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\begin{document}

\title{The relation between instant and light-front formulations of quantum
field theory} %Title of paper

\author{W.~N.~Polyzou}
\email{polyzou@uiowa.edu}
\thanks{This work supported by the U.S. Department of Energy,
  Office of Science, Grant \#DE-SC0016457}
\affiliation{Department of Physics and Astronomy\\ The University of
Iowa\\ Iowa City, IA 52242, USA}

\date{\today}

\begin{abstract}

The scattering equivalence of quantum field theories formulated with
light-front and instant-form kinematic subgroups is established using
non-perturbative methods.  The difficulty with field theoretic
formulations of Dirac's forms of dynamics is that the free and
interacting unitary representations of the Poincar\'e group are
defined on inequivalent representations of the Hilbert space, which
means that the concept of kinematic transformations must be modified
on the Hilbert space of the field theory.  This work addresses this
problem by assuming the existence of a field theory with the expected
properties and constructs equivalent representations with instant and
front form kinematic subgroups.  In this construction both the
light-front and instant-form formulations share the same vacuum and
one-particle states.  The free field Fock space plays no role.  There
is no ``quantization'' of a classical theory.  The property that
survives from the perturbative approach is the notion of a kinematic
subgroup, which means kinematic Poincar\'e transformations can be
trivially implemented by acting on suitable basis vectors.  This
non-perturbative approach avoids dealing with issues that arise in
perturbative treatments where is it necessary to have a consistent
treatment of renormalization, rotational covariance, and the structure
of the light-front vacuum.  While addressing these issues in a
computational framework is important for applications, this work
suggests the nature of the expected resolution and identifies some
differences between the perturbative and non-perturbative approaches.
  
\end{abstract}

\maketitle 

\section{Introduction}\label{sec1}

This paper discusses the relation between light-front and instant-form
formulations of quantum field theory.  The identification of different
forms of dynamics is due to Dirac \cite{Dirac:1949cp}.  In a
relativistic quantum theory the invariance of quantum probabilities in
different inertial coordinate systems requires that equivalent states
in different inertial coordinate systems are related by a unitary ray
representation of the subgroup of the Poincar\'e group continuously
connected to the identity \cite{Wigner:1939cj}.  The Poincar\'e Lie
algebra has 3 independent commutators involving rotationless boost and
translation generators that have the Hamiltonian on the right
\beq
[K^i,P^j]= i\delta_{ij} H.
\label{I:1}
\eeq
If $H=H_0+V$ for some interaction $V$ then the operators on the left
side of each commutator must also know about the interaction.  Dirac
identified three representations of the Poincar\'e Lie algebra with
the minimum number (3-4) of interaction-dependent Poincar\'e generators.  He
called these the instant, point and front-forms of the dynamics.  In
the instant form the generators of space translations and rotations
are free of interactions, in the point form, the generators of
rotationless Lorentz transformations are free of interactions and in
the front form the generators of transformations that leave a hyperplane
tangent to the light cone invariant are free of interactions.

The equivalence of these different representations of relativistic
quantum mechanics was settled by Sokolov and Shatnyi
\cite{Sokolov:1977im}\cite{Keister:1996bd}.  In quantum field theory
the problem is more complicated because the free and interacting
dynamics are formulated on different inequivalent representations of
the Hilbert space \cite{Haag:1955ev}, so the decomposition of the
Hamiltonian into the sum of a free Hamiltonian plus interaction is not
defined on the Hilbert space of the field theory.  Such a
decomposition makes
%\cite{JvNeumann2}\cite{JvNeumann3}\cite{JvNeumann}\cite{MHStone1}\cite{MHStone2}
sense in perturbative quantum field theory with cutoffs, so the notion
of instant- and light-front formulations
\cite{Chang1069}\cite{Soper}\cite{Chang:1972xt}\cite{Yan:1973.2}\cite{Yan:1973.3}\cite{Yan:1973.4} of
quantum field theory make sense perturbatively.  The price paid is
that as the cutoffs are removed the theory has infinities that have to
be renormalized.  The relation between the different forms of dynamics
depends on a consistent treatment of the renormalization.  The
light-front formulation of quantum field theory is of particular interest for
applications.  The most appealing property of the theory is the
apparent triviality of the light-front vacuum, which reduces the
solution of the field theory to linear algebra on a Hilbert space,
like non-relativistic quantum mechanics.  In addition to the
computational challenges of implementing this program in a theory with
an infinite number of degrees of freedom, there are a number of
puzzles that appear in comparing the two approaches.  These include:

\begin{itemize}
\item[$\bullet$] Is the light-front vacuum the same as the Fock vacuum?

