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

\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}

%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 dynamics of particles - finite number of DOF.
\vfill
\item[$\bullet$] Particles $\to$ fields.
\vfill
\item[$\bullet$] Light-front operator algebra.
\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}
\[
X \to 
\left (
\begin{array}{cc}
{\color{red}0} & x_{\perp}^*\\
x_{\perp} & x^- 
\end{array}
\right )
\qquad \tilde{\mathbf{x}}= (x^-,\mathbf{x}_{\perp})
\]
\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}
\begin{center}
{\bf 
Relativistic particles
}
\end{center}
\[
U(\Lambda ,a)
\]
\[
\Downarrow
\]
\[
\{P^{\mu},M^{\mu\nu} \}
\]
\[
\Downarrow
\]
\begin{center}
{\bf 
Commuting observables:
}
\end{center}
\[
\vert (m,j)\tilde{\mathbf{p}},\mu \rangle_f  \qquad 
\tilde{\mathbf{p}} : = (p^+,\mathbf{p}_{\perp}) \qquad
p^- = {\mathbf{p}_{\perp}^2 + m^2 \over p^+} 
\]
\[
j_{fi} = {1 \over 2} \epsilon_{ijk} B_f^{-1}(P/M)^i{}_{\mu}
B_f^{-1}(P/M)^j{}_{\nu}M^{\mu \nu}    
\]
\begin{center}
{\bf Mass $m$ spin $j$ irreducible unitary representation of P.G.:\\}
\end{center}
\[
\Downarrow
\]
\[
U(\Lambda ,a) 
\vert (m,j)\tilde{\mathbf{p}},\mu \rangle_f 
=
\]
\[
\sum 
e^{i \Lambda p\cdot a}
\vert (m,j)\tilde{\pmb{\Lambda}}p,\nu \rangle_f 
D^j_{\nu \mu}[B_f^{-1} (\Lambda p/m) \Lambda B_f(p/m)]
\sqrt{{ (\Lambda p)^+ \over p^+}}
\]

\end{frame}


\begin{frame}

\begin{center} 
{\bf 
Free-particle dynamics: $U(I, x^+)$.
}
\end{center}

\vfill
  
\begin{center}
{\bf $(\Lambda,b) \in $ kinematic subgroup  \\}
\end{center}
\[
\Downarrow
\]
\[
\{
e^{i \Lambda p\cdot b}, 
R_{wf}(\Lambda,p) :=  B_f^{-1} (\Lambda p/m) \Lambda B_f(p/m)],
\sqrt{{ (\Lambda p)^+ \over p^+}}
\}
\]

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

\vfill

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

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

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

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

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


\end{frame} 

%slide 7 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


%slide 8 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf Systems of free particles:}
\end{center}

\[
\vert (m_1,j_1) \tilde{\mathbf{p}}_1,\mu_1 \rangle_f 
\cdots \vert (m_n,j_n) \tilde{\mathbf{p}}_n,\mu_n \rangle_f 
\]

\begin{center}
{\bf Infinitesimal generators: $G_0=\sum_{n} G_n \otimes I_{!n} $.
}
\end{center}


%\begin{center}
%{\bf System 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}
\[
\vert (m_0,j_0) \tilde{\mathbf{p}}_0,\mu_0; \mathbf{d}  \rangle_{0f}
\]
\[
\mathbf{d} = \mbox{\bf invariant degeneracy parameters}
\]

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


\end{frame}


%slide 10 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

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

\begin{itemize}
{\bf 
\item[] Problem: 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)\\
set $\mathbf{j}=\mathbf{j}_0$,  
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}
\[
\vert (m_0,j_0 ) \tilde{\mathbf{p}}_0, \mu_0 ; l_0,s_0\rangle
\qquad \mathbf{d}=\{l_0,s_0\}
\]

\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:   
This gives a complete set of irreducible eigenstates with a light-front
kinematic subgroup that is
consistent with 
the dynamics of $M^2$.}
\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}


\begin{frame}
\[
N\geq 3
\]
\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[${\color{red}\bullet}$] The construction can be
generalized to any number of particles.

\vfill
 
\item[${\color{red}\bullet}$] Cluster properties require
a dynamical spin:
      
\[ U_{ij}(\Lambda,a) \otimes U_k(\Lambda ,a) \quad \to \quad
        \mathbf{j}_{(ij)\otimes
          k}\not=\mathbf{j}_0  
\] 


\item[${\color{red}\bullet}$] Key tool: Tensor product representations are
related to representations with non-interacting spin by
S-matrix-preserving unitary transformations.

\vfill

\item[${\color{red}\bullet}$] Cluster
properties require all dynamical operators are many-body
operators.  


