c\} $ - (Instant form)  
} 
\end{center}
\vfill
\[
\delta (\mathbf{p}' -\pmb{\Lambda}(p))
\vert {\omega_m ( {p}')) \over \omega_m ( {p})} \vert^{1/2}
e^{i (\Lambda p) \cdot a}
D^s_{\mu \nu} [\Lambda_c^{-1} (\Lambda p)\Lambda \Lambda_c(p)]
\]

\end{frame}  

%slide 11 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf
Dynamics non-trivial
}
\end{center}

\vfill
\[
[K^i,P^j] = i \delta_{ij}H = i \delta_{ij}(H_0 +{\color{red}V})    
\]

\vfill

\begin{center}
{\bf
Kinematic subgroups ${\cal K}$
}
\end{center}


\vfill  
\[
(\Lambda,a) \in {\cal K} \Leftrightarrow
{\cal D}^{m,s}_{b'b}(\Lambda  ,a) \qquad \mbox{\bf independent of }m
\]
\vfill
\begin{center}
{\bf
${\cal K}$ depends on basis
}
\end{center}

\vfill
\begin{itemize}
{\bf  
\item[] $b = \{\mathbf{p},\hat{s}_c\cdot \hat{\mathbf{n}}\}
\qquad {\cal K} = $ 3 dimensional Euclidean group
\vfill
\item[] $b = \{\mathbf{v},\hat{s}_c\cdot \hat{\mathbf{n}}\}
\qquad {\cal K} = $ Lorentz group
\vfill
\item[] $b = \{p^+,\mathbf{p}_{\perp},\hat{s}_f\cdot \hat{\mathbf{n}}\}
\qquad {\cal K} = $ subgroup preserving $x^+=0$ 
}
\end{itemize}
\end{frame}  

%slide 12 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Lorentz covariant relativistic quantum mechanics\\
(Hilbert space representations with kernels)
}
\end{center}
\[
D^s_{\mu \nu} [R] = \langle s,\mu \vert e^{i{\pmb{\theta} \over 2}
\cdot \pmb{\sigma}} \vert s,\nu \rangle \qquad
\mbox{\bf  entire function of $\pmb{\theta}$}
\]
\vfill
\[
\sum_{\alpha}D^s_{\mu \alpha} [R_2] D^s_{\alpha \nu} [R_1] - D^s_{\mu \nu} [R_2R_1] =0
\]
\vfill
\[
\langle s, \mu \vert s_1, \mu_1, s_2, \mu_2 \rangle 
D^{s_1}_{\mu_1 \nu_1} [R] D^{s_2}_{\mu_2 \nu_2} [R]\langle s_1, \nu_1, s_2, \nu_2 \vert s, \nu \rangle - D^s_{\mu \nu} [R] =0
\]
\vfill
\[
D^{s_1}_{\mu_1, \nu_1} [R] D^{s_2}_{\mu_2 ,\nu_2} [R] -\sum_{s\mu\nu}
\langle s_1, \mu_1, s_2, \mu_2 \vert s, \mu \rangle
D^s_{\mu \nu} [R] \langle s, \nu \vert s_1, \nu_1, s_2, \nu_2 \rangle =0
\]
\begin{center}
{\bf
Valid for complex angles by analytic continuation
}
\end{center}

\end{frame} 

\begin{frame}

\[
U(\Lambda ,a)\vert (m,s,d)\mathbf{p},\mu \rangle=
\]
\[
\sum_{\nu} \vert (m,s,d)\pmb{\Lambda}p,\nu \rangle
e^{i \Lambda p \cdot a}
D^{s}_{\nu \mu}[\Lambda_x^{-1} (\Lambda p)\Lambda \Lambda_x(p)]
\sqrt{{\omega_m (\pmb{\Lambda} p)\over \omega_m (\mathbf{p})}}
\]
\begin{center}
{\bf
Decompose Wigner rotation
}
\end{center}
\[
\sum_{\nu\alpha \beta} \vert (m,s,d)\pmb{\Lambda}p,\nu \rangle
e^{i \Lambda p \cdot a}
{\color{red}D^{s}_{\nu \alpha}[\Lambda_x^{-1} (\Lambda p)]
D^{s}_{\alpha \beta} [\Lambda]
D^{s}_{\beta \mu}[\Lambda_x(p)]}
\sqrt{{\omega_m (\Lambda p)\over \omega_m (\mathbf{p})}}
\]
\begin{center}
{\bf
Multiply on right by $\sqrt{\omega_m(\mathbf{p})}
D^{s}_{\alpha \mu} [\Lambda_x^{-1}(p)]$
}
\end{center}
\[
\Downarrow  
\]
\[
U(\Lambda ,a)\underbrace{
\vert (m,s,d)\mathbf{p},\alpha \rangle \sqrt{\omega_m(\mathbf{p})}
D^{s}_{\alpha \mu} [\Lambda_x^{-1}(p)]}
_{{\color{red}\vert (m,s)p,\mu, d \rangle_{cov}}}=
\]
\[
\sum_{\nu}
\underbrace{\sum_{\alpha}
\vert (m,s,d)\pmb{\Lambda}p,\alpha \rangle \sqrt{\omega_m(\pmb{\Lambda}p)}
D^{s}_{\alpha \mu} [\Lambda_x^{-1}(\Lambda p)]}_{{\color{red}\vert (m,s)\Lambda p, \nu, d \rangle_{cov}}} e^{i \Lambda p \cdot a}
{\color{red}D^{s}_{\nu \mu} [\Lambda]}
\]

\end{frame}

\begin{frame}
\begin{center}
{\bf
Lorentz covariant representations of the Poincar\'e group
}
\end{center}
\[
U(\Lambda ,a)\vert (m,s)p,\mu, d \rangle_{cov}=
\vert (m,s)\Lambda p, \nu, d \rangle_{cov}
e^{i \Lambda p \cdot a}
D^{s}_{\nu \mu} [\Lambda]
\]
\begin{center}
{\bf
Hilbert space inner product 
}
\end{center}
\begin{center}
{\bf
non-trivial {\color{red}dynamical} kernel
}
\end{center}

\[
\langle \psi \vert \phi \rangle
= \sum \int
\langle \psi \vert (m,s) \mathbf{p} ,\mu \rangle d\mathbf{p} 
\langle (m,s) \mathbf{p} ,\mu \vert \phi \rangle =
\]
\[
\sum \int \langle \psi \vert (m,s) \mathbf{p} ,\mu \rangle_{cov}
{\color{red}{d\mathbf{p} \over \omega_m (\mathbf{p})}
D^s_{\mu \nu} [\Lambda_x(p) \Lambda_x^{\dagger}(p)]}
_{cov}\langle (m,s) \mathbf{p} ,\nu \vert \phi \rangle
\]

\[
\sum \int \langle \psi \vert (m,s) \mathbf{p} ,\mu \rangle_{cov}
{\color{red}2 d^4p \delta (p^2+m^2)\theta (p^0)
D^s_{\mu \nu} [\sigma \cdot p]}
_{cov}\langle (m,s) \mathbf{p} ,\nu \vert \phi \rangle
\]

\[
\Lambda_x(p) \Lambda_x^{\dagger}(p)=
\Lambda_c(p)R_m(p) R_m^\dagger (p) \Lambda_c^{\dagger}(p) = 
\Lambda^2_c(p) = e^{\pmb{\rho} \cdot \pmb{\sigma}} = p\cdot \sigma
\]
\end{frame}  


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

\[
(R^{\dagger})^{-1} = R \qquad  (\Lambda^{\dagger})^{-1} \not= \Lambda \qquad  
\]

\begin{center}
{\bf
Leads to two {\color{red}inequivalent} Hilbert space kernels:
}
\end{center}
\vfill
\[
K^{m,s}_r(p,\mu, \nu):=  
2  \delta (p^2+m^2)\theta (p^0)
D^s_{\mu \nu} [\sigma \cdot p]
\]
\vfill
\[
K^{m,s}_l(p,\mu, \nu):=   
2  \delta (p^2+m^2)\theta (p^0)
D^s_{\mu \nu} [\sigma \cdot {\color{red}\Pi} p]
\]
\vfill
\[
{\color{red}\Pi} = \mbox{\bf space reflection}
\]
\vfill
\begin{itemize}
  {\bf
\item[] The kernels are familiar - 
$K^{m,s}_{l/r}(p,\mu \nu)=$ two-point Wightman function for a
spin-$s$ right or left handed free field.
}
\end{itemize}

\end{frame}

\begin{frame}
\begin{center}
{\bf
Summary: Poincar\'e covariant reps.  $\Rightarrow$ Lorentz covariant reps.
}
\end{center}

\begin{itemize}
{\bf 

\item[${\color{red}\bullet}$] Start with irreducible representations of the
Poincar\'e group in an $L^2$  Hilbert space of functions of eigenvalues
commuting functions of the Poincar\'e generators.  
\vfill

\item[${\color{red}\bullet}$] Use analyticity of the Wigner
rotations to factor the Wigner rotations
to construct Lorentz $(SL(2,\mathbb{C}))$ covariant
representation of the Poincar\'e
group. 

\vfill

\item[${\color{red}\bullet}$] Hilbert spaces have kernels - {\color{red}space reflection does not commute with the
kernels} - the direct sum of right- and left-handed kernels
can be used to get a Lorentz covariant representation of space reflection.

\vfill

\item[${\color{red}\bullet}$] Kernels are {\color{red}independent of original basis choice}.

\vfill
  
\item[${\color{red}\bullet}$] Kernels are familiar Wightman distributions.

}  
\end{itemize}  

\end{frame}  

\begin{frame}
\begin{itemize}
{\bf
\item[] Relation of Poincar\'e covariant wave functions
to Lorentz covariant wave functions  - 
spin choice $``x''$ enters in boost:
}
\end{itemize}
  
\vfill
\[
\boxed{    
_x\langle (m,s,d) \mathbf{p} , \mu \vert \psi \rangle = \sum_{\nu}
{D^s_{\mu \nu}[\Lambda_x^{\dagger}(p)] \over \sqrt{\omega_m (\mathbf{p})}}{}
_{cov,r} \langle (m,s,d) {p} , \nu \vert \psi \rangle_{\vert_{p^0=\omega_m(\mathbf{p})}} 
}
\]
\vfill
\[
\boxed{    
_x\langle (m,s,d) \mathbf{p} , \mu \vert \psi \rangle = \sum_{\nu}
{D^s_{\mu \nu}[\Lambda_x^{-1}(p)] \over \sqrt{\omega_m (\mathbf{p})}}{}
_{cov,l} \langle (m,s,d) {p} , \nu \vert \psi \rangle_{\vert_{p^0=\omega_m(\mathbf{p})}} 
}
\]
\vfill
\end{frame}  

\begin{frame}

\begin{center}
{\bf
Euclidean covariant representations of the Poincar\'e group
}
\end{center}

\begin{center}
{\bf
Some definitions:
}
\end{center}

\[
\sigma_{e\mu} = (i \sigma_0,\pmb{\sigma})
\]
\[
p^{\mu}_e= (-i p^0,\mathbf{p})
\]  
\[
P_e := p_e^{\mu}\sigma_{e\mu} =
\left (
\begin{array}{cc}
ip_e^0 +p^3 & p^1-i p^2\\
p^1+i p^2 & i p_e^0-p^3
\end{array}
\right ) 
\]  
\[
\mbox{det}(P_e) = - (p_e^0)^2 -\mathbf{p}^2  
\]

\[
P := p^{\mu}\sigma_{\mu} =
\left (
\begin{array}{cc}
p^0 +p^3 & p^1-i p^2\\
p^1+i p^2 &  p^0-p^3
\end{array}
\right ) 
\]  
\[
\mbox{det}(P) = (p^0)^2 -\mathbf{p}^2  
\]

\end{frame}

\begin{frame}

\begin{center}
{\bf
Complex orthogonal group $\sim$ complex Lorentz group
}
\end{center}
\vfill
\[
P_e' =A P_e B^t    \qquad \mbox{det}(A) = \mbox{det}(B)=1
\qquad \mbox{\bf orthogonal}
\]
\[
P' = A P B^t   \qquad \mbox{det}(A) = \mbox{det}(B)=1
\qquad \mbox{\bf Lorentz}
\]
\[
O_{\mu,\nu} = {1 \over 2}\mbox{Tr} (\sigma_{e\mu}^{\dagger} A \sigma_{e\nu} B^T) 
\qquad
\Lambda_{\mu,\nu} = {1 \over 2}\mbox{Tr} (\sigma_{\mu} A \sigma_{\nu} B^T) 
\]

\begin{itemize}
{\bf
\item[${\color{red}\bullet}$] $B=A^*= $ real Lorentz transformations

\vfill
  
\item[${\color{red}\bullet}$] $A,B$ unitary $=$ real orthogonal transformation

\vfill  
\item[${\color{red}\bullet}$] $A,B \in SL(\mathbb{C}) =$
complex orthogonal transformation;  complex Lorentz transformation.

}
\end{itemize}
\end{frame}


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

\item [${\color{red}\bullet}$] Real Lorentz group $\sim$ complex subgroup of complex orthogonal group

\vfill
  
\item [${\color{red}\bullet}$] Real orthogonal group $\sim$ complex subgroup of complex Lorentz group  

\vfill
  
\item[${\color{red}\bullet}$] Euclidean and Poincar\'e
generators related by related by
\[
H_m= i H_e \qquad \mathbf{K}_c \cdot \hat{\mathbf{n}} =
-i \mathbf{J}_{e\hat{\mathbf{n}},x_e^0}
\]

\vfill

\item[${\color{red}\bullet}$] Euclidean and Poincar\'e
generators {\color{red}cannot both be self-adjoint on the same representation
of the Hilbert space}.

}    
\end{itemize}  
\end{frame}


\begin{frame}
\begin{center}
{\bf 
Define the irreducible Euclidean covariant kernels
}  
\end{center}  

\[
K^{m,s}_{e,r/l}(x_e-y_e;\mu, \nu):= \int d^4p_e K^{m,s}_{e,r/l}(p_e,\mu, \nu)
e^{i p_e \cdot (x_e-y_e)}
\]
\vfill
\[
K^{m,s}_{e,r}(p_e,\mu, \nu) := {1 \over \pi} {D^s_{\mu \nu}[p_e \cdot \sigma_e]
\over p_e^2 + m^2}
\]
\vfill
\[
K^{m,s}_{e,l}(p_e,\mu, \nu) := {1 \over \pi} {D^s_{\mu \nu}[\Pi p_e \cdot \sigma_e]
\over p_e^2 + m^2}
\]

\end{frame}

\begin{frame}

\begin{itemize}
{\bf 
\item [${\color{red}\bullet}$] These kernels transform covariantly
under spin $s$ representations of the
4-d orthogonal transformations. 

\vfill

\item [${\color{red}\bullet}$] Define $\theta (x_e^0,\mathbf{x})
= (-x_e^0,\mathbf{x}) = $ Euclidean time reflection.

\vfill
  
\item [${\color{red}\bullet}$] Replacing $K^{m,s}_{e,r/l}(x_e-y_e;\mu, \nu)$
by $K^{m,s}_{e,r/l}({\color{red}\theta} x_e-y_e;\mu, \nu)$ makes the
$H_m$ and $\mathbf{K}_c \cdot \hat{\mathbf{n}}$ {\color{red}Hermitian} with
respect to the quadratic form with this kernel.

\vfill

\item [${\color{red}\bullet}$] The resulting sesquilinear form has {\color{red}negative norm
vectors} on the space of Euclidean test functions ($x^0_e$ even and
odd functions cannot both have positive norm).
}
\end{itemize} 
\end{frame}


\begin{frame}

\begin{itemize}
{\bf 
\item [${\color{red}\bullet}$] The sesquilinear form can be made positive by
  projecting on a suitable subspace.

\vfill  
  
\item [] (O.S.) Subspace $=$ Schwartz functions in Euclidean space-time variables
  with support for positive relative Euclidean times.
  

\vfill

\item [${\color{red}\bullet}$] The projection of $K^{m,s}_{e,r/l}({\color{red}\theta} x_e-y_e;\mu, \nu)$ on spinor functions with {\color{red}positive Euclidean time
support} is non-negative.  It can be completed to construct a {\color{red}new
Hilbert space representation}, ${\cal H}$

\vfill

\item [${\color{red}\bullet}$] The Euclidean kernels, 
$K^{m,s}_{e,r/l}(x_e-y_e;\mu, \nu)$, are {\color{red} reflection  positive}:

\[
\Pi_+ \Theta K \Pi_+ \geq 0 \qquad =\qquad \mbox{\bf reflection positivity}   
\]
  
}
\end{itemize} 
\end{frame}

\begin{frame}

\begin{center}
{\bf
On ${\cal H}$ the Poincar\'e generators ($s=0$ case) are self-adjoint:    
}  
\end{center}    
\[
{\color{red}
H\Psi (x_e)= {\partial \over \partial x^0_e}}  \Psi (x_e)
\]
\[
\mathbf{P}\Psi (x_e)= -i {\partial \over \partial \mathbf{x}_e }  \Psi (x_e)
\]
\[
\mathbf{J} \Psi (x_e)= -i \mathbf{x}
\times \pmb{\nabla}_x  \Psi (x_e)
\]
\[
K^j \Psi (x_e)={\color{red}( x^j {\partial \over \partial x^0_e}-
x^0_e {\partial \over \partial x^j})} \Psi (x_e).
\]

\begin{itemize}
{\bf 
\item [${\color{red}\bullet}$] In this representation the wave functions
involve Euclidean times, there is {\color{red}no analytic continuation!}
\vfill
\item [${\color{red}\bullet}$] Generators have the same form in dynamical
theories - only the Euclidean Green functions change.
}
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
{\bf
These generators are self-adjoint on ${\cal H}$.
}
\end{center}

\begin{itemize}
{\bf 
\item [${\color{red}\bullet}$] $\mathbf{P},\mathbf{J}$ generate one parameter
unitary groups.
}
\end{itemize}


\begin{center}
{\bf
Positive Euclidean time translations and rotations in Euclidean space-time
planes are used to construct the Hamiltonian and Lorentz boost generators 
}
\end{center}

\begin{itemize}
{\bf 
\item [${\color{red}\bullet}$] $H$ generates a contractive Hermitian
semigroup under $x_e^0$ translations. 
%\vfill

\item [${\color{red}\bullet}$] $\mathbf{K}$ generates a local
symmetric semigroup under rotations in $x_e^0,x^i$ planes.     
%\vfill

\item [${\color{red}\bullet}$] All have self-adjoint generators by
Stone's theorem and extensions.