\item[$\bullet$] How to understand $P^+=0$ (zero mode) singularities?

\item[$\bullet$] How to formulate spontaneous symmetry breaking in a light-front
  dynamics?
  
\item[$\bullet$] How to renormalize the theory consistent with rotational covariance.

\item[$\bullet$] What is the relation between light-front and canonical quantization of
  a quantum field theory?

\item[$\bullet$] Are both approaches equivalent?  
\end{itemize}
While there is a general consensus that the two formulations are the
equivalent, the answers to the above questions are not as clean as
desired.  These issues have been discussed extensively in the
literature
\cite{Chang1069}
\cite{Soper}
\cite{Leutwyler:1970wn}
\cite{Rohrlich:1971zz}
\cite{Chang:1972xt}
\cite{Yan:1973.2}
\cite{Yan:1973.3}
\cite{Yan:1973.4}
\cite{Schlieder:1972qr}
\cite{Maskawa:1976}
\cite{Nakanishi:1976yx}
\cite{Nakanishi:1977}
\cite{Leutwyler:1977vy}
\cite{Coester:1992}
\cite{Coester:1993fg}
\cite{Coester:1994}
\cite{Bylev:1996}
\cite{Brodsky:1998}
\cite{Yamawaki:1998}
\cite{choi:1998}
\cite{Tsujimaru:1997jt}
\cite{Lenz:2000}
\cite{Heinzl:2001}
\cite{Burkhardt:2002}
\cite{Heinzl:2003}
\cite{Werner:2006}
\cite{Bakker:2011zza}
\cite{Choi:2011xm}
\cite{Choi:2013ira}
\cite{Beane:2013oia}
\cite{sofia:2014}
\cite{Herrmann:2015dqa}
\cite{Ji:2017vfu}
\cite{Collins:2018aqt}
\cite{Mannheim:2020rod}.
%\cite{Ekstein:1960}
%\cite{Bargmann:1946me}
%\cite{araki-bk}
Most of these discussions are based on perturbation theory; in
particular the assumption that the light-front dynamics can be
formulated on the Fock space of a free field theory.  The continued
interest in these questions is because compelling resolutions of these
questions are obscured by the need to perform a non-perturbative
renormalization of the theory.  In this work these complications are
avoided by assuming the existence of the theory with the expected
properties.
The existence of an asymptotically complete
scattering theory is also assumed, based on the Haag-Ruelle
formulation of scattering theory
\cite{Haag:1958vt}\cite{Brenig:1959}\cite{Ruelle:1962}\cite{jost:1966}.
Haag-Ruelle scattering has the advantage that it can be formulated in
terms of wave operators defined by strong limits.  The construction in
this paper makes use of the strong limits. A two-Hilbert space
representation is used, which does not require the existence of a free
dynamics on the Hilbert space of the theory.  The second Hilbert space
is the direct sum of tensor products of single-particle Hilbert spaces
with physical masses.  In the simplest case it is a Fock space with
physical particle masses.  The mapping from the second Hilbert space
to the Hilbert space of the field theory puts in the internal
structure of the physical particles when they are asymptotically
separated. There is a natural unitary representation of the Poincar\'e
group on the second Hilbert space, which treats the particles as free
particles.  The second Hilbert space is called the asymptotic Hilbert
space.  There is sufficient freedom to choose the mappings from the
asymptotic Hilbert space to the Hilbert space of the field theory so
the unitary representation of the instant or light-front kinematic
subgroup of the Poincar\'e group on the asymptotic space maps on to
the dynamical unitary representation of the kinematic subgroup on the
field-theory Hilbert space without changing the scattering operator.

%This equivalence is the justification
%for the invariance of the scattering matrix under field redefinition.

A theorem due to Ekstein \cite{Ekstein:1960} gives necessary and
sufficient conditions for unitary transformations on a Hilbert space
to preserve the scattering operator.  This result assumes the
scattering is formulated in terms of wave operators defined as strong
limits.  In the two-Hilbert space formulation Ekstein's condition is the
requirement that if unitary transformation is applied to the mapping
from the asymptotic Hilbert space to the field theory Hilbert space, it
is asymptotically ``equivalent'' to the original mapping
(see eq. (\ref{S1:33})).
This
freedom can be exploited to construct the most general class of
S-matrix preserving unitary transformations that are invariant under a
kinematic subgroup of the dynamical representation of the Poincar\'e
group.  The proof uses the unitary of the wave operators which only
holds if the vacuum and one-body channels are included in the
asymptotic Hilbert space.  Ekstein's condition then implies that
acceptable unitary transformations must leave the vacuum and one-particle
states unchanged.  When these unitary transformations are
applied to the unitary representation of the Poincar\'e group on the
physical Hilbert space, they result in two new scattering equivalent
representations; with different kinematic subgroups that intertwine
with the kinematic subgroups on the asymptotic Hilbert space.