%\item[${\color{red}\bullet}$] Dynamics requires specifying a $P^-$, but 
%it is well-defined - no issues with ill-posed initial value problems.
%
%\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[${\color{red}\bullet}$] Dirac's other forms of dynamics - same 
construction; start with different sets of commuting observables.

\vfill

\item[${\color{red}\bullet}$] Dynamics 
in all of Dirac's other forms of dynamics related by an $S$-matrix preserving 
unitary transformation.

%\vfill
%\item[${\color{red}\bullet}$] Construction generic - always possible to
%transform any light-front $U(\Lambda, b)$ to an S-matrix preserving
%representation with $\mathbf{j}_f=\mathbf{j}_{0f}$.
  
%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}

\begin{frame}
  \begin{center}
{\bf
Particles $\to$ fields: 
} 
\end{center}
\begin{center}
{\bf
Lorentz covariant particle states: 
} 
\end{center}

\[
U(\Lambda ,0) 
\vert (m,j)\tilde{\mathbf{p}},\mu \rangle_f 
=
\]
\[
\sum 
\vert (m,j)\tilde{\pmb{\Lambda}}p,\nu \rangle_f 
D^j_{\nu \mu}[B_f^{-1} (\Lambda p) \Lambda B_f(p)] 
\sqrt{{ (\Lambda p)^+ \over p^+}}
\]
\[
\Downarrow
\]
\[
U(\Lambda ,0) 
%\underbrace{
{\color{blue}\vert (m,j)\tilde{\mathbf{p}},\mu \rangle_f 
D^j_{\nu \mu}[B_f^{-1} (p)]\sqrt{p^+}}
%}_{\mbox{\bf right hand covariant state}} 
=
\]
\[
\sum 
%\underbrace{
{\color{blue}\vert (m,j)\tilde{\pmb{\Lambda}}p,\nu \rangle_f 
D^j_{\nu \alpha}[B_f^{-1} (\Lambda p)]\sqrt{ (\Lambda p)^+ }}
%}_{\mbox{\bf right hand covariant state}}
{\color{red}D^j_{\alpha \mu}[\Lambda]} 
\]
\vfill
\[
U(\Lambda ,0) 
%\underbrace{
{\color{blue}
\vert (m,j)\tilde{\mathbf{p}},\mu \rangle_f 
D^j_{\nu \mu}[B_f^{\dagger} (p)]\sqrt{p^+}}
%}_{\mbox{\bf left hand covariant state}} 
=
\]
\[
\sum 
%\underbrace{
{\color{blue}\vert (m,j)\tilde{\pmb{\Lambda}}p,\nu \rangle_f 
D^j_{\nu \alpha}[B_f^{\dagger} (\Lambda p)] \sqrt{(\Lambda p)^+ }}
%}_{\mbox{\bf left hand covariant state}}
{\color{red}D^j_{\alpha \mu}[(\Lambda^{\dagger})^{-1}]} 
\]


\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




%slide 19 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

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

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

\end{frame}


%slide 20 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Free field dynamics:
}
\end{center}

%\item[] Free fields transform covariantly
\[    
U(\Lambda ,a) \Psi_{rf\mu} (x) U^{\dagger} (\Lambda ,a) =
\sum D^j_{\mu \nu}[\Lambda^{-1}] \Psi_{rf\nu} (\Lambda x+a )
\]
\vfill
\[
U(\Lambda ,a) \Psi_{lf\mu} (x) U^{\dagger} (\Lambda ,a) =
\sum D^j_{\mu \nu}[\Lambda^{\dagger}] \Psi_{lf\nu} (\Lambda x+a )
\]
  
\vfill 
\begin{itemize}
{\bf 
\item[${\color{red}\bullet}$] Free fields constructed using different 
Lorentz covariant irreducible bases (instant form, point front) 
are {\color{red}identical} to these. 
}
\end{itemize}

 
%\vfill
%
%\item[] These fields are operator-valued distributions.

\end{frame}


%slide 21 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf
Interacting fields
}
\end{center}
\begin{center}
{\bf
Light-front quantization:
}
\end{center}

\begin{itemize} 
{\bf 
\item[${\color{red}\bullet}$] Free field operators
restricted to the light-front hyperplane are irreducible (i.e. they can
be used to represent any operator).  

\vfill

\item[${\color{red}\bullet}$]
The spectral condition on $P^+$ {\color{red}suggests} that the physical
vacuum is identical to the free-field vacuum.  

\vfill

\item[${\color{red}\bullet}$] Noether's theorem
gives {\color{red}formal} expressions for the Poincar\'e generators as
operators integrated over the light-front
hyperplane.  