}
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
{\bf
Domains for local symmetric semigroups
}
\end{center}
  
\begin{figure}
%\begin{minipage}[t]{.2\linewidth}
\centering
\includegraphics[angle=000,scale=.9]{landau.pdf}
%\caption{Full matrix, $\lambda$=0, mass=1}
%\label{fig:8}
%\end{minipage}
%\begin{minipage}[t]{.2\linewidth}
%\centering
%\includegraphics[angle=000,scale=.3]{doe.pdf}
%\caption{Full matrix, $\lambda$=0.2, mass=1}
%\label{fig:9}
%\end{minipage}
%\begin{minipage}[t]{.2\linewidth}
%\centering
%\includegraphics[angle=000,scale=.35            ]{u_logo.pdf}
%\caption{Full matrix, $\lambda$=0.2, mass=1}
%\label{fig:9}
%\end{minipage}
\end{figure}

\end{frame}  

\begin{frame}
\begin{center}
{\bf
Hilbert space inner product on ${\cal H}$:
}
\end{center}
\vfill
\[
\langle f \vert g \rangle := (f,\Theta K g) =
\]
\[
{1 \over \pi} \int f^*(x_e,\mu ) {D^s_{\mu \nu}[p_e \cdot \sigma_e]
\over p_e^2 + m^2} e^{i p_e \cdot (\theta x_e-y_e)} g(y_e,\nu)
d^4 p_e d^4x_e d^4y_e
\]
\vfill
\begin{center}
{\bf
Integrate over $p^0_e$ - use the support condition to close the contour in
the lower half plane
}
\end{center}
\vfill
\[
\langle f \vert g \rangle = 
\]
\vfill
\[
\int d^4x_e d^4y_e f^*(x_e,\mu) {e^{-\omega_m (\mathbf{p})(x_e^0+y^0_e) +
i \mathbf{p} \cdot (\mathbf{x}-\mathbf{y})}
\over
(2 \pi)^{3}}
{d\mathbf{p}\over \omega_m(\mathbf{p})}
D^{s}_{\mu \nu} [p\cdot \sigma ]
g(y_e,\nu) 
\]

\end{frame}  

\begin{frame}
\begin{center}
{\bf
Relation to Poincar\'e covariant and Lorentz
covariant representations
}
\end{center}

\[
\int f^*(x_e,\mu) K^{m,s}_{e,r}(\theta x-y;\mu \nu) g(y_e,\nu) d^4x_e d^4y_e
=
\]
\[
\int 
\underbrace{\left (\int d^4x_ef^*(x_e,\mu) e^{-\omega_m (\mathbf{p})x_e^0}
{e^{i \mathbf{p} \cdot \mathbf{x}}\over
  (2 \pi)^{3/2}} \right )}_{{\color{red}\langle \psi \vert (m,s) p, \mu \rangle_{cov}}}
\]\
\[
{d\mathbf{p}\over \omega_m(\mathbf{p})}
D^{s}_{\mu \nu} [p\cdot \sigma ]
\]
\[
\underbrace{\left (\int d^4y_e g(y_e,\nu) e^{-\omega_m (\mathbf{p})y_e^0}
{e^{-i \mathbf{p} \cdot \mathbf{y}}\over
(2 \pi)^{3/2}} \right )}_{{\color{red}_{cov}\langle (m,s) p, \nu \vert \phi \rangle }}
\]

\end{frame}

\begin{frame}
\begin{itemize}
{\bf
\item[] The right and left handed wave functions in Lorentz covariant,
Euclidean covariant and Poincar\'e covariant  
representations are related by:
}
\end{itemize}
    
\vfill

\[  
{\color{red}_{cov,r}\langle (m,s) p, \nu \vert \phi \rangle}_{\vert_{p^0=\omega_m(\mathbf{p})}} =
\]
\[
\int d^4y_e {\color{red}g(y_e,\nu)} e^{-\omega_m (\mathbf{p})y_e^0} 
{e^{-i \mathbf{p} \cdot \mathbf{y}}\over
 (2 \pi)^{3/2}} =
\]
\[
\sqrt{\omega_m (\mathbf{p})} D^s_{\mu \nu} [\Lambda_x^{-1}(p)]
{\color{red}_x\langle (m,s) p, \nu \vert \phi \rangle}
\]

\vfill
\[  
{\color{red}_{cov,l}\langle (m,s) p, \nu \vert \phi \rangle}_{\vert_{p^0=\omega_m(\mathbf{p})}} =
\]
\[
\int d^4y_e {\color{red}g(y_e,\nu)} e^{-\omega_m (\mathbf{p})y_e^0}
{e^{-i \mathbf{p} \cdot \mathbf{y}}\over
 (2 \pi)^{3/2}} =
\]
\[
\sqrt{\omega_m (\mathbf{p})} D^s_{\mu \nu} [\Lambda_x(p)]
{\color{red}_x\langle (m,s) p, \nu \vert \phi \rangle}
\]


\end{frame}

\begin{frame}
\[
 _x\langle (m,s) p, \nu \vert \phi \rangle, \qquad 
{}_{cov}\langle (m,s) p, \nu \vert \phi \rangle, \qquad \mbox{and} \qquad
g(y_e,\nu)
\]
\begin{itemize}
{\bf   
\item [${\color{red}\bullet}$] All represent the same physical state in different Hilbert space representations.

\vfill
\item [${\color{red}\bullet}$] These relations hold for free particles of irreducible basis states
of an interacting theory.
}
\end{itemize}
\end{frame}



\begin{frame}
\begin{center}
{\bf
Euclidean representation - properties
}
\end{center}

\begin{itemize}
{\bf 
\item [${\color{red}\bullet}$] Calculations of physical inner products
can be performed directly in the Euclidean representation {\color{red}without
analytic continuation}.
\vfill
\item [${\color{red}\bullet}$] Reflection positivity gives both
a positive norm and the spectral condition $H \geq 0$.  
\vfill
\item [${\color{red}\bullet}$] The representation of the
Poincar\'e generators are the same for dynamical or
non-interacting theories.  The dynamics enters in the {\color{red}Euclidean}
kernel.
\vfill 
\item [${\color{red}\bullet}$] For gauge theories the above considerations
only apply to gauge invariant states.  
\vfill
\item [${\color{red}\bullet}$] Because of the time reflection
in the kernel,
{\color{red} delta functions are normalizable} states on
${\cal H}$!
}
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
{\bf
Dynamics - models
}
\end{center}

\begin{center}
{\bf
General considerations
}
\end{center}

\begin{itemize}
  {\bf
  \item [${\color{red}\bullet}$] The general
    dynamical problem is to decompose the unitary representation of
    the Poincar\'e group into a direct integral of irreducible
    representations.

  \item [${\color{red}\bullet}$] The first step is to construct the dynamical
    unitary representation of the Poincar\'e group.

  \item
    [${\color{red}\bullet}$] The second step is to
    decompose $U(\Lambda,A)$ into a direct integral of representations
    labeled by
    mass, spin and degeneracy parameters.

%  \item
%    [${\color{red}\bullet}$] How these steps are carried out depends
%    on how $U(\Lambda,a)$ is represented.

  \item
    [${\color{red}\bullet}$] It is natural to  start with a
    model of non-interacting particles.  Interactions are then added
    that should the preserve positivity of the Hilbert space norm,
    the unitarity of the representation
    of Poincar\'e group, the spectral condition,  cluster properties. 
    and should allow scattering.
  }
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
{\bf
  Poincar\'e covariant dynamics  ($N=2$)\\
  (generalized Bakamjian-Thomas construction)
}    
\end{center}

\begin{itemize}
{\bf
\item[${\color{red}\bullet}$] Use Clebsch-Gordan
    coefficients of the Poincar\'e group
\[
\langle (m_1,s_1)b_1, (m_2,s_2) \vert (m_{12},s_{12})b_{12},d_{12} \rangle
\]
\item[] to decompose the
    two-body Hilbert space into a direct intergal of irreducible
    representations in the same basis ($b$).
\item[${\color{red}\bullet}$] Add interactions
    to the mass that (1) commute with
    $b$, (2) are independent of $b$ (3) commute with the free two-body
    spin and (4) preserve the spectral condition.

\item[${\color{red}\bullet}$] Diagonalize the interacting mass in
  the free irreducible representation.

\item[${\color{red}\bullet}$] Simultaneous
    eigenstates of interactng mass, free spin and free
    vector variables ($b$) transform irreducibly, defining a
    relativistic two-body dynamics
}
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
{\bf
Realistic model interactions = ($N=2$)
}    
\end{center}
\[
\vert (m_{12},s_{12})b_{12},d_{12} \rangle \to
\vert (k,j)\mathbf{p},\mu; l,s \rangle
\]
\[
\langle (k',j')\mathbf{p}',\mu'; l',s' \vert V_{nn}
\vert (k,j)\mathbf{p},\mu; l,s \rangle =
\]
\[
\delta (\mathbf{p}'-\mathbf{p})\delta_{j'j}\delta_{\mu'\mu}
\langle k',l',s' \Vert V^j_{nn} \Vert k,l,s \rangle
\]
\[
  M = \sqrt{m_1^2 +2m_r ({\color{red}k^2/2m_r + V_{nn}})} +
  \sqrt{m_2^2 +2m_r({\color{red}k^2/2m_r + V_{nn}})}
\]
\begin{itemize}
{\bf
\item[${\color{red}\bullet}$] Eigenstates of $H_{nr}$ are eigenstates of $M$.
\vfill
\item[${\color{red}\bullet}$] Relativistic $S$-matrix fits same 2-body CM data
 as $NR$ 2-body $S$-matrix (invariance principle)! 
\[
S_{exp}= S_{nr} = \lim_{t \to \infty} e^{iH_0t}e^{2i H_{nr}}e^{iH_0t} =
\lim_{t \to \infty} e^{iM_0t}e^{2i Mt}e^{iM_0t} = S_r
\]
\item[${\color{red}\bullet}$] Construction violates cluster properties for $N\geq 3$.
}
\end{itemize}
\end{frame}


 
\begin{frame}
\begin{center}
{\bf
Poincar\'e covariant dynamics (Sokolov construction) \\
($2 \to 3$ - cluster properties - fixed up to 3BF) 
}    
\end{center}

\begin{itemize}
{\bf  
\vfill    
\item
[${\color{red}\bullet}$] N=3 Take tensor products of N=2 with N=1.
    Use unitary scattering equivalence to transform to representations
    with free three-body spin.

\vfill        
\item [${\color{red}\bullet}$] Add the transformed 2+1 interactions to the
  free three-body mass.  Use Poincar\'e group Racah coefficients to put is
  a common basis.

\vfill    
\item [${\color{red}\bullet}$] Multiply the result by a symmetrized product of the inverse of the 2+1 unitary scattering equivalences.

\vfill      
\item[${\color{red}\bullet}$] This construction gives a dynamical unitary
representation of the Poincar\'e group satisfying cluster properties,
spectral condition, and fits experimental two-body data.


}
\end{itemize}

\end{frame} 

\begin{frame}

\begin{center}
{\bf
Sokolov construction ($N=3$ example)
}    
\end{center}
  
\[  
H =
\]
\[
e^{\sum \ln A^{\dagger}_{ij,k}}
\left (\sum(A_{ij,k} H_{ij}\otimes H_k
A_{ij,k}^{\dagger}- 2H_1 \otimes H_2 \otimes H_3 +V_{123} \right ) e^{\sum \ln A_{ij,k}} 
\]
\begin{itemize}
{\bf
\item[${\color{red}\bullet}$] Method preserves Poincar\'e invariance, cluster properties and
spectral condition (for suitable interactions).
\vfill
\item[${\color{red}\bullet}$] Preserves kinematic subgroup (for suitable interactions).
\vfill
\item[${\color{red}\bullet}$] $A_{ij,k}$ generates frame-dependent many interactions.
\vfill
\item[${\color{red}\bullet}$] Resulting spin is dynamical.
\vfill
\item[${\color{red}\bullet}$] Recursive construction is messy,
result depends on choice of $A_{ij,k}$, connection with QFT not direct.
  
}
\end{itemize}

\end{frame}  

\begin{frame}
\begin{center}
{\bf
Lorentz covariant constraint dynamics (method I)
}    
\end{center}

\begin{center}
{\bf
Add interactions to Wightman functions
}    
\end{center}


\begin{itemize}
{\bf

\item[${\color{red}\bullet}$] Treat $C_i:= p_i^2+m_i^2 =0, \quad p_i^0 >0$
as first class constraints.
\[
K = \prod_i \delta (C_i)
\]  
\vfill
\item[${\color{red}\bullet}$] Add Lorentz covariant
interactions $C_i \to C_i'=C_i + \sum_j V_{ij}$ so
$C_i'$ are compatible constraints (first class)
\[
[C'_i,C'_j] = f_{ijk} C'_k \qquad \Rightarrow \qquad
K'= \Pi \delta (C_i')
\]
\item[${\color{red}\bullet}$] Two-body interactions satisfying first class condition:
\[
[V_{ij} ,p_i^2-p_j^2 ]=0  
\]

\item[${\color{red}\bullet}$] $K'$ are model Wightman functions - normally
imposed by solving coupled Dirac or Klein Gordon equations.  The first class
condition is an integrability condition.

\vfill

\item[${\color{red}\bullet}$] First class constraints satisfying
cluster properties for $N>2$ not known.

}
\end{itemize}
  
\end{frame} 



\begin{frame}
\begin{center}
{\bf
Lorentz covaraint Schwinger-Dyson equations (method II) 
}    
\end{center}

\begin{itemize}
{
\bf
\item[${\color{red}\bullet}$] Solutions are matrix elements of 
time-ordered products of fields.

\vfill

\item[${\color{red}\bullet}$] Key requirement - $\exists$ complete sets
of {\color{red}positive energy intermediate states} (plus vacuum) bewteen all
operators and states.

\vfill
  
\item[${\color{red}\bullet}$] This is where the direct integral of irreducible
representations appears.  

\item[${\color{red}\bullet}$] Using the integral representation of the
Heaviside function with the direct integral leads to poles associated
with the energies of the intermediate states:
\[
\theta (t) = {1 \over 2 \pi i} \int {e^{ist} \over s-i \epsilon}  
\]
}
\end{itemize}
\end{frame}

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

\item[${\color{red}\bullet}$] Residues of the (double)
poles have the general strufture
\[
\langle 0 \vert T(\Pi\phi_i(x_i))\vert (m,s)b \rangle 
{\color{red}\langle (m,s) b \vert O \vert (m',s')b' \rangle}  
\langle  (m',s')b' \vert T(\Pi \phi_j(x_j) ) \vert 0 \rangle
\]

\item[${\color{red}\bullet}$] The matrix elements of any operator
$\langle (m,s) b \vert O \vert (m',s')b' \rangle$ in irreducible
eigenstates can be extracted
using a normalization condition to eliminate the ``wave functions''
$\langle  (m,s)b \vert T(\Pi\phi_j(x_j) ) \vert 0 \rangle$.
\vfill
\item[${\color{red}\bullet}$] Equations are non-linear - but the solutions
can be expressed as moments of a path integral.
\vfill
\item[${\color{red}\bullet}$] The connection with the general requirements of
relativistic quantum theory is not direct.  This makes it difficult to
determine what properties are preserved under truncation.
\vfill
\item[${\color{red}\bullet}$] The method has a
natural connection with the $S$ matrix.  Bogoliubov's S-matrix
axioms may be easier to check.

}
\end{itemize}
\end{frame}


\begin{frame}
\begin{center}
{\bf
Euclidean covaraint dynamics
}    
\end{center}

\begin{itemize}
{\bf
\item[${\color{red}\bullet}$] Direct connection with Lagrangian
field theory.  The Euclidean kernels (before time reflection) are
moments of a Euclidean path integral.
\vfill

\item[${\color{red}\bullet}$] They satisfy non-linear
Euclidean Schwinger-Dyson equations.
\vfill

\item[${\color{red}\bullet}$] The connection with a relativistic
quantum field theory is more straightforward in the Euclidean case -
reflection positivity, Euclidean covariance and cluster properties
are the requirements for a relativistic quantum theory.
\vfill

\item[${\color{red}\bullet}$] For gauge theories reflection positivity
is only needed for gauge invariant intermediate states.
%\item[${\color{red}\bullet}$]

}
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
{\bf
Things to think about
}    
\end{center}
\begin{itemize}

{\bf
\item[${\color{red}\bullet}$] The Euclidean representation has a
{\color{red}Minkowski interpretation} provided we use the Euclidean time
reflection operator and Euclidean states with positive relative
Euclidean time support.

\vfill

\item[${\color{red}\bullet}$] {\color{red}Analytic continuation is not needed}, and
there are explicit representations of all 10 Poincar\'e generators
in the Euclidean representaion.  These {\color{red}generators are self adjoint}.

\vfill

\item[${\color{red}\bullet}$] Four dimensional {\color{red}delta functions are
square integrable} in the Euclidean representation - this may facilitate
numerical calculations.

\vfill

\item[${\color{red}\bullet}$] Locality logically independent of the other
Euclidean Axioms.
}
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}

{\bf 
\item[${\color{red}\bullet}$] The requirements of reflection positivity
 need to be better understood - for example what are sufficient
 conditions on Euclidean Bethe-Salpeter kernels for
 G to be reflection positive 
 \vfill
\[
G^{-1} = \prod {G_i}^{-1} - \sum K_{ij} - \sum K_{ijk}  
\]
\vfill 
\item[${\color{red}\bullet}$] Conditions on $K_{ij}$, $K_{ijk}$ so $G$ is reflection positive?
 
}
\end{itemize}
\end{frame}

\begin{frame}

\begin{itemize}
{\bf


\item[${\color{red}\bullet}$] By giving up locality it is possible to
have different N-point Green functions (i.e. $ G(1:3), G(2:2), G(3:1)$)
corresponding to different numbers of initial and final particles).
This makes the reflection positivity constraint easier. In a local
theory these are related by analytic continuation.

\item[${\color{red}\bullet}$] Haag-Ruelle scattering can be directly
formulated in the Euclidean representation.

\item[${\color{red}\bullet}$] Light front generators are given by
complex differential operators,
$p^+ := {\partial \over \partial x^0_e} -i {\partial \over \partial x^3}$
\item[] these should be self adjoint on ${\cal H}$. 
}
\end{itemize}
\end{frame}


\begin{frame}
\vfill  
\begin{center}
{\bf Thanks to the organizers!}
\end{center}
\vfill
\end{frame}
  
\end{document}  
\begin{frame}

\begin{itemize}
{\bf

\item[${\color{red}\bullet}$] Key question - how to define robust class of
pertrubations of free Euclidean green functions that preserve (1) Euclidean
covariance (2) cluster properties and (3) reflection positivity. (1) and (2) are
easy. 
  
\item[${\color{red}\bullet}$] By using reflection positivity
a relativistic quantum theory can be defined in the Euclidean
representation without explicit analytic continuation.

\item[${\color{red}\bullet}$] In coordinate space 4-dimensional
delta functions are square integrable.


\item[${\color{red}\bullet}$] Locality can be relaxed without
violating any of the other axioms of quantum field theory.


}
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
{

\item[${\color{red}\bullet}$] Direct formulation of scattering theory
in the Euclidean representation.
\vfill  
\item[${\color{red}\bullet}$] Inner product, Poincar\'e generators, cluster
properties - all that is needed to compute strong limits for scattering.




\[
\langle \psi_f \vert S \vert \psi_i \rangle =\lim_{t \to \infty}
\langle \psi_f \vert e^{iH_\alpha t} \Phi_\alpha^{\dagger}e^{-2iHt}\Phi_\beta e^{iH_\beta t} \vert \psi_i \rangle =
\]
\[
\lim_{t \to \infty}
\langle \psi_f \vert e^{if(H_\alpha)t} \Phi_\alpha^{\dagger}
e^{-2if(H)t}\Phi_\beta e^{if(H_\beta)t} \vert \psi_i \rangle  
\]
\[
f(H) = - e^{-a H} \qquad \sigma( e^{-a H})\in [0,1] 
\]
\[
  P(e^{-a H}) = \sum c_n e^{-n a H}  \to e^{if(H )t} \qquad \mbox{\bf uniformly}    \]

\item[${\color{red}\bullet}$] $\Phi_\alpha$ Products of asymptotic one-particle
creation operators  (Haag-Ruelle quasi-local fields).

}
\end{itemize}
\end{frame}


\begin{frame}
\begin{center}
{\bf
Remarks
}    
\end{center}
\begin{itemize}
{\bf
\item[${\color{red}\bullet}$] All methods for including dynamics
involve phenomenological input that cannot be controlled mathematically.

\vfill

\item[${\color{red}\bullet}$] Results must satisfy reflection positivity
(cluster properties and covariance).

\vfill

\item[${\color{red}\bullet}$] Locality is not required - by giving
up locality there can be different N-point kernels for different
$K+M=N$ reactions.  Leads to a simpler version of reflection positivity.

}
\end{itemize}  
\end{frame}
\end{document}
\begin{itemize}
{\bf   

\item [${\color{red}\bullet}$] Replacing $K^{m,s}_{e,r/l}(x-y;\mu, \nu)$ with
$K^{m,s}_{e,r/l}(\theta x-y;\mu, \nu)$ where $\theta$ is Euclidean
time reflection gives a kernel where the complex subgroup of the
orthogonal group corresponding to real Lorentz transformation
is ``norm'' preserving.

\item[${\color{red}\bullet}$] The problem is that this ``norm'' is not positive. 
\item[${\color{red}\bullet}$] Projection on the subspace of
functions with support for positive
Euclidean time gives a non-negative kernel.
  
\[
\int f^*(x_e,\mu) K^{m,s}_{e,r}(\theta x-y;\mu \nu) g(y_e,\nu) d^4x_e d^4y_e
=
\]
\[
\int 
\left (\int d^4x_ef^*(x_e,\mu) e^{-\omega_m (\mathbf{p})x_e^0}
{e^{i \mathbf{p} \cdot \mathbf{x}}\over
  (2 \pi)^{3/2}} \right )
\]\
\[
{d\mathbf{p}\over \omega_m(\mathbf{p})}
D^{s}_{\mu \nu} [p\cdot \sigma ]
\]
\[
\left (\int d^4y_e g(y_e,\mu) e^{-\omega_m (\mathbf{p})y_e^0}
{e^{-i \mathbf{p} \cdot \mathbf{y}}\over
(2 \pi)^{3/2}} \right )
\]

\item[${\color{red}\bullet}$] Which is the real Lorentz covariant
inner product.  