The only thing that is missing is the absence of a non-interacting
representation of the Poincar\'e group on the Hilbert space of field
theory.  In the event that the mapping from the asymptotic space to
the field theory Hilbert space has dense range, and the unitarized
mapping gives the same asymptotic condition, then it can be used to map
the unitary representation of the Poincar\'e group on the asymptotic
Hilbert space to a unitary representation of the Poincar\'e group on
the field theory Hilbert that plays the role of a free-particle
dynamics, however this is not a true free-particle representation
because the vacuum is unchanged and the particle masses are physical.

The results can be summarized as follows. The construction assumes a
unitary representation of the Poincar\'e group on the Hilbert space of
the field theory and constructs scattering equivalent representations
with light -front and instant-form kinematic symmetries.  One
consequence of the construction is that it requires that the
transformations relating these representations leave the vacuum and
one-particle states unchanged.  Because the one-body states are
eigenstates of the mass operator, there is no mass renormalization.
This condition is not compatible with a $1\leftrightarrow 2$ vertex
interaction in either of these representations.

This paper has three sections and two appendices.  Section two
summarizes the assumptions used in this work and discusses a
two-Hilbert space formulation of Haag-Ruelle scattering that has
kinematically invariant injection operators that map an asymptotic
many-particle Hilbert space to the Hilbert space of the field theory.
Haag-Ruelle scattering is the natural generalization of the usual
formulation of time-dependent scattering theory.  The construction of
equivalent instant and light-front representations of the field theory
is given in section 3.  Section 4 has a summary of the results and a
discussion of the implications.  There are two appendices.  The first
one contains a discussion of Ekstein's treatment of scattering
equivalences \cite{Ekstein:1960} that is used in section 3.  The
second appendix discusses the formulation of Haag-Ruelle injection
operators with the kinematic symmetry properties that are used in
sections two and three.

\section{General considerations}

This paper examines the relation between light-front and instant-form
formulations of quantum field theory from a more abstract perspective
that does not assume that the Hamiltonian can be decomposed as the sum of 
free and interacting operators acting on one representation of the
Hilbert space.  The starting assumptions are typical assumptions about
abstract properties of a quantum field theory.  These include:

\begin{itemize}
  
\item[$\bullet$] The Hilbert space of the field theory is generated by
  applying bounded functions, $F$, of smeared field operators to a
  unique vacuum vector, denoted by $\vert 0 \rangle$.  The smearing is
  assumed to be over Schwartz test functions in four space-time
  dimensions.  It is not assumed that fields
  restricted to a light-front or fixed-time manifold make sense.  A
  dense set of vectors, $\vert \psi \rangle$, can be taken to have the
  form:

\beq
\vert \psi \rangle = F \vert 0 \rangle
\qquad 
F=\sum_{n=1}^N c_n e^{i\phi_n (f_n)}
\label{S1:1}
\eeq
\item[]where the $c_n$ are complex coefficients, the $f_n$ are Schwartz
functions, $\phi_n$ are field operators and $N$ is finite.

\item[$\bullet$] There is a unitary representation of the Poincar\'e group,
$U(\Lambda,a)$, on this representation of the Hilbert space which is given by covariance.  In the spinless case
the action of $U(\Lambda ,a)$ on states of the form (\ref{S1:1}) is
\beq
U(\Lambda ,a) \vert \psi \rangle = \sum_c c_n e^{i\phi (f_n')} \vert 0 \rangle
\qquad
f_n'(x) = f_n(\Lambda^{-1}(x-a)).
\label{S1:2}
\eeq

\item[$\bullet$] The vacuum, $\vert 0 \rangle$, is Poincar\'e invariant and normalized to unity:
\beq
U(\Lambda ,a)\vert 0 \rangle  = \vert 0 \rangle \qquad \langle 0 \vert 0 \rangle =1 .
\label{S1:3}
\eeq

\item[$\bullet$] The unitary representation of the Poincar\'e group acts like the
identity on the vacuum subspace and can be decomposed into a direct integral
of positive-mass positive-energy irreducible representations of the
Poincar\'e group \cite{Wigner:1939cj}\cite{Bargmann:1946me} on the orthogonal complement of the
vacuum subspace.  The mass Casimir operator is assumed to have point-spectrum
eigenstates representing particles of the theory.