%\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
The conventional wisdom about the spectral condition on $P^+$ in
interacting theories:
}    
\end{center}

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

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

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

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

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

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

\end{frame}

\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 $O$ is proportional to the 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's theorem - kinematic generators  
}
\end{center}
\[
P^+ = 4
\int_{x^+=0} d\tilde{\mathbf{x}}: {\partial \phi(x) \over \partial x^-}
{\partial \phi(x) \over \partial x^-} :
\]
\[
P^i = 2 
\int_{x^+=0} d\tilde{\mathbf{x}} :{\partial \phi(x) \over \partial x^-}
{\partial \phi(x) \over \partial x^i}: 
\]
\[
J^3=
2 \int_{x^+=0} d\tilde{\mathbf{x}} 
:{\partial \phi(x) \over \partial x^-}
(
x^j {\partial \phi(x) \over \partial x^i}
-
x^i
{\partial \phi(x) \over \partial x^j}):
\]
\[
E^i =
2 \int_{x^+=0} d\tilde{\mathbf{x}} :{\partial \phi(x) \over \partial x^-}
(2
x^i
{\partial \phi(x) \over \partial x^-} -
x^+ {\partial \phi(x) \over \partial x^i}
):
\]
\[
K^3=
4 \int_{x^+=0} d\tilde{\mathbf{x}} :{\partial \phi(x) \over \partial x^-}
{\partial \phi(x) \over \partial x^-}: x^- 
\]


\end{frame}

%slide 25 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}
\begin{center}
{\bf
Noether's theorem - dynamical generators  
}
\end{center}
\[
P^- =
4 
\int_{x^+=0} d\tilde{\mathbf{x}} :\phi(x) 
{\color{black}(
-\nabla_{\perp}^2 \phi (x)
+ m^2 \phi (x) +
{d{\cal V} \over d\phi}(x)
)} :
\]
\[
F^i =
\]

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

%slide 26 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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

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

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

\begin{itemize}
{\bf
\item[] $F_m:$ local 4d field algebra $\to$ light-front field algebra.
}
\end{itemize}

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

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

\begin{itemize}
{\bf
\item[] Gives correct 2-point Wightman function using light-front Fock vacuum.
}
\end{itemize}

\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 (Schlieder and Seiler).
\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\\
($F_m(x,\tilde{\mathbf{y}})$ for interacting fields) 
}
\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 local Heisenberg field algebra 
into light-front Fock algebra. 

%\item [${\color{red}\bullet}$] Suggest that calculations could still
%use the algebra of free fields on a light front. 
}
\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   
Observations:
}
\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 [${\color{red}\bullet}$] Changing the orientation of the light front  
should not change the physics.

\vfill

\item [${\color{red}\bullet}$] Invariance under change of light-front 
orientation $=$ 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 [${\color{red}\bullet}$] Dynamical generators in local field theories
involve products of fields at the same point, which are not well defined.

\vfill

\item [${\color{red}\bullet}$] Ultraviolet and infrared renormalization 
are constrained by rotational covariance. 


%\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 in local field theories
%involve products of 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 $p^+=0$ infinities with a different 
%light front.
%
%\vfill
%
%\item [${\color{red}\bullet}$] Ultraviolet and infrared renormalization 
%are not independent, they are constrained by rotational covariance. 
%     
%\vfill 
%
%
%
%
%}
%\end{itemize}
%\end{frame}

%slide 34 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


%slide 35 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{frame}

\begin{center}
{\bf 
Is the light front vacuum really the Fock vacuum?
}
\end{center} 

\begin{center}
{\bf 
Example $: \phi^4 (\tilde{\mathbf{x}}):$
}
\end{center} 
\[
V\vert 0 \rangle = 
\int d\tilde{\mathbf{x}} :\phi^4 (\tilde{\mathbf{x}}):\vert 0 \rangle  =
\]
\[
{2 (2\pi)^2}\int {\color{red}p^+ \theta (p^+) \delta (p^+) dp^+}
\prod d\mathbf{p}_{i\perp}
{d \xi_1 d\xi_2 d\xi_3 d\xi_4 (1 -\sum_n \xi_n ) \over 
\sqrt{\xi_1 \xi_2 \xi_3 \xi_4}}
\times
\]
\[
\delta (\sum \mathbf{p}_{i \perp}) 
\prod a^{\dagger}(\xi_i\mathbf{p}^+,\mathbf{p}_{_i\perp })\vert 0 \rangle + \cdots 
\]
\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 [] is not defined of the free field Fock vacuum!   

}
\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 even
      distributions.  The singular terms ocurr at $p^+=0$.

\vfill 


\item [{\color{red}$\bullet$}] The light front Hamiltonian should
be a well-defined operator of the physical Hilbert space.

\[
[V,P^+]=0 \qquad \to \qquad 
V=V(P^+)  
\]

\item [] it could still have a complicated kernel in transverse relative
momenta and momentum fractions. 