}
\end{itemize} 
\end{frame}




\end{document}



\begin{frame}
\begin{center}
{\bf
Hilbert space representations: ${\cal H}= L^2(m,s,b,d)$     
}
\end{center}

\begin{itemize}
{\bf 
\item[${\color{red}\bullet}$] $\{ (m,s;d) \mathbf{p},\hat{s}_c \cdot \hat{\mathbf{n}}\}$ \qquad (instant form)
  
\item[${\color{red}\bullet}$] $\{ (m,s;d) \mathbf{v},\hat{s}_c \cdot
  \hat{\mathbf{n}}\}$ \qquad (point form)  
  
\item[${\color{red}\bullet}$] $\{ (m,s;d) \mathbf{p},\hat{s}_h \cdot 
\hat{\mathbf{z}}\}$ \qquad (Jacob Wick helicity )

\item[${\color{red}\bullet}$] $\{ (m,s;d) p^+,\mathbf{p}_{\perp},\hat{s}_f
\cdot \hat{\mathbf{n}}\}$ \qquad (light front)

\item[${\color{red}\bullet}$] $\vdots$

}
\end{itemize}
\end{frame}
\end{document}
%slide 5 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Hilbert space representations: ${\cal H}= L^2(m,s,b,d)$     
}
\end{center}

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




%slide 5 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
\end{document}
\begin{frame}
\begin{center}
{\bf
Contributors
}
\end{center}  

\begin{itemize}
{\bf
\item [] Victor Wessels
\vfill
\item [] Philip Kopp
\vfill
\item [] Gordon Aiello
\vfill
\item [] Gohin Samad
}    
\end{itemize}

\begin{center}
{\bf
Useful discussions with
}
\end{center}  

\begin{itemize}
{\bf
\item [] F. Coester
\vfill
\item []  P. J{\o}rgensen
}
\end{itemize}
\end{document}

\end{frame}

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

\begin{itemize}
{\bf
\item [${\color{red}\bullet}$] Motivation
\vfill
\item [${\color{red}\bullet}$] Construction of a relativistic
quantum theory
\vfill  
\item [${\color{red}\bullet}$] Haag-Ruelle scattering
\vfill
\item [${\color{red}\bullet}$] One body solutions - Stiltjes moment problem  
\vfill
\item [${\color{red}\bullet}$] Computational tricks
\vfill
\item [${\color{red}\bullet}$] Model tests
\vfill  
\item [${\color{red}\bullet}$] Unfinished business  
}  
\end{itemize}  
  
\end{frame}



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

\begin{itemize}
{\bf
\item [${\color{red}\bullet}$] Goal: Formulate relativistic quantum
models satisfying cluster properties.

\item [${\color{red}\bullet}$] Observation: locality is logically
independent of the rest of the Euclidean axioms of QFT.  This allows
for the construction of a relativistic quantum theory.

\item [${\color{red}\bullet}$] Reconstruction can be done without
analytic continuation. Relativistic quantum mechanics
can be formulated entirely in Euclidean space,

\item [${\color{red}\bullet}$] Given a Hilbert space representation,
a Hamiltonian satisfying cluster properties and particles - it should
be possible to formulate scattering directly in a Euclidean representation.

\item [${\color{red}\bullet}$] Euclidean phenomenology is related to
Lagrangian dynamics.
}
\end{itemize}
\end{frame}

\end{document}

\begin{frame}

\begin{center}{\bf
Elements of relativistic quantum mechanics}
\end{center}
\[
\langle \psi \vert  \phi \rangle  \qquad \mbox{\bf Hilbert space}
\]
\[
U(\Lambda ,a) \to \{P^{\mu}, J^{\mu\nu}\} \qquad \mbox{\bf Relativity} 
\]
\[
P^0= H \qquad \mbox{\bf Dynamics}
\]
\[
P^0 = H \geq 0 \qquad \mbox{\bf Spectral condition} \quad \to \quad \mbox{\bf stability} 
\]
\[
[U(\Lambda,a)-\otimes U_i(\Lambda ,a)]\vert \psi \rangle \to 0
\qquad (x_i-x_j)^2 \to \infty
\]
\begin{center} {\bf
Cluster properties: scattering asymptotic conditions}
\end{center}

\end{frame}

\begin{frame}

\begin{center}
{\bf
Osterwalder-Schrader (Euclidean) reconstruction
}
\end{center}

\[
\mbox{\bf Input:} \qquad  
\{ G_{en} (\x_1 \cdots \x_n) \}
\]

\begin{center}
{\bf
Properties
}
\end{center}

\begin{itemize}
{\bf
\item [${\color{red}\bullet}$] Distributions
\vfill
\item [${\color{red}\bullet}$] Euclidean covariant (invariant) 
\vfill
\item [${\color{red}\bullet}$] Cluster property
\vfill
\item [${\color{red}\bullet}$] Reflection positivity
}
\end{itemize}

\end{frame}

%\begin{frame}
%\begin{itemize}
%{\bf 
%\item [${\color{red}\bullet}$] Gives all elements of a relativistic quantum theory.
%\vfill  
%\item [${\color{red}\bullet}$] Analytic continuation not needed.
%\vfill  
%\item [${\color{red}\bullet}$] Above properties logically independent of locality.  
%}
%\end{itemize}  
%\end{frame}

\begin{frame}
\begin{center}{\bf 
Hilbert space ${\cal H}_m$
}
\end{center}

\begin{center}
{\bf  
Vectors (dense set) 
}
\end{center}
\[
\psi (x)  :=  ( \psi_1 (\x_{11}), \psi_2 (\x_{21},\x_{22}),\cdots )
\]
\[
\psi_n (\x_{n1}, \x_{n2},\cdots , \x_{nn}) =0
\qquad
\mbox{\bf unless} 
\qquad
0 < \x_{n1}^0 <  \x_{n2}^0  <\cdots <  \x^0_{nn} .
\]
\[
\theta x := \theta (\tau,\mathbf{x})= (-\tau,\mathbf{x}) \qquad 
\mbox{\bf Euclidean time reflection}
\]

\begin{center}
{\bf  
Physical Hilbert space inner product 
}
\end{center}

\[
\langle \psi \vert \phi \rangle_m = (\theta \psi ,G_e \phi)_e =
\sum_{kn} \int d^{4k}x d^{4n}y
\psi_n^* ( \theta \x_{n1}, \theta \x_{n2},\cdots ,\theta \x_{nn})
\times
\]
\[
G_{e,n+k}
( \x_{kk}, \cdots , \x_{1k};\y_{1n} , \cdots , \y_{nn})
\phi_k (\y_{k1}, \y_{k2},\cdots , \y_{kk}) 
\]

\begin{center}
{\bf  
All variables Euclidean!
}
\end{center}


\end{frame}

\begin{frame}
\begin{center}
{\bf     
Reflection positivity - property of $\{ G_{en} \}$
}
\end{center}
 
\[
\langle \psi \vert \psi \rangle_m = (\psi , \Pi_{+>} \Theta  G_e \Pi_{+>} \psi )_e \geq 0
\]

\begin{center}
{\bf  
No free lunch - most challenging aspect of model construction
}
\end{center}


\begin{center}
{\bf 
Euclidean invariance 
}
\end{center}
\[
G_{en} (\x_1 \cdots \x_n) =
G_{en} (O\x_1+\a, \cdots O\x_n+\a) 
\]
\begin{center}
{\bf 
Euclidean covariance 
}
\end{center}
\[
G_{en} (x_1 \cdots x_n) =
S_{en}(0^{-1})G_{en} (Ox_1+a, \cdots Ox_n+a) 
\]
\end{frame}

\begin{frame}
\begin{center}
{\bf 
Euclidean invariance $\to$ Poincar\'e invariance
}
\end{center}
  
\begin{center}
{\bf     
Relativity and $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$
}
\end{center}

\[
X_m := \left (
\begin{array}{cc}
t+ z & x-iy\\
x+iy & t-z
\end{array}
\right )
\qquad
X_e := 
\left (
\begin{array}{cc}
i\tau + z & x-iy\\
x+iy & i\tau-z
\end{array}
\right )
\]
\[
\mbox{\bf det} (X_m) = t^2 -\mathbf{x}^2 \qquad 
\mbox{\bf det} (X_e) = - (\tau^2 +\mathbf{x}^2)
\]
\[
X \to X' = A X B^t \qquad \mbox{\bf det}(A) = \mbox{\bf det} (B)=1
\]
\begin{center}{\bf 
Complex Lorentz group $=$ complex orthogonal group
}
\end{center}

\begin{center}{\bf 
Real orthogonal group $=$ subgroup of complex Lorentz group
}
\end{center}

\begin{center}
{\bf   
  Real Lorentz $(A,B)=(A,A^*)$;\\
  Real orthogonal $(A,B)\in SU(2)\times SU(2)$
}
\end{center}
\end{frame}

\begin{frame}
\begin{itemize}
{\bf 
\item[${\color{red}\bullet}$] The relation between the real orthogonal
group and a subgroup of complex Lorentz group relates the
infinitesimal generators of both groups.

\item[${\color{red}\bullet}$] Poincar\'e commutation relations follow from
the Euclidean commutation relations and the resulting relation between
the generators of real
Poincar\'e and Euclidean transformations.

\item[${\color{red}\bullet}$] Time reflection in the physical Hilbert space ${\cal H}_m$ makes
Poincar\'e generators Hermitian.

\item[${\color{red}\bullet}$] One-parameter group/semigroup properties
  of real Euclidean transformations on ${\cal H}_m$  lead to self-adjoint
  Poincar\'e generators.  
}
\end{itemize}  
\end{frame}

\begin{frame}
\begin{center}
{\bf 
Relation between Euclidean and Poincar\'e generators
}
\end{center}
\begin{itemize}
{\bf
\item[${\color{red}\bullet}$] Euclidean time translations $\to$ contractive Hermitian semigroup
on ${\cal H}_m$.
\[
H_e = P^0_e = - i H_m = -i P^0_m   
\]  
\item[${\color{red}\bullet}$] Euclidean space-time rotations $\to$ local symmetric semigroup   
on ${\cal H}_m$.
\[
J^{0j}_{e} = i J^{0j}_{m}= i K^j
\]  
\item[${\color{red}\bullet}$] Euclidean space rotations $\to$ unitary one parameter groups
\[
J_{e}^{ij} = J_{m}^{ij}
\]  
\item[${\color{red}\bullet}$] Euclidean space translations $\to$ unitary one parameter groups  
\[
P_{e}^i = P_{m}^i
\]  
\item[ ] $\{ P^{\mu}_m ,J^{\mu \nu}_m\} = $  10 self-adjoint generators
satisfying Poincar\'e CR on ${\cal H}_m$
}
\end{itemize}

\end{frame}

\begin{frame}
\begin{center}{\bf
Spinless case
}   
\end{center}   
\[
H \psi_n (\x_{n1}, \x_{n2},\cdots , \x_{nn})
=
\sum_{k=1}^n {\partial \over \partial \x_{nk}^0} \psi_n (\x_{n1}, \x_{n2},\cdots , \x_{nn})
\]
\[
\mathbf{P}  \psi_n (\x_{n1},\x_{n2},\cdots , \x_{nn})
=
-i \sum_{k=1}^n {\partial \over \partial \mathbf{x}_{nk}} \psi_n (\x_{n1}, \x_{n2},\cdots ,\x_{nn})
\]
\[
\mathbf{J} \psi_n ( \x_{n1}, \x_{n2},\cdots ,
\x_{nn}) = -i \sum_{k=1}^n \mathbf{x}_{nk} \times {\partial \over
\partial \mathbf{x}_{nk}} \psi_n (\x_{n1},
\x_{n2},\cdots ,\x_{nn})
\]
\[
\mathbf{K} \psi_n ( \x_{n1}, \x_{n2},\cdots ,
\x_{nn}) = \sum_{k=1}^n (\mathbf{x}_{nk} {\partial \over
\partial {x}^0_{nk}}-  {x}^0_{nk}  {\partial \over
\partial \mathbf{x}_{nk}}) \psi_n ( \x_{n1},
\x_{n2},\cdots , \x_{nn}).
\]
\begin{center}{\bf
All integration variables Euclidean; Minkowski time is a parameter
}
\end{center}  
\end{frame}

\begin{frame}

\begin{center}
{\bf
Cluster properties
}
\end{center}

\[
G_{n+m} \to G_n G_m \to
\]

\begin{center}
{\bf
Generators become addidive a asymptotically separated subsystems
}
\end{center}

\[
(\chi_n \chi_m, \theta G_{n+m}H\psi_m\psi_n)_e  \to
\]
\[
(\chi_n \theta, G_nH\psi_n)_e(\chi_m,\theta  G_m\psi_m)_e  +
(\chi_n \theta, G_n\psi_n)_e(\chi_m, \theta  G_mH\psi_m)_e
\]


\end{frame}  
  

\begin{frame}
\begin{center}{\bf 
Two point Green function: Euclidean $\to$ Minkowski
}
\end{center}

\[
\langle \phi \vert \psi\rangle_m =   
\int 
\phi^* (-\tau_x, \mathbf{x})
{\color{red} {d^4p dm \over (2\pi)^4}{e^{i p \cdot (x-y)} \over p^2 + m^2}}
\psi (\tau_y, \mathbf{y})   
d^4 x d^4y  =
\]
\[ 
\int \xi_m(\mathbf{p})^*
{d\mathbf{p} dm  \over 2\omega_m (\mathbf{p})}
\chi_m(\mathbf{p}) 
\]

\begin{center}{\bf 
Euclidean wave function $\to$ Minkowski wave function
}
\end{center}

\[
\chi(\mathbf{p}) := \int {d^4y \over (2\pi)^{3/2}}
e^{ - \omega_m (\mathbf{p})\tau_y -i \mathbf{p}\cdot \mathbf{y}}
\psi_m (\tau_y, \mathbf{y})
\]
\[
\xi_m(\mathbf{p}) := \int {d^4x \over (2\pi)^{3/2}}
e^{ - \omega_m (\mathbf{p})\tau_x -i \mathbf{p}\cdot \mathbf{x}}
\phi_m (\tau_x, \mathbf{x})
\]

\end{frame}

\begin{frame}
\begin{center}{\bf 
Mass $m$ spin $j$ irreducible representations of PG (particles) 
}
\end{center}

\begin{center}{\bf
Right handed
}
\end{center}
 

\[
\langle \phi \vert \psi\rangle_m   =
\]
\[
\int 
\phi^* (-\tau_x, \mathbf{x},
\mu ){\color{red} {e^{i p \cdot (x-y)}
D^j_{\mu \nu}(p\cdot \sigma_e/m)  \over p^2 + m^2}}
{{\color{red}d^4p} d^4 x d^4y  \over {\color{red}(2\pi)^4}} \psi (\tau_y, \mathbf{y},\nu)  =
\]
\[ 
\int \xi(\mathbf{p},\mu)^*
{d\mathbf{p} dm D^j_{\mu \nu}(p_m\cdot \sigma/m) \over 2 \omega_m (\mathbf{p})}
\chi(\mathbf{p},\nu) 
\]

\begin{center}{\bf
Left handed
}
\end{center}

\[
\langle \phi \vert \psi\rangle_m =
\]
\[
\int 
\phi^* (-\tau_x, \mathbf{x},\mu ) {\color{red}{e^{i p \cdot (x-y)}
D^j_{\mu \nu}(Pp\cdot \sigma_e/m)  \over p^2 + m^2}}
{{\color{red} d^4p} d^4 x d^4y \over {\color{red}(2\pi)^4}}
\psi (\tau_y, \mathbf{y},\nu )   
  =
\]
\[ 
\int \xi(\mathbf{p},\mu)^*
{d\mathbf{p}  D^j_{\mu \nu}(P p_m\cdot \sigma/m) \over 2\omega_m (\mathbf{p})}
\chi(\mathbf{p},\nu) 
\]

\begin{center}{\bf
$P$ space reflection; Both L and R reflection positive
}
\end{center}

\end{frame}

\begin{frame}
\begin{center}{\bf 
Elements of scattering theory - 2 Hilbert space formulation
}
\end{center}

\[
\mbox{\bf Scattering probability} =
\vert \langle \psi_+ \vert \psi_- \rangle\vert^2   
\]
\[
\vert \psi_{\pm} \rangle = \Omega_{\pm} \vert \psi_{0\pm} \rangle
\]
\[
\Omega_{\pm} \vert \psi_{0\pm} \rangle =
\lim_{t \to \pm \infty}\sum
e^{iHt}\underbrace{\prod_n\vert \phi_n, \mathbf{p}_n, \mu_n \rangle}_{J}
\underbrace{e^{-ie_nt}}_{e^{-iH_0 t}} \underbrace{f_n(\mathbf{p}_n,\mu_n)}_{\vert \psi_0\pm\rangle} d\mathbf{p}_n  =
\]
\[
\lim_{t \to \pm \infty} e^{iHt}J e^{-iH_0t} \vert \psi_0\pm\rangle
\]


\begin{center}
{\bf
Elements: Cluster properties, subsystem bound states: $\vert \phi_n\rangle$,
wave packets: $f_n$, dynamics: $H$, strong limits.
}
\end{center}
\end{frame}

\begin{frame}
\begin{center}{\bf 
Field theoretic implementation:  Haag-Ruelle scattering
}
\end{center}

\begin{center}{\bf 
Subsystem bound states (Minkowski case)
}
\end{center}

\begin{itemize}
{\bf   
\item[${\color{red}\bullet}$] Construct interpolating field.

\item[${\color{red}\bullet}$] Fourier transform interpolating field.

\item[${\color{red}\bullet}$] Multiply the Fourier transform by function of
mass that is 1 on the physical mass and vanishes on the rest of the
mass spectrum.

\item[${\color{red}\bullet}$] Inverse Fourier the product.

\item[${\color{red}\bullet}$] Select the creation operator part of the resulting operator.  
}  
\end{itemize}  
\end{frame}

\begin{frame}

\[
\Phi (x)  = \mbox{\bf interpolating field}
\]
\[
\tilde{\Phi} (p) = {1 \over (2 \pi)^2}\int e^{-i p \cdot x} \Phi (x) d^4 x
\]
\[
\tilde{\Phi}_m (p) =  h(p^2) \tilde{\Phi} (p), \,
h(-m^2)=1, \,  h(p^2)=0, \,  -p^2\not\in
(m^2-\epsilon, m^2 + \epsilon)   
\]
\[
\Phi_m (x) = {1 \over (2 \pi)^2}\int e^{i p \cdot x} \tilde{\Phi}_m (p) d^4 p
\]
\[
f_m (x) = {i \over {(2 \pi)}^{3/2}}\int e^{-i \sqrt{\mathbf{p}^2 + m^2}t+i
\mathbf{p}\cdot \mathbf{x}} \tilde{f}(\mathbf{p}) d\mathbf{p}  
\]
\[
A_m^{\dagger}(f_m,t) =
- i \int d\mathbf{x} 
\left (
{\partial \Phi_m (t,\mathbf{x})\over \partial t}
f (t,\mathbf{x})-
\Phi_m (t,\mathbf{x})
{\partial f (t,\mathbf{x}) \over \partial t}-
\right )
\]
%\[
%A^{\dagger} \vert f \rangle =
%U(I,-t)(H \Phi_m - \Phi_m H_m) U_m(I,t)\vert f \rangle  
%\]
%\begin{center}
%Asymptotically this looks like a creation operator for a particle of mass $m$,
%but it is time dependent (Ruelle) 
%\end{center}
\[
\Omega_{\pm} \vert \psi_{0\pm} \rangle =
s-\lim_{t \to \pm \infty}
\Pi_i A_{m_i}^{\dagger}(f_{m_i},t)\vert 0 \rangle
\]
\end{frame}

\begin{frame}
\begin{center}{\bf 
Euclidean formulation - Technical issues
}
\end{center}

\begin{itemize}
{\bf  
\item[${\color{red}\bullet}$] One-body solutions must satisfy the
time support condition.

\item[${\color{red}\bullet}$] Products of one-body solutions must
satisfy the relative time support condition ($n=2$, no spin) .
}
\end{itemize}
\[
J :
\langle \x_1 \vert  \phi_1, \mathbf{p}_1 \rangle 
\langle \x_2 \vert  \phi_2, \mathbf{p}_2 \rangle =
\]
\[
h_1 (\nabla_1^2)\delta (\x_1^0 - \tau_1 )
h_2 (\nabla_2^2)\delta (\x_2^0 - \tau_2 )
{1 \over (2 \pi)^3} e^{i \mathbf{p}_1\cdot  \mathbf{x}_1
+ i \mathbf{p}_2\cdot  \mathbf{x}_2}
\]

\[
h_i(\lambda_i^2)=1 \qquad  h_i(\lambda_i^2)=0 \qquad  m^2\not\in
(\lambda_i^2-\epsilon_i, \lambda_i^2 + \epsilon_i)
\qquad M^2 = \nabla_e^2
\]

\begin{itemize}
{\bf  
\item[${\color{red}\bullet}$] Integration by parts with two-point Green
function - $h_i(\lambda_i^2)\to h_i(m^2)$
}
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
{\bf     
\item[${\color{red}\bullet}$] Delta functions are square integrable in ${\cal H}_m$!
\vfill
\item[${\color{red}\bullet}$] A sufficient condition for $ h_i (\nabla^2)$ to preserve the support
  condition is for polynomials in $\nabla^2$ to be complete with
  respect to the inner product on ${\cal H}_m$
\vfill
\item[${\color{red}\bullet}$] The above representation can be used to satisfy the support condition
  and control the shape of the momentum dependence in the free initial
  and final wave packets

}  
\end{itemize}
\end{frame}

\begin{frame}
\begin{center}
{\bf     
Technical observations 
}
\end{center}
{\bf
\begin{center}  
Widder's theorem, reflection positivity and the Lehmann representation
\end{center}
}
\[
\int f(\theta s) k(s-t) f(t) ds dt =
\int f(s) k(-s-t) f(t) ds dt
\geq 0
\]
\[
\Downarrow
\]  
\[
k(-\tau'-\tau) = \int e^{- \lambda (\tau'+\tau) } \rho (\lambda ) d\lambda
\]
\[
\Downarrow
\]  
\[
k(-\tau'-\tau) =  \int_0^{\infty} {\lambda \over \pi} 
\rho (\lambda ) d\lambda \int_{-\infty}^\infty ds  
{e^{-is ( \tau'+\tau) }\over s^2 + \lambda^2 }.  
\]
\end{frame}

\begin{frame}
\begin{center}
{\bf   
Reflection positivity constrains the structure of $G_{en}$
}
\end{center}

\[
 G_{2e} (x) =
{1 \over (2\pi)^4} \int {d^4 p \over p^2 + m^2}  \rho(m) dm
e^{i p \cdot (x-y)} = 
\]  
\[
{1 \over 4 \pi^2 } \int {m \over \sqrt{ x_0^2 + \mathbf{x}^2}}
K_1 ({m \over \sqrt{ x_0^2 + \mathbf{x}^2}}) \rho(m) dm
\]

\begin{center}
{\bf   
Singular at $x_e=0$ but falls off exponentially - polynomials
in Laplacian defined
}
\end{center}

\end{frame}

\begin{frame}

\begin{center}
{\bf   
Completeness of $P_n (\nabla^2)$ sufficient to construct $h(m^2)$ without
violating positive Euclidean time support condition.  
}
\end{center}
\begin{center}
{\bf 
Stiltjes moment problem \\
$G_{e2}$ moments
}
\end{center}
\[
\gamma_n := \int_0^\infty {e^{- \sqrt{m^2 + \mathbf{p}^2} \tau} 
\over 2 \sqrt{m^2 + \mathbf{p}^2}} \rho(m) 
m^{2n} dm 
\]
{\bf where} 
$\tau = \tau_1 + \tau_2 >0$.

\begin{center}
{\bf 
Carleman's condition
}
\end{center}
\[
\sum_{n=0}^\infty  \vert \gamma_n \vert^{-{1\over 2n}} > \infty
\]

{\bf Satisfied the  for $\rho(m^2)$ tempered
distribution $\Rightarrow$\quad $P(\nabla^2)$ complete.}
\[
\vert \gamma_n \vert^{-{1\over 2n}}  \sim {1 \over n+c} 
\]

\end{frame}

\begin{frame}
\begin{center}{\bf 
Existence - sufficient condition (Cook) 
}
\end{center}

\[
\int_a^\infty \Vert  (HJ - J H_0)
U_{0} (\pm t) \vert \psi_{0} \rangle
\Vert < \infty
\]

\[
\Vert  (HJ - J H_0)
\Phi U_{0} (\pm t) \vert \psi_{0} \rangle
\Vert^2=
\]
\[
(\psi_0 U_0 (\mp t) (J^{\dagger} H - H_0(J^{\dagger})
\theta G_e (HJ - J H_0)
U_{0} (\pm t) \vert \psi_{0} )
\]
\begin{center}{\bf
{\color{red} The effect of using one-body solutions for 2-2 scattering is that
the contribution from the disconnected part of $G_e$ to the above is zero.}
The connected part is expected to behave like $ct^{-3}$ for large $t$.
This fails for LSZ scattering.
}
\end{center}
\end{frame}


\begin{frame}
\begin{center}
{\bf 
Computational tricks for scattering \\
Invariance principle
}
\end{center}
\[
\lim_{t \to \pm \infty} e^{iHt} J e^{-iH_0t } \vert \psi \rangle =
\lim_{t \to \pm \infty} e^{if(H)t} J e^{-if(H_0)t } \vert \psi \rangle 
\]
\[
f(x) = -e^{-\beta x}
\]
\[
\lim_{t \to \pm \infty} e^{iHt} J e^{-iH_0t } \vert \psi \rangle =
\lim_{n \to \infty} e^{\mp i ne^{-\beta H}} J e^{i \pm ne^{-\beta H_0}} \vert \psi \rangle
\]

\[
\sigma(e^{-\beta H}) \in [0,1] \to
\]

\[
\vert e^{inx} - P_m(x) \vert < \epsilon \qquad x \in [0,1]
\]

\[
\vert\Vert e^{ine^{-\beta H}} - P_m(e^{-\beta H}) \vert\Vert < \epsilon
\]

\begin{center}{\bf 
Matrix elements of $e^{-n\beta H}$ are easy to calculate:
$\psi(\tau) \to \psi(\tau-n\beta)$
}
\end{center}

\end{frame}



\begin{frame}
\begin{center}{\bf 
Model tests
}
\end{center}
\[
M^2= 4(k^2+ m^2) -\lambda \vert g \rangle \langle g \vert 
\]
\[
\langle k \vert g \rangle = {1 \over k^2 + m_{pi}^2} 
\]
\begin{center}
{\bf   
Attractive - one pion exchange range, bound state with deuteron mass.
}
\end{center}
\[
\lim e^{-n e^{-\beta H}} \approx P_N(e^{-\beta H})
\]

\[
\langle \phi_f \vert
(e^{in e^{-\beta H_0}} e^{i2n e^{-\beta H}} e^{in e^{-\beta H_0}}-I)
\rangle \psi_i \rangle 
\]

\end{frame}

\begin{frame}
\begin{itemize}
{\bf
\item [${\color{red}\bullet}$] Choose sufficiently narrow initial and final wave packets.
\vfill
\item [${\color{red}\bullet}$] Choose sufficiently large $n$
\vfill
\item [${\color{red}\bullet}$] Replace $e^{i2n e^{-\beta H}}$ by a polynomial approximation  
\vfill
\item [${\color{red}\bullet}$] Calculations formally independent
of $\beta$, adjust $\beta$
for faster convergence.
\vfill
\item [${\color{red}\bullet}$] Model allows independent
  tests of each approximation.
\vfill
\item [${\color{red}\bullet}$] Approximations must be done in the proper order.
}
\end{itemize}
\end{frame}

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

\begin{itemize}
{\bf
\item [${\color{red}\bullet}$] Converges to exact sharp momentum transition
matrix elements.
\vfill
\item [${\color{red}\bullet}$] Tests converge for $.050-2$ GeV.
\vfill
\item [${\color{red}\bullet}$] Biggest source of error is the wave packet
width.  
\vfill
\item [${\color{red}\bullet}$] Model may be too simple?
}
\end{itemize}
\end{frame}

\begin{frame}
\vbox{
\begin{center}
Parameters  
\end{center}
\begin{center}
\begin{tabular}{llll}
\hline
p [GeV] & $\beta[$GeV$^{-1}$] & $p\times\beta$ &  $n$ \\
\hline
0.05 &	80.0    &  4.0 &  630  \\
%0.1  &	40.0    &  4.0 &  450  \\
0.2  &	10.0    &  2.0 &  470  \\
%0.3  &	5.0     &  1.5 &  330  \\
0.4  &	4.0     &  1.6 &  235  \\
%0.5  &	3.0     &  1.5 &  205  \\
0.6  &	2.5   &  1.5 &  225  \\
%0.7  &	1.6   &  1.2 &  200  \\
0.8  &	1.4   &  1.12&  200  \\
%0.9  &	1.05  &  .945&  190  \\
1.0  &	1.0     &  1.0 &  200  \\
%1.1  &	0.95  &  1.045 & 200  \\
1.2  &	0.9   &  1.08 & 200  \\
%1.3  &	0.85  &  1.105 & 200  \\
1.4  &	0.7   &  0.98 & 200  \\
%1.5  &	0.63  &  0.945 & 200  \\
1.6  &	0.57  &  0.912 & 200  \\
%1.7  &	0.5   &  0.85 & 200  \\
1.8  &	0.45  &  0.81 & 200  \\
%1.9  &	0.42  &  0.798 & 200  \\
2.0  &	0.4   &  0.8 & 200  \\
\hline                
\end{tabular}           
\end{center}
}
\end{frame}

\begin{frame}
\vbox{
\begin{center}
Sharp momentum T calculation vs exact
\end{center}
\begin{center}
\begin{tabular}{llll}
\hline
$p(GeV/c) $ & Real T & Im T & \% error \\
\hline
0.05  &   2.18499e-1    &  -1.03160e+0	          &     0.0982   \\
%0.1   &  -2.30337e-1    &  -4.09325e-1	          &     0.0956   \\
0.2   &  -1.01512e-1    &  -4.61420e-2	          &     0.0981   \\
%0.3   &  -3.46973e-2    &  -6.97209e-3	          &     0.0966   \\
0.4   &  -1.39007e-2    &  -1.44974e-3	          &     0.0997   \\
%0.5   &  -6.44255e-3    &  -3.86459e-4	          &     0.0986   \\
0.6   &  -3.34091e-3    &  -1.24434e-4	          &     0.0952   \\
%0.7   &  -1.88847e-3    &  -4.63489e-5           &     0.0977   \\
0.8   &  -1.14188e-3    &  -1.93605e-5           &     0.0965   \\
%0.9   &  -7.28609e-4    &  -8.86653e-6           &     0.0982   \\
1.0   &  -4.85708e-4    &  -4.37769e-6           &     0.0967   \\
%1.1   &  -3.35731e-4    &  -2.30067e-6           &     0.0987   \\
1.2   &  -2.39235e-4    &  -1.27439e-6           &     0.0968   \\
%1.3   &  -1.74947e-4    &  -7.38285e-7           &     0.0985   \\
1.4   &  -1.30818e-4    &  -4.44560e-7           &     0.0955   \\
%1.5   &  -9.97346e-5    &  -2.76849e-7           &     0.0956   \\
1.6   &  -7.73390e-5    &  -1.77573e-7           &     0.0992   \\
%1.7   &  -6.08794e-5    &  -1.16909e-7           &     0.0964   \\
1.8   &  -4.85672e-5    &  -7.87802e-8           &     0.0956   \\
%1.9   &  -3.92110e-5    &  -5.42037e-8           &     0.0967   \\
2.0   &  -3.20000e-5    &  -3.80004e-8           &     0.0966   \\
\hline                                                                           
\end{tabular}
\end{center}
}
\end{frame}


\begin{frame}
\vbox{
\begin{center}
Table 3: $p=100$[MeV], $\alpha=105000[$GeV$^{-2}]$  
\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  &-8.73395186664514e-1 &4.95616337213744e-1 \\ 
100 &-1.34576615227520e+0 &7.59199494502660e-1 \\
150 &-1.49091126760062e+0 &8.39700869213905e-1 \\
200 &-1.51566533604846e+0 &8.53352070852317e-1 \\
250 &-1.51799902547669e+0 &8.54631681615040e-1 \\
300 &-1.51811943431498e+0 &8.54697554653043e-1 \\
350 &-1.51812278309620e+0 &8.54699376334219e-1 \\
400 &-1.51812288480017e+0 &8.54699423334768e-1 \\
450 &-1.51812290596551e+0 &8.54699435479409e-1 \\
500 &-1.51812290955424e+0 &8.54699438102000e-1 \\
550 &-1.51812290968227e+0 &8.54699438639971e-1 \\
600 &-1.51812288857871e+0 &8.54699427676778e-1 \\
650 &-1.51812275938123e+0 &8.54699356334623e-1 \\
\hline   
ex  &-1.51812291315971e+0 &8.54699438329052e-1 \\
\hline
\end{tabular}
\end{center}
}        
\end{frame}

\begin{frame}
\vbox{
\begin{center}
Table 8: $k_0=2.0$[GeV], $\alpha=135$[GeV$^{-2}$]  
\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  & -2.60094316473225e-6 & 1.94120750171791e-3 \\ 
100 & -2.82916859895010e-6 & 2.35553585404449e-3 \\
150 & -2.83171624670953e-6 & 2.37471383801820e-3 \\ 
200 & -2.83165946257657e-6 & 2.37492460997990e-3 \\ 
250 & -2.83165905312632e-6 & 2.37492527186858e-3 \\ 
300 & -2.83165905257121e-6 & 2.37492527262432e-3 \\ 
350 & -2.83165905190508e-6 & 2.37492527262493e-3 \\
400 & -2.83165905234917e-6 & 2.37492527262540e-3 \\
\hline                                             
ex  & -2.83165905227843e-6 & 2.37492527259701e-3 \\
\hline
\end{tabular}
\end{center}
}
\end{frame}

\begin{frame}
\vbox{
\begin{center}
Table 10: Polynomial convergence
\end{center}
\begin{center}
\begin{tabular}{llll}
\hline
x     &  n  &    deg   &    poly error \% \\
\hline
0.1  &	 200  &  150 &	 7.939e+00     \\
0.1  &   200  &  200 &   3.276e+00    \\
0.1  &   200  &  250 &   1.925e-11  \\
0.1  &   200  &  300 &   4.903e-13  \\
\hline
0.1  &   630  &  580 &   3.573e+00    \\
0.1  &   630  &  630 &   2.069e+00    \\
0.1  &   630  &  680 &   5.015e-08   \\
0.1  &   630  &  700 &   7.456e-11   \\
\hline                         
0.5  &   200  &  150 &   1.973e-13  \\
0.5  &   200  &  200 &   1.627e-13  \\
0.5  &   200  &  250 &   3.266e-13  \\
\hline0.5  &   630  &  580 &   1.430e-14  \\
0.5  &   630  &  630 &   3.460e-13  \\
0.5  &   630  &  680 &   9.330e-13  \\
\hline                
0.9  &   200  &  150 &   7.939e+00      \\
0.9  &   200  &  200 &   3.276e+00      \\
0.9  &   200  &  250 &   1.950e-11  \\
0.9  &   200  &  300 &   9.828e-13  \\
\hline
0.9  &   630  &  580 &   3.573e+00  \\
0.9  &   630  &  630 &   2.069e+00  \\
0.9  &   630  &	 680 &   5.015e-08  \\
0.9  &	 630  &  700 &   7.230e-11  \\
\hline
\end{tabular}
\end{center}
}
\end{frame}  

\begin{frame}
\begin{center}
{\bf
Unfinished business
}  
\end{center}

\begin{itemize}
{\bf
\item[] Structure theorem for reflection positive Euclidean
invariant n>2 point functions?

\item[] Numerical implementation of polynomial approximation to
to the Haag Ruelle function $h(m^2)$?  

\item[] Bethe-Salpeter Kernels that lead to reflection positivity? 

\item[] Hilbert space method in lattice approximation?  
}
\end{itemize}  

\end{frame}  

\end{document}
\begin{frame}

\begin{itemize}
{\bf
\item[] $\beta E \approx 1$

\item[] Except for $p=50$MeV, $n=250$ is good for convergence 

\item[] degree 10-20\% larger than $n$

\item[]  0.1\% error for wave packet width less than 3\% of the on-shell momentum.
}    
\end{itemize}  
\end{frame}

\begin{frame}
\begin{center}
{\bf 
Particles, fields, dynamics and the light-front vacuum
}
\end{center}

\vfill

\begin{center}
{\bf 
W. N. Polyzou
}
\end{center}
\vfill

\begin{center}
{\bf
The University of Iowa 
}
\end{center}

\begin{figure}
\begin{minipage}[t]{.3\linewidth}
\centering
\includegraphics[angle=000,scale=.3]{jlab.pdf}
%\caption{Full matrix, $\lambda$=0, mass=1}
%\label{fig:8}
\end{minipage}
\begin{minipage}[t]{.4\linewidth}
\centering
\includegraphics[angle=000,scale=.5]{doe.pdf}
%\caption{Full matrix, $\lambda$=0.2, mass=1}
%\label{fig:9}
\end{minipage}
\end{figure}


%\begin{figure}
%\centering
%\includegraphics[scale=.3]{mumbai.pdf}
% \caption{V(x)}
%\label{figure 1}
%\end{figure}  

\end{frame}

\[
f (x)  :=  ( f_1 (\x_{11}), f_2 (\x_{21},\x_{22}),\cdots )
\]

\[
f_n (\x_{n1}, \x_{n2},\cdots , \x_{nn}) =0
\qquad
\mbox{unless} 
\qquad
0 < \x_{n1}^0 <  \x_{n2}^0  <\cdots <  \x_{nn} .
\]

\[
\Theta f (x) = f(\theta x) =
\left (f_1 (\theta \x_{11}), f_2 (\theta \x_{21}, \theta \x_{22}), \cdots
\right )
\]

\[
\langle f \vert g \rangle =
\]
\[
\sum_{mn} \int d^{4n}x d^{4m}y
f_n^* ( \x_{n1}, \x_{n2},\cdots , \x_{nn})
S_{m+n}
(\theta \x_{nn}, \cdots , \theta \x_{1n},\y_{1m} , \cdots , \y_{mm})
g_m (\y_{m1}, \y_{m2},\cdots , \y_{mm}) 
\]

\[
(f , \Pi \Theta  S \Pi g)
\]

\[
\langle f \vert f \rangle = (f ,\Pi \Theta S \Pi f) \geq 0 .
\]

\[
f_n (\x_{n1}, \x_{n2},\cdots , \x_{nn}) \to
f_n^{\prime } (\x_{n1}, \x_{n2},\cdots , \x_{nn}) =
f_n (O \x_{n1}-a , O\x_{n2}-a ,\cdots , O\x_{nn}-a ), 
\]

\[
H f_n (\x_{n1}, \x_{n2},\cdots , \x_{nn})
=
\sum_{k=1}^n {\partial \over \partial \x_{nk}^0} f_n (\x_{n1}, \x_{n2},\cdots , \x_{nn})
\]
\[
\mathbf{P}  f_n (\x_{n1},\x_{n2},\cdots , \x_{nn})
=
-i \sum_{k=1}^n {\partial \over \partial \mathbf{x}_{nk}} f_n (\x_{n1}, \x_{n2},\cdots ,\x_{nn})
\]
\[
\mathbf{J} f_n ( \x_{n1}, \x_{n2},\cdots ,
\x_{nn}) = -i \sum_{k=1}^n \mathbf{x}_{nk} \times {\partial \over
\partial \mathbf{x}_{nk}} f_n (\x_{n1},
\x_{n2},\cdots ,\x_{nn})
\]
\[
\mathbf{K} f_n ( \x_{n1}, \x_{n2},\cdots ,
\x_{nn}) = \sum_{k=1}^n (\mathbf{x}_{nk} {\partial \over
\partial {x}^0_{nk}}-  {x}^0_{nk}  {\partial \over
\partial \mathbf{x}_{nk}}) f_n ( \x_{n1},
\x_{n2},\cdots , \x_{nn}).
\]