\vfill

\item [{\color{red}$\bullet$}] $P^+ \geq 0$ implies that any modifications
to the vacuum are associated with $p^+=0$.
      
%\vfill 
%
%\item [{\color{red}$\bullet$}] This implies that some
%      dynamical modifications of the light-front vacuum associated
%      with $p^+=0$ is 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 the free field light-front algebra
({\color{red}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   
Light-front 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 - free light front algebra
}
\end{center}

\begin{center}{\bf 
Spectral condition: 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}$\bullet$}] Translational invariance of vacuum $\to$
$P^-$ densely defined self-adjoint operator satisfying:  
\[
P^- \vert 0 \rangle =0
\]
\[
::\phi^n (\tilde{x}):: \to  \lim_{y_i\to 0} (\prod \phi (\tilde{x}+y_i) - \sum B_k (\tilde{x}) C_k(y_1 \cdots y_n))
\]

\item[{\color{red}$\bullet$}] The requirement $::\phi^n(\tilde{x}) ::\vert 0 \rangle =0 $ relates the coefficients $C_k(y_1\cdots y_n)$ to the vacuum.

\vfill

\item[{\color{red}$\bullet$}] Non-trivial contributions are due to $p^+=0$

\vfill
  
\item[{\color{red}$\bullet$}] The structure of vacuum,  the definition of $P^-$,
and rotational covariance must be treated simultaneously.
}
\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. All forms of dynamics are equivalent.

\vfill

\item [{\color{red}$\bullet$}] Free fields are independent of Lorentz
covariant basis.

\vfill

\item [{\color{red}$\bullet$}] Both particle and field dynamics require
constructing self-adjoint dynamical generators on the Hilbert space. 
%which is the
%direct integral of Poincar\'e irreducible subspaces.    

\vfill

\item[] For fields on the light front this requires (1) extending the
Noether generators to operators in the light-front algebra, and (2)
a consistent treatment of the vacuum, renormalization of local
operator products on the light front in the dynamical generators and
rotational covariance.

\vfill

\item [{\color{red}$\bullet$}] Extension should lead to same theory as
canonical quantization.
}
\end{itemize}
\end{frame}


%slide 41 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx  


%slide 42 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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

\begin{itemize} {\bf

\item [{\color{red}$\bullet$}] Can light-front field theories be
formulated in terms of the free field algebra (with a non-Fock 
vacuum)?

\vfill

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

\vfill

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

\vfill

\item [{\color{red}$\bullet$}] More information (answers?): 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}


\begin{frame}
\begin{center}
{\bf
Extensions to more than two particles
}
\end{center}
\[
U_{(ij)\otimes k}(\Lambda, a) :=
U_{(ij)}(\Lambda,a) \otimes U_{k}(\Lambda,a)
\]
\[
\mathbf{j}_{(12)\otimes(3)f} \not= 
\mathbf{j}_{(23)\otimes(1)f} \not= 
\mathbf{j}_{(31)\otimes(2)f} \not= \mathbf{j}_{0f}
\]
\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}

\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}
\[
M^{\prime 2} :=M^{\prime 2}_{(12)(3)}+ M^{\prime 2}_{(23)(1)} + M^{\prime 2}_{(31)(2)}
-2M^{2}_{0}
\]
\[
\Downarrow
\]
\[
U'(\Lambda,a) {\color{red}\not\to}  U_{ij}(\Lambda,a)\otimes  U_k(\Lambda,a)
\]
\begin{itemize}
{\bf 

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

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

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

\end{frame}

\begin{frame}
\begin{center}
{\bf
Result: Three body light-front dynamics
}
\end{center}
\vfill
\[
U(\Lambda,a) = 
A U'(\Lambda,a) A^{\dagger} 
\]
\vfill
\begin{center}
{\bf
Cluster properties
}
\end{center}
\vfill
\[
U(\Lambda,a) \to  U_{ij}(\Lambda,a)\otimes  U_k(\Lambda,a)
\]
\begin{center}
{\bf
Light-front preserving subgroup kinematic
}
\end{center}
%\[
%S=S'
%\]
\vfill
\[
\mathbf{j}_{fI} = A \mathbf{j}_{f0} A^{\dagger} \qquad
\mbox{\bf dynamical spin}
\]

\end{frame}

%slide 18 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 Generators of light-front preserving transformations}
\end{center}
\[
\{P^+=P^0+P^3,\mathbf{P}_{\perp},J^3=M^{12},  
\]
\[
K^3=M^{03},
E^1 =M^{01} + M^{31},
E^2 =M^{02} - M^{23}
\}
\]
\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 Invariant functions of generators}
\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 mass $m$ spin $j$\\ 
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 of P.G.\\}
\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{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 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}

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

\begin{itemize} {\bf 

\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.


\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}