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



\[
\lim_{t \to \pm \infty}
\Vert U(t,I) \vert \psi_{\pm} \rangle - J_{\cal A} U_{\cal A}(t,I) 
\vert \psi_{\cal A} \rangle \Vert =0 .
\]
Using the unitarity of $U(t,I)$ this can be expressed as
\[
\lim_{t \to \pm \infty}
\Vert \vert \psi_{\pm} \rangle - U(-t,I) J_{\cal A} U_{\cal A}(t,I) 
\vert \psi_{\cal A} \rangle \Vert =0 
\]


\[
\int_a^{\pm \infty} 
\Vert (H J_{\cal A}  - J_{\cal A} H_{\cal A} )
U_{\cal A}(t,I) \vert \psi_{\cal A} \rangle \Vert dt < \infty  
\]

\[
\Vert (H J_{\alpha} - J_{\alpha} H_{\alpha} )
U_{\alpha}(t,I) \vert \psi_\alpha \rangle \Vert =
\]
\[
( \psi_\alpha \vert U_{\alpha}(-t,I) 
(J^{\dagger}_{\alpha}H - H_{\alpha} J^{\dagger}_{\alpha}) 
\Theta  S 
(H J_{\alpha} - J_{\alpha} H_{\alpha} ) U_{\alpha}(t,I)
\vert \psi_\alpha )^{1/2} .
\]

\[
S_4(\x_1, \x_2, \y_2, \y_1) = 
S_2(\x_1,y_1) S_2(\x_2, \y_2 ) +
S_2(\x_1,y_2) S_2(\x_2, \y_1 ) +
S_c (\x_1, \x_2, \y_2, \y_1)  
\]

\]
\langle \x_1, \x_2  \vert  J_\alpha \vert \mathbf{p}_1, \mathbf{p}_2 \rangle
=
\]
\[
h_1 (\nabla_1^2) h_2 (\nabla_2^2)
\delta (\x_1^0 - \tau_1 )\delta (\x_2^0 - \tau_2 )
{1 \over (2 \pi)^3} e^{i \mathbf{p}_1\cdot  \mathbf{x}_1
+ i \mathbf{p}_2\cdot  \mathbf{x}_2}
\]

\[
H_\alpha = \omega_{m_1}(\mathbf{p}_1) + \omega_{m_2}(\mathbf{p}_2)
\qquad
\omega_{m_1}(\mathbf{p}_1) := \sqrt{m_i^2 + \mathbf{p}_i^2} 
\]
\[
U_\alpha (t,I) = 
e^{-i (\omega_{m_1}(\mathbf{p}_1) + \omega_{m_2}(\mathbf{p}_2))t}
\label{f.6}
\]
\[
H = {\partial \over \partial x^{0}_1} + {\partial \over \partial x^{0}_2} .
\]

\[
\Vert (H J_{\alpha} - J_{\alpha} H_{\alpha} )
U_{\alpha}(t,I) \vert \psi_\alpha \rangle \Vert^2 =
\]
\[
\int f_1^* (\mathbf{p}_1) f_2^* (\mathbf{p}_2) 
e^{i  (\omega_{m_1}(\mathbf{p}_1) + \omega_{m_2}(\mathbf{p}_2))t}
(
{\partial \over \partial x^{0}_1} + {\partial \over \partial x^{0}_2}
- 
\omega_{m_1}(\mathbf{p}_1) - \omega_{m_2}(\mathbf{p}_2))
\langle \mathbf{p}_1 , \mathbf{p}_2  \vert J_\alpha^{\dagger} 
\vert \x_1 , \x_2 \rangle
\times 
\]
\[ 
\left (
S_2(\theta \x_1,y_1) S_2(\theta\x_2, \y_2 )+
S_2(\theta \x_1,y_2) S_2(\theta\x_2, \y_1 )
+ S_c (\theta \x_1, \theta \x_2, \y_2, \y_1)
\right ) 
\]
\[
(
{\partial \over \partial y^{0}_1} + {\partial \over \partial y^{0}_2}
- 
\omega_{m_1}(\mathbf{p}_1') - \omega_{m_2}(\mathbf{p}_2'))
\langle \y_1 , \y_2 \vert J_\alpha  \vert \mathbf{p}_1' , \mathbf{p}_2'  \rangle
\]
\[
e^{-i  (\omega_{m_1}(\mathbf{p}_1') + \omega_{m_2}(\mathbf{p}_2'))t}
f_1 (\mathbf{p}_1') f_2 (\mathbf{p}_2') 
d\mathbf{p}_1
d\mathbf{p}_2
d\mathbf{p}_1'
d\mathbf{p}_2'
d^4x_1 d^4 x_2 d^4 y_1 d^4 y_2 
\]


\[
\int f_1^* (\mathbf{p}_1) f_2^* (\mathbf{p}_2) 
e^{i  (\omega_{m_1}(\mathbf{p}_1) + \omega_{m_2}(\mathbf{p}_2))t}
(
{\partial \over \partial \x^{0}_1} + {\partial \over \partial \x^{0}_2}
- 
\omega_{m_1}(\mathbf{p}_1) - \omega_{m_2}(\mathbf{p}_2))
\times
\]
\[
\langle \mathbf{p}_1 , \mathbf{p}_2  \vert J^{\dagger} 
\vert \x_1 , \x_2 \rangle 
S_2(\theta \x_1,\y_1) S_2(\theta\x_2, \y_2 ) \times
\]
\[
(
{\partial \over \partial y^{0}_1} + {\partial \over \partial y^{0}_2}
- 
\omega_{m_1}(\mathbf{p}_1') - \omega_{m_2}(\mathbf{p}_2'))
\langle \y_1 , \y_2 \vert J  \vert \mathbf{p}_1' , \mathbf{p}_2'  
\rangle 
\times
\]
\[
e^{-i  (\omega_{m_1}(\mathbf{p}_1') + \omega_{m_2}(\mathbf{p}_2'))t}
f_1 (\mathbf{p}_1') f_2 (\mathbf{p}_2') 
d\mathbf{p}_1
d\mathbf{p}_2
d\mathbf{p}_1'
d\mathbf{p}_2'
d^4x_1 d^4 x_2 d^4 y_1 d^4 y_2  =
\]
\[
\int f_1^* (\mathbf{p}_1) f_2^* (\mathbf{p}_2) 
e^{i  (\omega_{m_1}(\mathbf{p}_1) + \omega_{m_2}(\mathbf{p}_2))t}
(
{\partial \over \partial \x^{0}_1} + {\partial \over \partial \x^{0}_2}
- 
\omega_{m_1}(\mathbf{p}_1) - \omega_{m_2}(\mathbf{p}_2)) \times
\]
\[
h_1 (\nabla_{\x1}^2) h_2 (\nabla_{\x2}^2)
\delta (\x_1^0 - \tau_1 )\delta (\x_2^0 - \tau_2 )
{1 \over (2 \pi)^3} e^{-i \mathbf{p}_1\cdot  \mathbf{x}_1
-i \mathbf{p}_2\cdot  \mathbf{x}_2}
\times 
\]
\[ 
{1 \over (2 \pi)^6} {d\mathbf{k}_1 d\mathbf{k}_2 \rho_1 (m) \rho_2' (m') dm dm'
\over 4 \omega_{m} (\mathbf{k}_1)\omega_{m'} (\mathbf{k}_2)}
e^{-\omega_{m} (\mathbf{k}_1)(\x_1^0 + \y_1^0)-   
\omega_{m'} (\mathbf{k}_2)(\x_2^0-\y_2^0)}
e^{i \mathbf{k}_1 \cdot (\mathbf{x}_1 - \mathbf{y}_1) +
i \mathbf{k}_2 \cdot (\mathbf{x}_2 - \mathbf{y}_2)} \times
\]
\[
(
{\partial \over \partial y^{0}_1} + {\partial \over \partial y^{0}_2}
- 
\omega_{m}(\mathbf{p}_1') - \omega_{m'}(\mathbf{p}_2'))
\times
\]
\[
h_1 (\nabla_{\y1}^2) h_2 (\nabla_{\y2}^2)
\delta (\y_1^0 - \tau_1 )\delta (\y_2^0 - \tau_2 )
{1 \over (2 \pi)^3} e^{i \mathbf{p}_1'\cdot  \mathbf{y}_1
+ i \mathbf{p}_2'\cdot  \mathbf{y}_2}
\times
\]
\[
e^{-i  (\omega_{m_1}(\mathbf{p}_1') + \omega_{m_2}(\mathbf{p}_2'))t}
f_1 (\mathbf{p}_1') f_2 (\mathbf{p}_2') 
d\mathbf{p}_1
d\mathbf{p}_2
d\mathbf{p}_1'
d\mathbf{p}_2'
x^4x_1 d^4 x^2 d^4 y_1 d^4 y_2 =
\]
\[
\int f_1^* (\mathbf{p}_1) f_2^* (\mathbf{p}_2) 
(
\omega_{m}(\mathbf{p}_1) - \omega_{m'}(\mathbf{p}_2) - 
\omega_{m_1}(\mathbf{p}_1) - \omega_{m_2}(\mathbf{p}_2)) \times
h_1 (m^2) h_2 (m^{\prime 2})
\times 
\]
\[ 
{d\mathbf{p}_1 d\mathbf{p}_2 \rho_1 (m) \rho_2' (m') dm dm'
\over 4 \omega_{m} (\mathbf{p}_1)\omega_{m'} (\mathbf{2}_2)}
e^{-2\omega_{m} (\mathbf{p}_1)\tau_1   
-2 \omega_{m'} (\mathbf{p}_2)\tau_2}
\]
\[
(
\omega_{m}(\mathbf{p}_1) - \omega_{m'}(\mathbf{p}_2)
- 
\omega_{m_1}(\mathbf{p}_1) - \omega_{m_2}(\mathbf{p}_2))
h_1 (m^2) h_2 (m^{\prime 2})
f_1 (\mathbf{p}_1) f_2 (\mathbf{p}_2) .
\]


Table 1

\begin{center}
\begin{tabular}{lllll}
\hline
$k_0$ & $\alpha$ &  $k_w$  &    \% error &  $k_w/k_0$ \\
\hline
[GeV] & [GeV$^{-2}]$ & [GeV] &   &  \\
\hline
0.05 &325000	&0.00175412	&0.1     &0.035 \\
0.1  &105000	&0.00308607	&0.1     &0.030 \\
0.2  &26000	&0.00620174	&0.1     &0.031 \\
0.3  &10500	&0.009759	&0.1     &0.032 \\
0.4  &5100	&0.0140028	&0.1     &0.035 \\
0.5  &3000	&0.0182574	&0.1     &0.036 \\
0.6  &2000	&0.0223607	&0.1     &0.037 \\
0.7  &1350	&0.0272166	&0.1     &0.038 \\
0.8  &1000	&0.0316228	&0.1     &0.039 \\
0.9  &750	&0.0365148	&0.1     &0.040 \\
1.0  &600	&0.0408248	&0.1     &0.040 \\
1.1  &475	&0.0458831	&0.1     &0.041 \\
1.2  &400	&0.05	        &0.1     &0.041 \\
1.3  &330	&0.0550482	&0.1     &0.042 \\
1.4  &290	&0.058722	&0.1     &0.041 \\
1.5  &250	&0.0632456	&0.1     &0.042 \\
1.6  &210	&0.0690066	&0.1     &0.043 \\
1.7  &190	&0.0725476	&0.1     &0.042 \\
1.8  &170	&0.0766965	&0.1     &0.042 \\
1.9  &150	&0.0816497	&0.1     &0.042 \\
2.0  &135	&0.0860663	&0.1     &0.043 \\
\end{tabular}
\end{center}

The last column of Table 1 suggests that a 0.1\% error will generally
be obtained if this width is less than 3\% of the on-shell momentum.
This property holds over a wide range of momenta in this model.  This
is a simple model transition operator; one may anticipate narrower
wave packets are needed for more realistic models.  This approximation
can be improved by decreasing the width of the wave packet; it is the
largest source of error in the calculations.  In any realistic
calculation the wave packet width does not have to be smaller than 
the experimental momentum resolution.

The second step is to approximate $S$-matrix elements in these
Gaussian wave packets using equation (\ref{e.24}).  It
is important to first pick the wave packets because the $n$ 
value needed for convergence depends on the width of the 
wave packet.  The approximate quantities are 
\beq
\langle \psi' \vert S_n \vert \psi \rangle :=
\int d\mathbf{k} d\mathbf {k}' \psi'(\mathbf{k}) 
e^{- i n e^{-\beta k^2/m}}\langle \mathbf{k} \vert
e^{2in e^{-\beta H}} \vert \mathbf{k}' \rangle     
e^{- i n e^{-\beta k^{\prime 2}/m}} \psi(\mathbf{k}').
\eeq
In Tables 2-8 these quantities are computed using the spectral
expansion for $H$.  In these calculations the bound-state contribution
is not included because it vanishes in the large $n$ limit.
Tables 2-8 show the real and imaginary parts of matrix elements
$\langle \psi' \vert( S_n-I) \vert \psi \rangle$ for different values
of $n$ for $k_0=50,100,200,500,1000,1500, 2000$ MeV.  The exact value is 
given at the
bottom of each table.  Table 9 shows the values of $\beta$, the
product $k_0\times \beta$ and the $n$-values used in our final
calculations.  Table 9 suggests that $\beta$ should be chosen so
$k_0\times \beta$ is of order unity.  Except for the $k_0=50$MeV case, 
$n=250$ or more gives errors for the $n$-limits that are smaller than 
the errors made in the factorization approximation, (\ref{e.33}).  

The $n$ dependence of the real and imaginary part of matrix elements of
$\langle \psi' \vert (S-I) \vert \psi \rangle$, computed using  
(\ref{e.24}), are plotted as a function of $n$ for different values of
$k_0$ in figures 1-10.  Figures 11 and 12 show how fast the
neglected bound state contribution to the spectral expansion
falls off with $n$ for $k_0=1$ GeV. 

\vbox{
\begin{center}
Table 2: $k_0=50$[MeV], $\alpha=325000[$GeV$^{-2}]$  
\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 &-4.61629554857848e-1 &-9.31358324913684e-2 \\ 
50 &-7.62976513315350e-1 & -1.52406978178214e-1 \\
%100 &-8.69931511831809e-1 &-1.80034893509867e-1 \\ 
100 & -1.33113144491104e+0 & -2.75546806155677e-1 \\
%150 &-1.21140381384123e+0 &-2.53103932623990e-1 \\ 
150 & -1.67324498184421e+0 &  -3.50392186517949e-1 \\
%200 &-1.47308854900661e+0 &-3.09402219154459e-1 \\ 
200 &-1.83391449191883e+0 & -3.86136590065981e-1 \\
%250 &-1.65673884380536e+0 &-3.49058703542241e-1 \\  
250 &-1.89273779093641e+0 & -3.99389077283171e-1 \\
%300 &-1.77476024360498e+0 &-3.74672757051976e-1 \\  
300 &-1.90951807884485e+0 & -4.03228425149703e-1 \\
%350 &-1.84424118543806e+0 &-3.89825868046778e-1 \\  
350 &-1.91324926322936e+0 & -4.04093826445785e-1 \\
%400 &-1.88168370990428e+0 &-3.98022623423276e-1 \\  
400 &-1.91389545311571e+0 & -4.04245969269804e-1 \\
%450 &-1.90011829823232e+0 &-4.02080778737878e-1 \\
450 & -1.91398265491004e+0 & -4.04266741575048e-1 \\
%500 &-1.90859253987081e+0 &-4.04011953492714e-1 \\
500 &-1.91399177708580e+0 & -4.04268944214980e-1 \\
%550 & -7.e-3 & 1.e-1 \\
550 &-1.91399249452470e+0 & -4.04269131167755e-1 \\
%600 & -7.e-3 & 1.e-1 \\
600 &-1.91399253056843e+0 & -4.04269144047622e-1 \\
%650 & -7.e-3 & 1.e-1 \\ 
650 &-1.91399252976074e+0 & -4.04269145893613e-1 \\
\hline
ex  &-1.91399253060872e+0 &-4.04269147714400e-1 \\   
\hline
\end{tabular}
\end{center}
}        
           
\vbox{
\begin{center}
Table 3: $k_0=100$[MeV], $\alpha=105000[$GeV$^{-2}]$  
\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  &-8.73395186664514e-1 &4.95616337213744e-1 \\ 
100 &-1.34576615227520e+0 &7.59199494502660e-1 \\
150 &-1.49091126760062e+0 &8.39700869213905e-1 \\
200 &-1.51566533604846e+0 &8.53352070852317e-1 \\
250 &-1.51799902547669e+0 &8.54631681615040e-1 \\
300 &-1.51811943431498e+0 &8.54697554653043e-1 \\
350 &-1.51812278309620e+0 &8.54699376334219e-1 \\
400 &-1.51812288480017e+0 &8.54699423334768e-1 \\
450 &-1.51812290596551e+0 &8.54699435479409e-1 \\
500 &-1.51812290955424e+0 &8.54699438102000e-1 \\
550 &-1.51812290968227e+0 &8.54699438639971e-1 \\
600 &-1.51812288857871e+0 &8.54699427676778e-1 \\
650 &-1.51812275938123e+0 &8.54699356334623e-1 \\
\hline   
ex  &-1.51812291315971e+0 &8.54699438329052e-1 \\
\hline
\end{tabular}
\end{center}
}        

\vbox{
\begin{center}
Table 4: $k_0=200$[MeV], $\alpha=26000$[Gev$^{-2}$]  
\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 &-2.08408481834932e-1 &4.56550768265380e-1 \\
100 &-3.11945696198071e-1 &6.85279276148059e-1 \\
150 &-3.38623394392403e-1 &7.44641333032490e-1 \\
200 &-3.42127100784575e-1 &7.52475859236454e-1 \\
250 &-3.42359208499266e-1 &7.52997722504423e-1 \\
300 &-3.42366821259122e-1 &7.53015087494957e-1 \\
350 &-3.42366956571344e-1 &7.53015382645461e-1 \\
400 &-3.42366963936415e-1 &7.53015400508140e-1 \\
450 &-3.42366965122021e-1 &7.53015404697748e-1 \\
500 &-3.42366965247355e-1 &7.53015405389574e-1 \\
550 &-3.42366962667700e-1 &7.53015400608940e-1 \\
600 &-3.42366938358008e-1 &7.53015348812648e-1 \\
650 &-3.42366884413759e-1 &7.53015247619997e-1 \\
\hline
ex & -3.42366967477707e-1 &7.53015410457076e-1 \\
\hline
\end{tabular}
\end{center}
}        

\vbox{
\begin{center}
Table 5: $k_0=500$[MeV], $\alpha=3000$[Gev$^{-2}$]  
\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  & -5.93330385580271e-3 & 9.65738963854834e-2 \\ 
100 & -7.12681692349637e-3 & 1.18415504225673e-1 \\ 
150 & -7.18129565909453e-3 & 1.19491569949089e-1 \\ 
200 & -7.18179706798405e-3 & 1.19502423190014e-1 \\ 
250 & -7.18179794641838e-3 & 1.19502444469971e-1 \\ 
300 & -7.18179794670870e-3 & 1.19502444477783e-1 \\ 
350 & -7.18179794671048e-3 & 1.19502444477784e-1 \\ 
400 & -7.18179794671081e-3 & 1.19502444477784e-1 \\ 
\hline                                              
ex  & -7.18179797016073e-3 & 1.19502444795275e-1 \\
\hline
\end{tabular}
\end{center}
}
 
\vbox{
\begin{center}
Table 6: $k_0=1$[GeV], $\alpha=600$[Gev$^{-2}$]  
\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  &-1.47024820732811e-4 &1.55922557816223e-2 \\ 
100 &-1.62726188649875e-4 &1.79713868930101e-2 \\ 
150 &-1.62967714125273e-4 &1.80219591282450e-2 \\ 
200 &-1.62968113934903e-4 &1.80220978916775e-2 \\ 
250 &-1.62968113982642e-4 &1.80220979403700e-2 \\ 
300 &-1.62968113982642e-4 &1.80220979403721e-2 \\ 
350 &-1.62968113982753e-4 &1.80220979403720e-2 \\ 
400 &-1.62968113982975e-4 &1.80220979403718e-2 \\ 
\hline                                            
ex  &-1.62968113982742e-4 &1.80220979403858e-2 \\
\hline
\end{tabular}
\end{center}
}

\vbox{
\begin{center}
Table 7: $k_0=1.5$[GeV], $\alpha=250$[GeV$^{-2}$]  
\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   &-1.40242356887477e-5 &4.66175982621713e-3 \\ 
100  &-1.54430995201738e-5 &5.52235009412764e-3 \\ 
150  &-1.54679181726403e-5 &5.55112445776045e-3 \\ 
200  &-1.54679281958447e-5 &5.55130663695727e-3 \\ 
250  &-1.54679280390813e-5 &5.55130688968718e-3 \\ 
300  &-1.54679280390813e-5 &5.55130688978369e-3 \\ 
350  &-1.54679280394143e-5 &5.55130688978374e-3 \\ 
400  &-1.54679280388592e-5 &5.55130688978377e-3 \\ 
\hline                                             
ex   &-1.54679280242191e-5 &5.55130688386830e-3 \\
\hline
\end{tabular}
\end{center}
}

\vbox{
\begin{center}
Table 8: $k_0=2.0$[GeV], $\alpha=135$[GeV$^{-2}$]  
\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  & -2.60094316473225e-6 & 1.94120750171791e-3 \\ 
100 & -2.82916859895010e-6 & 2.35553585404449e-3 \\
150 & -2.83171624670953e-6 & 2.37471383801820e-3 \\ 
200 & -2.83165946257657e-6 & 2.37492460997990e-3 \\ 
250 & -2.83165905312632e-6 & 2.37492527186858e-3 \\ 
300 & -2.83165905257121e-6 & 2.37492527262432e-3 \\ 
350 & -2.83165905190508e-6 & 2.37492527262493e-3 \\
400 & -2.83165905234917e-6 & 2.37492527262540e-3 \\
\hline                                             
ex  & -2.83165905227843e-6 & 2.37492527259701e-3 \\
\hline
\end{tabular}
\end{center}
}

\vbox{
\begin{center}
Table 9: Parameters  
\end{center}
\begin{center}
\begin{tabular}{llll}
\hline
k0 [GeV] & $\beta[$GeV$^{-1}$] & $k0\times\beta$ &  $n$ \\
\hline
0.05 &	80.0    &  4.0 &  630  \\
0.1  &	40.0    &  4.0 &  450  \\
0.2  &	10.0    &  2.0 &  470  \\
0.3  &	5.0     &  1.5 &  330  \\
0.4  &	4.0     &  1.6 &  235  \\
0.5  &	3.0     &  1.5 &  205  \\
0.6  &	2.5   &  1.5 &  225  \\
0.7  &	1.6   &  1.2 &  200  \\
0.8  &	1.4   &  1.12&  200  \\
0.9  &	1.05  &  .945&  190  \\
1.0  &	1.0     &  1.0 &  200  \\
1.1  &	0.95  &  1.045 & 200  \\
1.2  &	0.9   &  1.08 & 200  \\
1.3  &	0.85  &  1.105 & 200  \\
1.4  &	0.7   &  0.98 & 200  \\
1.5  &	0.63  &  0.945 & 200  \\
1.6  &	0.57  &  0.912 & 200  \\
1.7  &	0.5   &  0.85 & 200  \\
1.8  &	0.45  &  0.81 & 200  \\
1.9  &	0.42  &  0.798 & 200  \\
2.0  &	0.4   &  0.8 & 200  \\
\hline                
\end{tabular}           
\end{center}
}
The third approximation is the polynomial approximation to
$e^{i2n e^{-\beta H}}$.  In this application we use the freedom to shift
the zero of the energy of the free and interacting Hamiltonians 
by a constant (the binding energy) so the 
spectrum of $e^{-\beta H}$ is strictly between zero and one.
Then it is only necessary to find a polynomial approximation to
$e^{inx}$ for $x \in [0,1]$.

The polynomial approximation is made using the Chebyshev expansion:
\beq 
f(y) \approx P_N(y) := {1 \over 2} c_0 T_0 (y) + \sum_{k=1}^N c_k T_k (y) 
\label{e.28}
\eeq
\beq
c_j = {2 \over N+1} \sum_{k=1}^N f( \cos({2k-1 \over N+1}{\pi
\over 2})) \cos(j {2k-1 \over N+1}{\pi \over 2}). 
\label{e.29}
\eeq
The Chebyshev expansion
is designed for functions on the interval $[-1,1]$.  To
make full use of the information in this approximation 
we approximate $e^{-2inx} = f(2x-1)$ by the polynomial
approximation to $f(y)$. This gives the polynomial
\beq 
f(2 e^{-\beta H'}-1 ) \approx {1 \over 2} c_0 T_0 (
2e^{-\beta H'}-1) + \sum_{k=1}^N c_k T_k (2e^{-\beta H'}-1)
= P_N ( e^{-\beta H'})
\label{e.30}
\eeq
where $H'=H+e_b$. It follows that 
%\beq
%f(x) = e^{2inx}
%\label{e.31}
%\eeq
\beq
\vert e^{2inx} - P_N(x) \vert < 2 {{n}^{N+1}\over (N+1)!} 
\qquad
|\Vert e^{2ine^{-\beta H'}} - P_N(e^{-\beta H'}) |\Vert < 2 {{n}^{N+1}\over (N+1)!} 
\label{e.32}
\eeq
Chebyshev polynomials are known to be good approximations to
the best uniform polynomial approximation \cite{oxford2004}. 
Table 10 shows polynomial approximations to $e^{inx}$ for different 
values of $x\in [0,1]$, and $n$.  The errors are between 
$10^{-11}$ and $10^{-13}$ for the degree of the polynomial only slightly 
above $n$.
\vbox{
\begin{center}
Table 10: Polynomial convergence
\end{center}
\begin{center}
\begin{tabular}{llll}
\hline
x     &  n  &    deg   &    poly error \% \\
\hline
0.1  &	 200  &  150 &	 7.939e+00     \\
0.1  &   200  &  200 &   3.276e+00    \\
0.1  &   200  &  250 &   1.925e-11  \\
0.1  &   200  &  300 &   4.903e-13  \\
\hline
0.1  &   630  &  580 &   3.573e+00    \\
0.1  &   630  &  630 &   2.069e+00    \\
0.1  &   630  &  680 &   5.015e-08   \\
0.1  &   630  &  700 &   7.456e-11   \\
\hline                         
0.5  &   200  &  150 &   1.973e-13  \\
0.5  &   200  &  200 &   1.627e-13  \\
0.5  &   200  &  250 &   3.266e-13  \\
\hline0.5  &   630  &  580 &   1.430e-14  \\
0.5  &   630  &  630 &   3.460e-13  \\
0.5  &   630  &  680 &   9.330e-13  \\
\hline                
0.9  &   200  &  150 &   7.939e+00      \\
0.9  &   200  &  200 &   3.276e+00      \\
0.9  &   200  &  250 &   1.950e-11  \\
0.9  &   200  &  300 &   9.828e-13  \\
\hline
0.9  &   630  &  580 &   3.573e+00  \\
0.9  &   630  &  630 &   2.069e+00  \\
0.9  &   630  &	 680 &   5.015e-08  \\
0.9  &	 630  &  700 &   7.230e-11  \\
\hline
\end{tabular}
\end{center}
}
Table 10 suggests that a polynomial with degree 10-20\% larger than $n$ 
is needed for convergence to one part in $10^{12}$ ($10^{-10}$\%).

All three approximations are combined to get the approximations to the
on-shell transition operator for incident momenta between 50 MeV and 2
GeV.  These results are displayed in table 11.  The errors are all
better than .1\%.  The only significant source of error is the
approximate factorization of the sharp momentum matrix element from
the $S$ matrix elements.  This can be made as small as desired by
choosing sufficiently narrow wave packets, although there is no need
to improve accuracy to better than experimental resolution.
\vbox{
\begin{center}
Table 11: Final calculation
\end{center}
\begin{center}
\begin{tabular}{llll}
\hline
$k0$ & Real T & Im T & \% error \\
\hline
0.05  &   2.18499e-1    &  -1.03160e+0	          &     0.0982   \\
0.1   &  -2.30337e-1    &  -4.09325e-1	          &     0.0956   \\
0.2   &  -1.01512e-1    &  -4.61420e-2	          &     0.0981   \\
0.3   &  -3.46973e-2    &  -6.97209e-3	          &     0.0966   \\
0.4   &  -1.39007e-2    &  -1.44974e-3	          &     0.0997   \\
0.5   &  -6.44255e-3    &  -3.86459e-4	          &     0.0986   \\
0.6   &  -3.34091e-3    &  -1.24434e-4	          &     0.0952   \\
0.7   &  -1.88847e-3    &  -4.63489e-5           &     0.0977   \\
0.8   &  -1.14188e-3    &  -1.93605e-5           &     0.0965   \\
0.9   &  -7.28609e-4    &  -8.86653e-6           &     0.0982   \\
1.0   &  -4.85708e-4    &  -4.37769e-6           &     0.0967   \\
1.1   &  -3.35731e-4    &  -2.30067e-6           &     0.0987   \\
1.2   &  -2.39235e-4    &  -1.27439e-6           &     0.0968   \\
1.3   &  -1.74947e-4    &  -7.38285e-7           &     0.0985   \\
1.4   &  -1.30818e-4    &  -4.44560e-7           &     0.0955   \\
1.5   &  -9.97346e-5    &  -2.76849e-7           &     0.0956   \\
1.6   &  -7.73390e-5    &  -1.77573e-7           &     0.0992   \\
1.7   &  -6.08794e-5    &  -1.16909e-7           &     0.0964   \\
1.8   &  -4.85672e-5    &  -7.87802e-8           &     0.0956   \\
1.9   &  -3.92110e-5    &  -5.42037e-8           &     0.0967   \\
2.0   &  -3.20000e-5    &  -3.80004e-8           &     0.0966   \\
\hline                                                                           
\end{tabular}
\end{center}
}


\end{document}

%slide 2 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Outline
}
\end{center}
\vfill
\begin{itemize}
{\bf 
\item[$\bullet$] Light-front representations of particles
\vfill
\item[$\bullet$] Light-front representations of systems of free particles
\vfill
\item[$\bullet$] Light-front dynamics of particles - finite number of DOF
\vfill
\item[$\bullet$] Particles $\to$ fields
\vfill
\item[$\bullet$] Light-front operator algebra - Fock vacuum
\vfill
\item[$\bullet$] Noether's theorem and light-front dynamics of fields
\vfill
\item[$\bullet$] Zero modes and the light-front vacuum 
%\vfill
%\item[$\bullet$]
}
\end{itemize}
\end{frame}

%slide 3 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf Notation}
\end{center}

\begin{center}
{\bf Poincar\'e group - inhomogeneous $SL(2,\mathbb{C})$}
\end{center}
\[
X = x^{\mu} \sigma_{\mu} = 
\left (
\begin{array}{cc}
x^+ & x_{\perp}^*\\
x_{\perp} & x^- 
\end{array}
\right )
\qquad
x^{\mu} = {1 \over 2} \mbox{Tr}(\sigma_{\mu}X)
\]
\[
\mbox{det} (X) = -(x^0)^2 + \mathbf{x}^2 
\]
\[
X' = AXA^{\dagger} + B \qquad  B=B^{\dagger} \qquad \mbox{det} (A)=1
\]
%\[
%\Lambda^{\mu}{}_{\nu} =
%{1 \over 2} \mbox{Tr}(\sigma_{\mu}A \sigma_{\nu} A^{\dagger})
%\]
\begin{center}
{\bf 
Light-front
}
\end{center}
\[
X \to 
\left (
\begin{array}{cc}
0 & x_{\perp}^*\\
x_{\perp} & x^- 
\end{array}
\right )
\]
\begin{center}
{\bf 
Kinematic subgroup, preserves $x^+=0$, (7 parameters)
}
\end{center}
%\[
%\left (
%\begin{array}{cc}
%0 & x_{\perp}^*\\
%x_{\perp} & x^- 
%\end{array}
%\right ) \to
%\left (
%\begin{array}{cc}
%0 & x_{\perp}^{\prime *}\\
%x_{\perp}' & x^{\prime -} 
%\end{array}
%\right )
%\]
\[
A = 
\left (
\begin{array}{cc}
\alpha  e^{i\phi} & 0 \\
(\beta_1 + i \beta_2)  & e^{-i \phi}/\alpha 
\end{array}
\right )
\qquad 
B=
\left (
\begin{array}{cc}
0 & b_{\perp}^*\\
b_{\perp} & b^- 
\end{array}
\right )
\]

\end{frame}

%slide 4 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
%{
%\footnotesize
\begin{center}
{\bf 
Particles
}
\end{center}
\begin{center}
{\bf Single-particle light-front representations of the Poincar\'e group}
\end{center}
\[
U(\Lambda,a) \qquad \to \qquad \{P^{\mu},M^{\mu\nu} \} 
\]
\begin{center}
{\bf Kinematic subgroup generators\\}
\end{center}
\[
\{P^+=P^0+P^1,\mathbf{P}_{\perp},J^3=M^{12},  
\]
\[
K^3=M^{03},
E^1 =M^{01} + M^{31},
E^2 =M^{02} - M^{23},
\}
\]
\begin{center}
{\bf Dynamical generators}
\end{center}
%\end{frame}
%\end{document}

\[
\{ P^-= P^0-P^3,{J}^1= M^{23},J^2=M^{31} \}
\]
\begin{center}
{\bf Invariants}
\end{center}
\[
W^{\mu}= - {1 \over 2}\epsilon^{\mu \nu \alpha \beta}P_{\nu} M_{\alpha \beta}  
\]
\[
P^2 = P^{\mu} P_{\mu} = -{\color{red}M^2} \qquad
W^2 = W^{\mu} W_{\mu} = -{\color{red}M^2 j^2}  
\]

\end{frame}

%slide 5 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf Commutation relations $\to$ commuting observables\\}
\end{center}
\[
\{ M , \mathbf{j}^2,P^+ ,\mathbf{P}_{\perp}, \mathbf{j}_f \cdot \hat{\mathbf{z}}
\}
\]
\[
(0,\mathbf{j}_f) = -{1 \over M} B_f^{-1} (P/M)^{\mu}{}_{\nu} W^{\nu}  
\]

\[
B_f(p/m)  = 
\left (
\begin{array}{cc}
\sqrt{{P^+ / M}}  &  0 \\
{P_{\perp} / \sqrt{MP^+}} & \sqrt{{M/ P^+}}
\end{array}
\right )
\]

\begin{center}
{\bf Light-front basis - for irreducible representation space\\}
\end{center}

\[
\vert (m,j)\tilde{\mathbf{p}},\mu \rangle_f  \qquad 
\tilde{\mathbf{p}} : = (p^+,\mathbf{p}_{\perp}) \qquad
p^- = {\mathbf{p}_{\perp}^2 + m^2 \over p^+} 
\]

\begin{center}
{\bf Mass $m$ spin $j$ irreducible unitary representation\\}
\end{center}

\[
U(\Lambda ,b) 
\vert (m,j)\tilde{\mathbf{p}},\mu \rangle_f 
=
\]
\[
\sum 
e^{i \Lambda p\cdot b}
\vert (m,j)\tilde{\pmb{\Lambda}}p,\nu \rangle_f 
D^j_{\nu \mu}[B_f^{-1} (\Lambda p/m) \Lambda B_f(p/m)]
\sqrt{{ (\Lambda p)^+ \over p^+}}
\]

\end{frame}

%slide 6 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{itemize} 
{\bf 
\item[] Dynamics: $P^-$ self-adjoint generator of
a continuous one-parameter unitary group, $U(x^+)$.
}
\end{itemize}

\vfill
  
\begin{center}
{\bf $(\Lambda,b) \in $ kinematic subgroup: $\Rightarrow$ \\}
\end{center}

\[
\{
e^{i \Lambda p\cdot b}, 
R_{wf}(\Lambda,p/m) :=  B_f^{-1} (\Lambda p/m) \Lambda B_f(p/m)],
\sqrt{{ (\Lambda p)^+ \over p^+}}
\}
\]

\begin{center}
{\bf independent of mass $(m)$}
\end{center}

\vfill

\begin{center}
{\bf Special light-front properties }
\end{center}

\[
D^j_{\mu \nu}[R_{wf}(B_f(p'/m),p/m) = \delta_{\mu \nu} 
\qquad
D^j_{\mu \nu}[R_{wf}(R,p/m)] \not= D^j_{\mu \nu}[R]   
\]

%\[
%B_f(p/m) \qquad \mbox{\bf  kinematic } p/m \qquad \mbox{\bf dynamical} 
%\]

\[
P^+ \geq 0 \qquad \mbox{\bf kinematic}
\]

\[
P^+= P^- =M \qquad \mbox{\bf rest frame dynamical} 
\]


\end{frame} 

%slide 7 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf Relation to Dirac's other forms of dynamics}
\end{center}
\vfill
\begin{itemize} 
{\bf 
\item [$\bullet$] Different choices of commuting observables lead to {\color{red}unitary equivalent} single-particle representations of the Poincar\'e 
group with different
kinematic subgroups.
\vfill
{\color{blue}
\item [] Continuous variable choices: light-front momenta, three momenta,
four velocity, $\cdots$ 
\vfill
\item [] Spins choices: spins invariant under light-front boosts,
rotationless boosts, Jacob-Wick helicity boosts to rest frame $\cdots$.  
}
\vfill
\item [$\bullet$] Each basis choice leads to a unitarily equivalent 
representation of a mass $m$ spin $j$ particle (system) with {\color{red}different 
kinematic (mass-independent) subgroups.}

}
\end{itemize}

\end{frame}

%slide 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf Light-front representations - many free particles}
\end{center}

\[
\vert (m_1,j_1) \tilde{\mathbf{p}}_1,\nu_1 \rangle_f 
\cdots \vert (m_n,j_n) \tilde{\mathbf{p}}_n,\nu_n \rangle_f 
\]

\begin{center}
{\bf Infinitesimal generators: $G=\sum_{n} G_n \otimes I_{!n} $
}
\end{center}


\begin{center}
{\bf $P^-$: well-defined dynamics
}
\end{center}

\[
U_0(x^+) = \otimes_{m} U_m (x^+)  \qquad P_0^- = \sum_m P_m^-
\]


\begin{center}
{\bf Commuting observables (system)}
\end{center}
\[
\{ M_0 , \mathbf{j}^2,P^+_0 ,\mathbf{P}_{0\perp},
\mathbf{j}_{0f} \cdot \hat{\mathbf{z}}, \mathbf{d} 
\}
\]

\begin{center}
{\bf Irreducible basis states}
\end{center}
\[
\vert (m_0,j_0) \tilde{\mathbf{p}}_0,\nu_0; \mathbf{d}  \rangle_{0f}
\]
\[
\mathbf{d} = \mbox{\bf invariant degeneracy parameters}
\]

\end{frame}

%slide 9 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\[
U(\Lambda ,b) 
\vert (m_0,j_0)\tilde{\mathbf{p}}_0,\mu; \mathbf{d} \rangle_f 
=
\]
\[
\sum 
e^{i \Lambda P_0\cdot b}
\vert (m_0,j_0)\tilde{\pmb{\Lambda}}p_0,\nu; \mathbf{d} \rangle_f 
D^j_{\nu \mu}[R_{wf}(\Lambda,p_0)] 
\sqrt{{ (\Lambda p_0)^+ \over p_0^+}}
\]

\begin{itemize}
{\bf
\item[] Any dynamical unitary representation of the Poincar\'e group
can be decomposed as a direct integral/sum of irreducible 
representations. 

\[
U(\Lambda ,a) = \int_{\oplus} d\rho(m) \sum_j U_{mj} (\Lambda ,a)  
\]

\item[] Each irreducible subspace can be represented in a
light-front basis that transforms as above.

}
\end{itemize}

\end{frame}

%slide 10 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf Dynamics - Finite DOF}
\end{center}

\begin{itemize}
{\bf 
\item[] Problem: how to add non-trivial interactions to $M$ and $\mathbf{J}^2$ 
that satisfy
$[M,\mathbf{J}^2]=0$ and commute with all kinematic generators.
}
\end{itemize}

\begin{center}
{\bf Bakamjian-Thomas solution (finite DOF)\\
replace $\mathbf{J}$ by $\mathbf{j}=\mathbf{j}_0$,  
add interactions to $M$.
}
\end{center}
\[
M^2=M^2_0+V
\]
\[
{\color{red}[V,\mathbf{j}_{0}]}=[V,\mathbf{E}_{\perp 0}]=[V,K^3_0]=
[V,P^+_0]= [V,\mathbf{P}_{\perp 0 }]=0
\]
\vfill
\[
M>0 \qquad M=M^{\dagger}
\]
%\[
%\langle  (M_0,j_0 ) \tilde{\mathbf{P}}_0, \mu_0 ; l,s \vert V
%\vert (M_0',j_0' ) \tilde{\mathbf{P}}'_0, \mu_0' ; l',s'\rangle =
%\]
%\[
%\delta_{j_0j_0'}\delta_{\mu_0\mu_0'}
%\delta (\tilde{\mathbf{P}}_0 - \tilde{\mathbf{P}}_0')
%\langle M_0,l ,s \vert V^{j_0} \vert M_0' ,l' ,s' \rangle
%\]
\vfill
\begin{center}
{\bf
Diagonalize $M^2 = M^2_0+V$ in non-interacting irreducible basis
}
\end{center}
\[
\vert (m_0,j_0 ) \tilde{\mathbf{p}}_0, \mu_0 ; l_0,s_0\rangle
\]

\end{frame}

%slide 11 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

%\begin{center}
%{\bf
%Light-front Clebsch-Gordan coefficients\\
%used to construct irreducible free-particle basis}
%\end{center}
%\[
%_f\langle   
%(m_1,j_1) \tilde{\mathbf{p}}_1,\mu_1 ;
%(m_2,j_2) \tilde{\mathbf{p}}_2,\mu_2
%\vert (m_0,j_0 ) \tilde{\mathbf{P}}, \mu_0 ; l_0,s_0
%\rangle_f
%\]
%
%\vfill
%\[
%\vert ({\color{red}m_0},j_0 ) \tilde{\mathbf{p}}_0, \mu_0 ; l_0,s_0 \rangle_f
%\to
%\vert ({\color{red}m_I},j_0 ) \tilde{\mathbf{p}}_0, \mu_0 ; d \rangle_f
%\]

\begin{center}
{\bf
Dynamics: Diagonalize $M^2$ in free-particle irreducible basis.  
This gives a complete set of irreducible eigenstates consistent with 
the dynamics.}
\end{center}

\vfill

\[
U(\Lambda ,b) 
\vert (m_I,j_0)\tilde{\mathbf{p}}_0,\mu ;\mathbf{d}_I \rangle_f 
=
\]
\[
\sum 
e^{i \Lambda p\cdot b}
\vert (m_I,j_0)\tilde{\pmb{\Lambda}}p,\nu, ;\mathbf{d}_I \rangle_f 
D^j_{\nu \mu}[R_{wf}(\Lambda, p)] 
\sqrt{{ (\Lambda p)^+ \over p^+}}
\]

\begin{center}
{\bf
Interacting angular momentum
} 
\end{center}

\[
\mathbf{J}_{\perp} = 
{1 \over P_0^+} 
[{1\over 2}  (P_0^+-{\color{red}P^-}) (\hat{\mathbf{z}}\times \mathbf{E}_0) -
(\hat{\mathbf{z}}\times \mathbf{P}_0)K_0^3 +\mathbf{P}_{\perp 0} j_{f0}^3 + 
{\color{red}M} {\color{blue}\mathbf{j}_{f\perp 0}}]
\]

\end{frame}


%slide 12 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%\begin{frame}
%\[
%{\color{red}
%\Downarrow
%}
%\]
%\begin{center}
%{\bf
%Interacting angular momentum
%} 
%\end{center}
%
%
%\begin{center}
%{\bf
%Function of $M$, kinematic spins, and kinematic generators
%} 
%\end{center}
%
%
%\[
%\mathbf{J}_{\perp} = 
%{1 \over P_0^+} 
%[{1\over 2}  (P_0^+-{\color{red}P^-}) (\hat{\mathbf{z}}\times \mathbf{E}_0) -
%(\hat{\mathbf{z}}\times \mathbf{P}_0)K_0^3 +\mathbf{P}_{\perp 0} j_{f0}^3 + 
%{\color{red}M} {\color{blue}\mathbf{j}_{f\perp 0}}]
%\]
%
%\begin{center}
%{\bf
%Kinematic spin special to Bakamjian-Thomas construction.  
%% 
%\end{center}

%\begin{center}
%{\bf
%Defined by a light front boost of angular momentum to rest frame; 
%while the boost is kinematic, the parameters defining the rest 
%frame are not.  
%} 
%\end{center}



%\end{frame}

%slide 13 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 2-body light-front BT dynamics}
\end{center}
\begin{itemize}
{\bf
\item [${\color{red}\bullet}$] Unitary representation of the
Poincar\'e group - including transverse rotations.
\vfill
\item [${\color{red}\bullet}$] Kinematic subgroup has no
interactions.
\vfill
\item [${\color{red}\bullet}$] Spin is not interacting but the
total and orbital angular momenta are dynamical.
\vfill
\item [${\color{red}\bullet}$] Dynamical $x^+$ evolution is given by a
unitary one-parameter group - there are no issues with
the light-front being a characteristic surface.  
\vfill
\item [${\color{red}\bullet}$] $U_{BT}(\Lambda ,b) \to
U_1(\Lambda ,b)\otimes U_2(\Lambda ,b)$ when the interactions
are turned off (cluster properties)
}    
\end{itemize}
\end{frame}


%slide 14 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Extensions to more than two particles non-trivial
}
\end{center}
\[
U_{(12)\otimes 3}(\Lambda, a) :=
U_{(ij)}(\Lambda,a) \otimes U_{k}(\Lambda,a)
\]
\[
\mathbf{j}_{(12)\otimes(3)f} \not= 
\mathbf{j}_{(23)\otimes(1)f} \not= 
\mathbf{j}_{(31)\otimes(2)f} \not= \mathbf{j}_{0f}
\]
\begin{itemize}
{\bf 
\item[${\color{red}\bullet}$] Given  $V_{ij}$ there is a $V_{ij}'$,
satisfying $[V'_{ij},\mathbf{j}_{0f}]=0$ 
such that 
\[
P^-_{(ij)\otimes k} =
{{M_{ij0}^2 + V_{ij}} + \mathbf{P}_{ij0\perp}^2 
\over P_{ij}^+}\otimes I_k + I_{ij} \otimes 
{{ m_k^2} + \mathbf{P}_{k\perp}^2 
\over P_{3}^+} 
\]
\[
P^-_{(12)(3)} = {{M_{1230}^2 + V_{12}'} + \mathbf{P}_{1230\perp}^2 
\over P_{120}^+}   
\]
\item[] gives the same $2+1$ $S$-matrix ($S_{ij}\otimes S_k = S'_{(ij)(k)}$)  
%however the resulting unitary 
%representation of the Poincar\'e group fails to satisfy cluster 
%properties.
%
%\[
%U_{(12)\otimes 3}(\Lambda, a) =
%U_{(12)}(\Lambda,a) \otimes U_{3}(\Lambda,a) \not= 
%U_{(12)(3)I }(\Lambda,a) 
%\]
}
\end{itemize}
\[
M^{\prime 2} :=M^{\prime 2}_{(12)(3)}+ M^{\prime 2}_{(23)(1)} + M^{\prime 2}_{(31)(2)}
-2M^{2}_{0}
\]

\end{frame}

%slide 15 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%\begin{frame}
  
%\begin{itemize} 
%
%{\bf 
%\item[${\color{red}\bullet}$] Key difference
%      
%\[ 
%\mathbf{j}_{(12)\otimes(3)f} \not= \mathbf{j}_{f0} 
%\]
%
%\vfill 
%
%\[ 
%\mathbf{j}_{(12)(3)f} = \mathbf{j}_{f0} 
%\] 
%
%\item[]
%Cluster properties is {\color{red}not} compatible with a kinematic spin.
%
%\vfill
%%
%\item[${\color{red}\bullet}$] Adding interactions to {\color{red}different 
%pairs} in the tensor product representation breaks Poincar\'e invariance 
%but preserves cluster properties
%(comparable to truncated field theoretic case).
%
%\vfill
%
%\item[${\color{red}\bullet}$] Adding interactions to different pairs in the 
%3-body BT representation preserves Poincar\'e invariance but
%breaks cluster properties.

%\item[${\color{red}\bullet}$] Correcting errors in rotational
%covariance has dynamical consequences.
%
%
%}
%\end{itemize}
%\end{frame}

%slide 16 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%\begin{frame}
%\begin{itemize} 
%
%{\bf 
%\item[${\color{red}\bullet}$] The relative orbital angular momentum
%and total spin involve a light-front boost to the rest frame.
%
%\vfill
%
%\item[${\color{red}\bullet}$] While the boost is kinematic, the parameters 
%of the boost that identify the physical (vs kinematic) 
%rest frame of the interacting system are dynamical. 
%
%\[
%P^+=P^- = M = M_0+V 
%\]
%
%\[
%j_{if} = {1 \over 2} \epsilon_{ijk}
%B_f^{-1}(P/M)^i{}_{\mu} B_f^{-1}(P/M)^j{}_{\nu}M^{\mu \nu}  
%\]
%
%\[
%M_I \to \sqrt{M_{Ia}^2 +\mathbf{k}^2} + \sqrt{M_{Ib}^2 +\mathbf{k}^2}
%\]
%
%\vfill
%
%\item[${\color{red}\bullet}$] The physical relative orbital angular 
%momentum depends on the dynamics. 
%}
%\end{itemize}
%
%
%\end{frame}

%slide 17 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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

\item[${\color{red}\bullet}$] Ekstein's theorem:
$S_{ij}\otimes S_k = S'_{(ij)(k)} \to $ 
there exists kinematically-invariant, short-range, unitary operators $A_{(ij)(k)}$: 
\[
\lim_{x^+ \to \pm \infty} \Vert (I-A_{(ij)(k)}) U_{0}(I,x^+) \vert \psi \rangle
\Vert =0
\]
\item[] satisfying
\[
U_{ij}(\Lambda ,a)\otimes U_k(\Lambda, a)= A_{(ij)(k)}  
U'_{(ij)(k)} (\Lambda,a) A^{\dagger}_{(ij)(k)}
\]

\[
A := e^{\ln (A_{(12)(3)})+ \ln (A_{(23)(1)}) + \ln (A_{(31)(3)})}
\]

\[
M^2 = A M^{\prime 2} A^{\dagger} =
A ( \sum A^{\dagger}_{(ij)(k)} M^2_{(ij)\otimes k}A_{(ij)(k)}
- 2 M_{0}^2) A^{\dagger}  
\]

\[
\mathbf{j}_{fI} = A \mathbf{j}_{f0} A^{\dagger} \qquad
\mbox{\bf dynamical spin}
\]

}
\end{itemize}
\end{frame}

%slide 18 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{itemize}
{\bf
\item[${\color{red}\bullet}$] Generators
constructed from $M^2$, $\mathbf{j}_{fI}$ 
and kinematic generators
give a dynamical light-front representation of 
the Poincar\'e group satisfying cluster properties.  

\item[${\color{red}\bullet}$] Construction due to Sokolov.

\vfill

\item[${\color{red}\bullet}$] Cluster properties and full rotational 
covariance satisfied.
 
\vfill

\item[${\color{red}\bullet}$] Dynamics is well defined - no issues with
ill-posed initial value problems.  Generators are self-adjoint operators.

\vfill

%\item[${\color{red}\bullet}$] Corrections to rotational covariance have 
%dynamical consequences.
%
%\vfill
%
%
%\vfill
  
\item[${\color{red}\bullet}$] Light front spin
and mass are dynamical many-body operators.

\vfill

\item[${\color{red}\bullet}$] Construction generic - always possible to
transform any light-front $U(\Lambda, b)$ to an S-matix preserving
representation with a kinematic spin.  The transform destroys cluster
properties: 
\[
U(\Lambda ,a) \to U_{ij}(\Lambda ,a)
\otimes U_k(\Lambda ,a) \to 
U_1(\Lambda ,a) \otimes U_2(\Lambda ,a) \otimes U_3(\Lambda ,a)
\]
}
\end{itemize}

\end{frame}

%slide 19 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Particles $\to$ free fields
}
\end{center}

\begin{itemize}
{\bf 
\item[] Local right or left handed local, mass $m$ spin $j$ light-front {\color{red}
free} fields can 
be constructed from occupation number representation using Lorentz-covariant 
single particle-wave functions:
}
\end{itemize}
\vfill
\[
\Psi_{rf} (x) = 
{Z  \over (2 \pi)^{3/2}} \int 
{d \tilde{\mathbf{p}} \theta(p^+) \over \sqrt{p^+}}
(e^{ip \cdot x}
D^j_{\mu \nu}
(B_f(p/m ))
a^{}_f(\tilde{\mathbf{p}},\nu)
\]
\[
+
e^{-ip \cdot x} 
D^j_{\mu \nu}
({B}_f(p/m )\sigma_2 )
b^{\dagger}_f(\tilde{\mathbf{p}},\mu)) 
\]
\vfill
\[
\Psi_{lf} (x) = 
{Z  \over (2 \pi)^{3/2}} \int 
{d \tilde{\mathbf{p}} \theta(p^+) \over \sqrt{p^+}}
( e^{ip \cdot x}
D^j_{\mu \nu}
(B_f^{\dagger -1}(p/m ))
a^{}_f(\tilde{\mathbf{p}},\nu) +
\]
\[
e^{-ip \cdot x} 
D^j_{\mu \nu}
({B}_f^{\dagger -1}(p/m )\sigma_2 )
b^{\dagger}_f(\tilde{\mathbf{p}},\mu)) 
\]

\end{frame}

%slide 20 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Free fields
}
\end{center}

\begin{itemize}
{\bf 
\item[] Free fields transform covariantly
\[    
U(\Lambda ,b) \Psi_{rf\mu} (x) U^{\dagger} (\Lambda ,b) =
\sum D^j_{\mu \nu}(\Lambda^{-1}) \Psi_{rf\nu} (\Lambda x+b )
\]
\[
U(\Lambda ,b) \Psi_{lf\nu} (x) U^{\dagger} (\Lambda ,b) =
\sum D^j_{\mu \nu}(\Lambda^{\dagger}) \Psi_{lf\nu} (\Lambda x+b )
\]
  
\vfill 

\item[] Fields constructed from different irreducible
bases (instant form, point front) are related by variable
changes.

\vfill

\item[] These fields are operator-valued distributions.
}
\end{itemize}

\end{frame}

%slide 21 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf
Interacting fields
}
\end{center}
\begin{center}
{\bf
Dynamical issues in light-front quantization:
}
\end{center}

\begin{itemize} 
{\bf 
\item[${\color{red}\bullet}$] Noether's theorem
    gives formal expressions for the Poincar\'e generators as
    operators integrated over the light-front
    hyperplane.  
\item[${\color{red}\bullet}$] Free field operators
    restricted to the light-front hyperplane are irreducible (i.e. can
    be used to represent any operator).  
\item[${\color{red}\bullet}$]
  Spectral condition on $P^+$ {\color{red}suggests} that the physical
  vacuum is identical to the free-field vacuum.  
\item[${\color{red}\bullet}$]
    {\color{red}Problem 1:} Noether's theorem does not directly tell how
    to express the dynamical generators in terms of the irreducible
    algebra.
%\item[${\color{red}\bullet}$]
%How does one represent
%${\partial \phi (x) \over \partial x^+}_{\vert x^+=0}$ in terms of   
%$\phi (\tilde{\mathbf{x}})$?

}
\end{itemize}
\end{frame}

%slide 22 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Irreducibility - free scalar field case
}
\end{center}

\[
\phi(x) = {1 \over (2 \pi)^{3/2} }
\int
{d \tilde{\mathbf{p}}\theta (p^+)\over \sqrt{p^+}}
(e^{i p \cdot x} a(\tilde{\mathbf{p}})+
e^{-i p \cdot x} 
a^{\dagger}(\tilde{\mathbf{p}})
)
\]

\begin{center}
{\bf  
Restriction to light front irreducible
}
\end{center}

\[
\phi (\tilde{\mathbf{p}}) = {1 \over 2 (2 \pi)^{3/2}} 
\int_{x^+=0}  e^{-i \tilde{\mathbf{x}}
\cdot \tilde{\mathbf{p}}} \phi(\tilde{\mathbf{x}} ,0 ) d\tilde{\mathbf{x}}
\]

\[
a(\tilde{\mathbf{p}}) =\sqrt{p^+} \theta (p^+) \phi(\tilde{\mathbf{p}})
\]

\[
a^{\dagger}(\tilde{\mathbf{p}}) =\sqrt{p^+} \theta (p^+) \phi(-\tilde{\mathbf{p}})
\]
 
\begin{itemize}
{\bf
\item[] $[O,\phi (\tilde{\mathbf{x}},0)]=0$ $\forall \tilde{\mathbf{x}}$ 
implies $O$ proportional to identity (exception $p^+=0$ contributions) 

\item[] $p^+=0$ is strongly suppressed for operators constructed out
of (4d) smeared fields.
}
\end{itemize}

\end{frame}

%slide 23 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Noether generators - dynamical theory
}
\end{center}
\[
P^+ = 4
\int_{x^+=0} d\tilde{\mathbf{x}}: {\partial \phi(x) \over \partial x^-}
{\partial \phi(x) \over \partial x^-} :
\]
\[
P^i = 2 
\int_{x^+=0} d\tilde{\mathbf{x}} :{\partial \phi(x) \over \partial x^-}
{\partial \phi(x) \over \partial x^i}: 
\]
\[
J^3=
2 \int_{x^+=0} d\tilde{\mathbf{x}} 
:{\partial \phi(x) \over \partial x^-}
(
x^j {\partial \phi(x) \over \partial x^i}
-
x^i
{\partial \phi(x) \over \partial x^j}):
\]
\[
E^i =
2 \int_{x^+=0} d\tilde{\mathbf{x}} :{\partial \phi(x) \over \partial x^-}
(2
x^i
{\partial \phi(x) \over \partial x^-} -
x^+ {\partial \phi(x) \over \partial x^i}
):
\]
\[
K^3=
4 \int_{x^+=0} d\tilde{\mathbf{x}} :{\partial \phi(x) \over \partial x^-}
({\partial \phi(x) \over \partial x^-} x^- -
{\color{blue}{\partial \phi(x) \over \partial x^+} x^+}):
\]
\[
P^- = 4 
\int_{x^+=0} d\tilde{\mathbf{x}} :{\partial \phi(x) \over \partial x^-}
{\color{red}{\partial \phi(x) \over \partial x^+}} :
\]
\[
F^i =
2 \int_{x^+=0} d\tilde{\mathbf{x}} :{\partial \phi(x) \over \partial x^-}
(
2 x^i
{\color{red}{\partial \phi(x) \over \partial x^+}} -
x^- {\partial \phi(x) \over \partial x^i}):
\]

\end{frame}

%slide 24 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{itemize}
{\bf 
\item [${\color{red}\bullet}$] Noether's generators on light front
$+$ irreducible set of fields on the light front {\color{red}insufficient} to
define the dynamics (characteristic initial-value surface problem).

\vfill

\item [${\color{red}\bullet}$] Additional assumptions are needed to
define self-adjoint dynamical generators of unitary one-parameter subgroups.

\vfill

\item [${\color{red}\bullet}$] Physical constraint - dynamics 
should lead to the Wightman functions of a local field theory.

\vfill
  
\item [${\color{red}\therefore}$] Additional input: Use the field theory - field equations + 
integration by parts
}
\end{itemize}

\[
{\partial^2 \phi(x) \over\partial x^- \partial x^+} =
(\nabla_{\perp}^2 - m^2) \phi (x) - {d{\cal V}\over d\phi(x)} 
\]
\end{frame}

%slide 25 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\[
P^- = 4 
\int_{x^+=0} d\tilde{\mathbf{x}} :{\partial \phi(x) \over \partial x^-}
{\color{red}{\partial \phi(x) \over \partial x^+}} :=
\]
\[
-4 
\int_{x^+=0} d\tilde{\mathbf{x}} :\phi(x) 
{\color{red}{\partial^2 \phi(x) \over\partial x^- \partial x^+}} :=
\]
\[
4 
\int_{x^+=0} d\tilde{\mathbf{x}} :\phi(x) 
{\color{red}(
-\nabla_{\perp}^2 \phi (x)
+ m^2 \phi (x) +
{d{\cal V} \over d\phi}(x)
)} :
\]
\[
F^i =
2 \int_{x^+=0} d\tilde{\mathbf{x}} :{\partial \phi(x) \over \partial x^-}
(
2 x^i
{\color{red}{\partial \phi(x) \over \partial x^+}} -
x^- {\partial \phi(x) \over \partial x^i}):=
\]

\[
-2 \int_{x^+=0} d\tilde{\mathbf{x}} :\phi(x) 
(
2 x^i
{\color{red}{\partial^2 \phi(x) \over \partial x^-\partial x^+}} -
x^- {\partial^2 \phi(x) \over \partial x^-\partial x^i}):=
\]

\[
2 \int_{x^+=0} d\tilde{\mathbf{x}} :\phi(x)
(
2 x^i(
{\color{red} -\nabla_{\perp}^2 \phi (x)
+ m^2 \phi (x) +
{d{\cal V} \over d\phi}(x)))} -
x^- {\partial^2 \phi(x) \over \partial x^-\partial x^i}):
\]
\end{frame}

%slide 26 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{itemize}
{\bf 
\item[${\color{red}\bullet}$]  Generators now {\color{red} formally} 
defined as operators on the irreducible light-front algebra.
}  
\end{itemize}  

\begin{center} 
{\bf 
Free field case
}
\end{center}

\begin{itemize}
{\bf 
\item[${\color{red}\bullet}$]  Free field

\[
P^- = \int d\tilde{\mathbf{p}} a^{\dagger}(\tilde{\mathbf{p}})
{\mathbf{p}_{\perp}^2 + m^2 \over p^+}a (\tilde{\mathbf{p}})   
\]

\item[${\color{red}\bullet}$]  Light-front Heisenberg equations
generate $x^+$ dependence:
\[
{d \phi (x) \over dx^+}  = i [P^-, \phi(x)] 
\]

\item[${\color{red}\bullet}$] The solution can be expressed in terms
of the light-front creation and annihilation operators which can
be expressed in terms of the irreducible algebra of fields on the 
light front.
}  
\end{itemize}  
\end{frame}

%slide 27 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\[
\phi (x) = \int F_m (x,\tilde{\mathbf{y}}) \phi ( \tilde{\mathbf{y}},0)
d\tilde{\mathbf{y}}
\]

\[
F_m (x,\tilde{\mathbf{y}})= {1 \over 2(2\pi)^3}
\int d \tilde{\mathbf{p}} 
e^{-{i\over 2} {\mathbf{p}_{\perp}^2 + m^2 \over p^+}x^+ + i
\tilde{\mathbf{p}} \cdot (\tilde{\mathbf{x}}-\tilde{\mathbf{y}})}
\]


\[
{1 \over 2(2\pi)^3}
\int d^4x f(x^+,\tilde{\mathbf{x}})
e^{-{i\over 2} {\mathbf{p}_{\perp}^2 + m^2 \over p^+}x^+ + i
\tilde{\mathbf{p}} \cdot \tilde{\mathbf{x}}} =
{f}_{ft}({\mathbf{p}_{\perp}^2 + m^2 \over p^+},\tilde{\mathbf{p}}) 
\]

\begin{itemize}
{\bf 
\item[${\color{red}\bullet}$] 
$f_{ft} (\tilde{\mathbf{p}}, {\mathbf{p}_{\perp}^2 + m^2 \over p^+})$
vanishes strongly near $p^+ =0$.  

\item[] All spectral information is lost for $x^+=0$
}
\end{itemize}
\end{frame}

%slide 28 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

%\begin{frame}
%
%\[
%\phi (x) = \int F_m (x,\tilde{\mathbf{y}}) \phi
%  (\tilde{\mathbf{y}})d\tilde{\mathbf{y}}
%\]
%
%\[
%{1 \over (2\pi)^3} \int {d\tilde{\mathbf{p}} \over p^+} e^{-i
%    {\mathbf{p}_{\perp}^2 +m^2 \over p^+}x^+ + i
%    \tilde{\mathbf{p}} \cdot (\tilde{\mathbf{x}}- \tilde{\mathbf{y}})}
%  \phi (\tilde{\mathbf{y}})d\tilde{\mathbf{y}}
%\]
%
%\[
%\int_0^a {e^{ic/p^+}\over p^+} dp^+ =
%{\pi \over 2} - (Ci ({c \over a}) + i Si ({c \over a}))
%\]
%
%\begin{itemize}
%{\bf 
%\item[] All spectral information is lost without an integral over $x^+$
%}
%\end{itemize}
%\end{frame}

\begin{frame}
\[
\langle 0 \vert \phi(x) \phi(y) \vert 0 \rangle =
\]
\[
\int F_m (x,\tilde{\mathbf{z}})
F_m (y,\tilde{\mathbf{w}})
{\color{red} \langle 0 \vert \phi(\tilde{\mathbf{z}},0) \phi(\tilde{\mathbf{w}},0) \vert 0 \rangle}
d\tilde{\mathbf{z}}d\tilde{\mathbf{w}} =  W_{20}(s) =
\]
\[
-i {\epsilon (s^0) \delta (s_0^2 -\mathbf{s}^2) \over 4 \pi} 
\]
\[
+ {i m \theta (s_0^2 -\mathbf{s}^2)\over 
8 \pi \sqrt{(s_0^2 -\mathbf{s}^2)}}
(\epsilon (s^0)J_1 (m\sqrt{(s_0^2 -\mathbf{s}^2)})
\]
\[
- i N_1 (m\sqrt{(s_0^2 -\mathbf{s}^2)}))
\]
\[
- { m \theta (\mathbf{s}^2- s_0^2)
\over 4 \pi^2 \sqrt{\mathbf{s}^2- s_0^2 }}
K_1 (m \sqrt{\mathbf{s}^2- s_0^2 })
\]


\end{frame}

%slide 29 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{itemize}
{\bf
\item[] Limit $z^+ \to 0$:
}
\end{itemize}
\[
\langle 0 \vert \phi (x) \phi (y) \vert 0 \rangle  \to 
-i {\epsilon (z^-) \delta (\mathbf{z}_{\perp}^2) \over 4 \pi} 
- { m 
\over 4 \pi^2 \sqrt{\mathbf{z}_{\perp}^2 }}
K_1 (m \sqrt{\mathbf{z}^2_{\perp} }).
\]
\begin{center}
{\bf (Lorentz invariant - remembers mass)} 
\end{center}
\begin{itemize}
{\bf
\item[] Direct calculation on light front
}
\end{itemize}
\[
\langle 0 \vert \phi (\tilde{\mathbf{x}}) 
\phi (\tilde{\mathbf{y}}) \vert 0 \rangle =
{1 \over (2 \pi)^3}
\int {d\tilde{\mathbf{p}}\theta (p^+) \over 2p^+} 
e^{i \tilde{\mathbf{p}}\cdot   \tilde{\mathbf{z}}}.
\]
\begin{center}
{\bf (not Lorentz invariant - does not know about mass)}
\end{center}
\begin{center}
{\bf (all information about spectrum lost)}
\end{center}
\begin{center}
{\bf limit of integral $\not=$ integral of limit}
\end{center}
\end{frame}

%slide 30 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Remark on Haag's theorem
}
\end{center}

\begin{itemize}
{\bf 
\item[] Different mass free fields restricted to the light front
are unitarily equivalent.
\vfill
\item[] Extensions to local $4d$ algebra with different $F_m$ are inequivalent 
}
\end{itemize}
\[
f \sim_{m_1} g \qquad \phi_{m_1} (f) = \phi_{m_1} (g)
\]
\vfill
\[
f \not\sim_{m_2} g \qquad
\phi_{m_2} (f) \not= \phi_{m_2} (g)
\]

\begin{itemize}
{\bf 
\item[] Schlieder and Seiler show that different mass fields
restricted to fixed-time surfaces also become unitarily equivalent
for suitable restrictions of the space of test functions. 
}
\end{itemize}


\end{frame}

%slide 31 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf
Representations of Heisenberg fields in the light front algebra
}
\end{center}

\begin{center}
{\bf
Hagg expansion - irreducibility of in(out)  fields
}
\end{center}

\[
\phi (x) = \sum_{n=0}^\infty {1 \over n!}
\int d^4x_1 \cdots d^4x_n L_n (x; x_1, \cdots ,x_n) 
:\phi_{in} (x_1) \cdots \phi_{in} (x_n):.
\]
\[
\phi (x) = \sum_{n=0}^\infty {1 \over n!}
\int d^4x_1 \cdots d^4x_n L_n (x; x_1, \cdots ,x_n)
\times 
\]
\[
F_{m_1}(x_1^+;\tilde{\mathbf{x}}_1-\tilde{\mathbf{y}}_1) 
\cdots 
F_{m_n}(x_n^+;\tilde{\mathbf{x}}_n-\tilde{\mathbf{y}}_n)
\times
\]
\[
d\tilde{\mathbf{y}}_1 \cdots d\tilde{\mathbf{y}}_n
:\tilde{\phi}_{10} (\tilde{\mathbf{y}}_1)
\cdots
\tilde{\phi}_{n0} (\tilde{\mathbf{y}}_n):
\]

\begin{itemize}
{\bf 
\item [${\color{red}\bullet}$] Maps smeared fields into light front 
algebra. 
}
\end{itemize}

\end{frame}


%slide 32 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf   
{\color{red} Problem 2}: Interacting fields, zero modes and renormalization
}
\end{center}

\begin{center}
{\bf 
Fuda operator\\
(motivated by V. Karmanov)
}
\end{center}  
\[
U(\theta) = e^{i \mathbf{J} \cdot \pmb{\theta}} e^{-i \mathbf{J}_0 \cdot \pmb{\theta}} 
\]
\begin{itemize}
{\bf 
\item [${\color{red}\bullet}$] Changes the orientation of the light front.  
For finite $\#$ of DOF this is an $S$-matrix-preserving unitary transformation.

\vfill

\item [${\color{red}\bullet}$] Existence of Fuda operators = rotational 
invariance.

\vfill

\item [${\color{red}\bullet}$] A Fuda operator must exist for field 
operators - otherwise S-matrix elements would depend on the choice 
of light front.

\vfill

\item [${\color{red}\bullet}$] $p^+ \to 0$ with one light front corresponds 
to $p_i \to - \infty$
in a theory with a different light front.

%\vfill
%
%\item [${\color{red}\bullet}$]  Generators now {\color{red} formally} 
%defined as operators on the irreducible light-front algebra.
}  
\end{itemize}  
\end{frame}

%slide 33 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

%\begin{center}
%{\bf 
%Fuda operator - implications
%}
%\end{center}  

\begin{itemize}
{\bf 
\item [${\color{red}\bullet}$] Dynamical generators involve products of local fields at
the same point.

\vfill

\item[${\color{red}\bullet}$] Local operator products are not well-defined 
operator valued distributions.

\vfill

\item [${\color{red}\bullet}$] Rotational covariance implies ultraviolet infinities in one 
light front become $p^+=0$ infinites with a different light front.

\vfill

\item [${\color{red}\bullet}$] Renormalization is constrained by rotational covariance - 
ultraviolet and infrared renormalization are not independent.    

}
\end{itemize}
\end{frame}

%slide 34 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


\begin{frame}
\begin{center}
{\bf
Conventional wisdom on the spectral condition on $P^+$ in
interacting theories
}    
\end{center}

\[
[V,P^+]=0 
\]

\[
P^+ V \vert 0 \rangle = V P^+ \vert 0 \rangle  = 0
\]

\[
V \vert 0 \rangle =0 \qquad \mbox{or} 
\]

\[
V \vert 0 \rangle \not=0  
\]

\begin{center}
{\bf
and is an eigenstate of $P^+$ with eigenvalue 0
}
\end{center}

\begin{center}
{\bf
This is the origin of assertion that the light front vacuum is the 
free Fock vacuum.
}
\end{center}


\end{frame}

%slide 35 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf 
What really happens - example $: \phi^4 (\tilde{\mathbf{x}}):$
}
\end{center} 
\[
:V:\vert 0 \rangle = 
\int d\tilde{\mathbf{x}} :\phi^4 (\tilde{\mathbf{x}}):\vert 0 \rangle  =
\]
\[
{2 (2\pi)^2}\int {\color{red}p^+ \theta (p^+) \delta (p^+) dp^+}
\prod d\mathbf{p}_{i\perp}
{d \xi_1 d\xi_2 d\xi_3 d\xi_4 (1 -\sum_n \xi_n ) \over 
\sqrt{\xi_1 \xi_2 \xi_3 \xi_4}}
\times
\]
\[
\delta (\sum \mathbf{p}_{i \perp}) 
\prod a^{\dagger}(\xi_i\mathbf{p}^+,\mathbf{p}_{_i\perp })\vert 0 \rangle
\]
\begin{itemize}
{\bf 
\item [] which looks like the interaction should annihilate the vacuum, 
{\color{red}however}
%$\Vert :V:\vert 0 \rangle \Vert^2 =  \langle 0 \vert :V::V:\vert 0 \rangle$  
%is ill-defined.

\[
\Vert :V:\vert 0 \rangle \Vert^2 \to 
\int {\delta (p^+)^2 \theta(p^+) \over p^+} 
\cdots 
\]

\item [{\color{red}$\bullet$}] is not defined!   

}
\end{itemize}
\end{frame}

%slide 36 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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

\item [{\color{red}$\bullet$}] The assertion that $:V:$ annihilates 
the vacuum is misleading.  Quantities like $\delta (p^+)^2$, 
$\delta (p^+) \theta(p^+)$ are not distributions.

\vfill 
  
\item [{\color{red}$\bullet$}] Local operator products in the 
light-front Hamiltonian require renormalization due to singularities 
that arise from $p^+=0$ contributions.

\vfill

\item [{\color{red}$\bullet$}] This suggests some dynamical
modifications of the light front vacuum associated with $p^+=0$
are also needed.  

%\vfill
%
%\item [{\color{red}$\bullet$}] Normal ordering normally involves 
%recursively subtracting intermediate vacuum contributions from operator 
%products.  Algebraic normal ordering is ignorant of the contributions
%to the vacuum associated with $p^+=0$.  

}
\end{itemize}

\end{frame} 

%slide 37 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf  
The light-front vacuum generating functional
}
\end{center}

\[
\phi (\tilde{f}) = \phi_+(\tilde{f})+  \phi_-(\tilde{f})
\]
\begin{center}
{\bf 
$\phi_+(\tilde{f})$ increases $p^+$ \qquad 
$\phi_-(\tilde{f})$ decreases $p^+$
}
\end{center}

\begin{center}
{\bf   
{\color{red}Algebraic} normal ordering on free light-front algebra
(independent of mass spectrum)
}
\end{center}

\[
:e^{i \phi (\tilde{f}) }: = e^{i \phi_+(\tilde{f})}
e^{i\phi_-(\tilde{f})}
\]
\[
e^{i \phi (\tilde{f}) } = :e^{i \phi (\tilde{f}) }:e^{{1\over 2}(\tilde{f},\tilde{f})}
\]
\[
(\tilde{f},\tilde{g}) = \int {d\tilde{\mathbf{p}}\theta(p^+) \over
p^+}\tilde{f}(-\tilde{\mathbf{p}})\tilde{g}(\tilde{\mathbf{p}})
\]

\begin{center}
{\bf   
Fock vacuum
}
\end{center}

\[
\langle 0 \vert :e^{i \phi (\tilde{f}) }: \vert 0 \rangle =1
\]
\[
E_{Fock} [\tilde{f}] :=
\langle 0 \vert e^{i \phi (\tilde{f})} \vert 0 \rangle =
e^{{1\over 2}(\tilde{f},\tilde{f})}
\]

\end{frame}   

%slide 38 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf   
Non-Fock vacuum
}
\end{center}

\begin{center}{\bf 
Non-zero contributions from $p^+=0$
}
\end{center}
\[
\langle 0 \vert :e^{i \phi (\tilde{f}) }: \vert 0 \rangle =
1+\sum \langle 0 \vert : \phi (f)^n : \vert 0 \rangle 
\]
\begin{center}
{\bf 
Cluster properties imply:
}
\end{center}
\[
\langle 0 \vert :e^{i \phi (\tilde{f}) }: \vert 0 \rangle
= e^{\sum {i^n \over n!}w_{cn} (f)}
\]
\[
E[\tilde{f}] = E_{Fock} [\tilde{f}]e^{\sum {i^n \over n!}w_{cn} (f)}
\]

\begin{itemize}
{\bf
\item[{\color{red}$\bullet$}] $w_{cn}$ multilinear 
connected functionals.  They can only depend on  
$\mathbf{p}_{i\perp}$ and {\color{red}test functions and their derivatives
at $p^+=0$.}
}
\end{itemize}
\end{frame}

%slide 39 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{itemize}
{\bf 
\item[{\color{red}$\bullet$}] The dynamical requirement
\[
P^- \vert 0 \rangle =0
\]
\item[{\color{red}$\bullet$}] should relate the $p^+=0$ contibutions 
to the vacuum to 
the $p^+=0$ contibutions to the renormalized $P_-$.

\vfill

\item [{\color{red}$\bullet$}] Normal ordering normally involves 
recursively subtracting intermediate vacuum contributions from operator 
products.  Algebraic normal ordering is ignorant of the contributions
to the vacuum associated with $p^+=0$ and knows nothing about the
mass spectrum of the theory.

\vfill

\item [{\color{red}$\bullet$}] Refining the definition of normal ordering to
remove the singular $p^+=0$ contributions to the algebraically
normal ordered $:P^-$: results in a modification of the Fock vacuum
generating functional that is limited to $p^+=0$ contributions.

}
\end{itemize}
\end{frame}  

%slide 40 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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

\begin{itemize} {\bf 

\item [{\color{red}$\bullet$}] There are no
issues with ill-posed initial value problems with light-front
treatments of interacting particles.  

\item [{\color{red}$\bullet$}] Light-front dynamics is scattering
equivalent (unitary $+$ S-matrix preserving) in any of Dirac's forms
of dynamics.

\item [{\color{red}$\bullet$}] Dynamics requires constructing self-adjoint
generators on the Hilbert space, which is the
direct integral of Poincar\'e irreducible subspaces.  

\item [{\color{red}$\bullet$}] Free-particle states can be used to build
local free fields.  Restrictions of these field operators to the
light-front are irreducible.

}
\end{itemize}
\end{frame}
\begin{frame}

%slide 41 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx  

\begin{center}
{\bf
Summary - continued
}  
\end{center}

\begin{itemize} {\bf 

\item [{\color{red}$\bullet$}] Noether's theorem on the light front - 
plus irreducible fields on the light front insufficient to define the dynamics.
    
\item [{\color{red}$\bullet$}] Need a self-adjoint $P^-$.  Field equations 
define the dynamics. Need renormalization associated with $p^+=0$. 

\item [{\color{red}$\bullet$}] Solving $P^-\vert 0 \rangle = 0$ 
relates the problem of defining $P^-$ and constructing a non-Fock vacuum.

}
\end{itemize}
\end{frame}

%slide 42 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Open questions \\
(for the students in the audience) 
}
\end{center} 

\begin{itemize} {\bf
\item [{\color{red}$\bullet$}] Role of rotational covariance in 
relating infrared and ultraviolet divergences?

\item [{\color{red}$\bullet$}] Is there a systematic way to 
simultaneously construct a self-adjoint $P^-$ and compatible vacuum?

\item [{\color{red}$\bullet$}] Both regularization and non-trivial
vacuum contributions at $p^+=0$ tend to break light-front scale 
invariance.  How does it get recovered with a compatible choice of 
dynamics and vacuum?

\item [{\color{red}$\bullet$}] More information: See next four 
talks by Collins, Brodsky, Beane and Hiller 

}
\end{itemize}
\begin{center}
{\bf{ \color{blue}
Thanks to the conference organizers and sponsors!
}}
\end{center}

\end{frame}

\end{document}