\item[$\bullet$] There is a complete set of asymptotically
  complete Haag-Ruelle 1\cite{Haag:1958vt}\cite{Brenig:1959}\cite{Ruelle:1962}\cite{jost:1966}
scattering states.

\end{itemize}

A characteristic element of Dirac's forms of dynamics is the notion of
a {\it kinematic subgroup}.  Kinematic subgroups leave a manifold in
Minkowski space, that classically intersects every (massive)
particle's world line once, invariant.  In an instant-form dynamics
the invariant manifold is a fixed-time plane, which is preserved under
the group generated by rotations and space translations.  In a
light-front dynamics the manifold is a hyperplane that is tangent to the
light cone.  This is invariant under a seven parameter subgroup of the
Poincar\'e group.  These manifolds are 
prominent in Dirac's work; in this work the focus will be on the subgroups of
the Poincar\'e group that leave these manifolds invariant.  These
subgroups will be referred to as kinematic subgroups.

For quantum mechanical systems of a finite number of degrees of
freedom kinematical and dynamical unitary representations of the
Poincar\'e group exist in the same representation of the Hilbert
space.  In the field theoretic case the representation of the Hilbert
space depends on the dynamics, so {\it there is no free dynamics} that
acts on the Hilbert space of the theory.  This is one of the
distinctions with the perturbative approach, which attempts to define
the interacting theory as a perturbation of a free field theory.  The
incompatibility of these formulations leads to the infinities that
complicate the analysis of the relation between light-front and
instant-form treatments of field theory.

The first step in the abstract  formulation is to define what is meant
by  the  kinematic  subgroup  in  the  absence  of  a  non-interacting
representation of the Poincar\'e group.  The unitary representation of
the Poincar\'e group has a set of 10 infinitesimal generators that are
self-adjoint  operators  on the  Hilbert  space  of the  field  theory
satisfying the Poincar\'e Lie algebra.   This follows because each one
is  the  generator of  a  unitary  one-parameter group.   The  unitary
representation of the Poincar\'e group can be decomposed into a direct
integral  of  irreducible representations.   In  this  work a  typical
spectral  condition is assumed,  where the  representations that
appear  in  the  direct  integral  are  assumed  to  be  positive-mass
positive-energy  representations,  and  the  identity  on  the  vacuum
subspace. 

The direct integral of irreducible representations of the Poincar\'e
group results in a direct integral decomposition of the Hilbert space into
irreducible subspaces or slices.  The next step is to choose a basis
on each irreducible subspace.  Two choices will be considered in this
work.  The first basis consists of simultaneous eigenstates of the
mass, spin, linear momentum, and the projection of the canonical spin
on the $z$ axis.  These are all functions of the infinitesimal
generators of the unitary representation of the Poincar\'e group.  There
will also be Poincar\'e invariant degeneracy parameters, which can be
taken as discrete quantum numbers, since any continuous parameters can
be replaced by a basis of square integrable functions of the
continuous degeneracy parameters.  The other basis on the same
irreducible subspace consists of simultaneous eigenstates of the mass,
spin, the three light-front components of the four momentum, and the
projection of the light-front spin on the $z$ axis.  These are also
functions of the infinitesimal generators with the same degeneracy
parameters.

These bases can be formally expressed as
\beq
\vert 0 \rangle \cup \{
\vert (m,s,d_n) \mathbf{p},\mu_c \rangle \} \qquad \mbox{and} \qquad
\vert 0 \rangle \cup \{ \vert (m,s,d_n) \tilde{\mathbf{p}},\mu_f
\rangle \}
\label{S1:4}
\eeq
where the light-front components of the four momentum are defined by
\beq
\tilde{\mathbf{p}} =
(\mathbf{p}\cdot \hat{\mathbf{x}}, \mathbf{p}\cdot \hat{\mathbf{y}}, p^0+\mathbf{p}\cdot \hat{\mathbf{z}}) :=
(\mathbf{p}_{\perp},p^+).
\label{S1:5}
\eeq
The variables (\ref{S1:5}) are eigenvalues of the generators of
translations tangent to the light-front hyperplane.  The first basis
in (\ref{S1:4}) will be referred to as the ``instant-form'' basis and
the second basis will be referred to as the ``light-front'' basis.

These basis states are related by a unitary change of basis.  The
change of basis is block diagonal in the direct integral.  The
different basis vectors are related by
\beq
\vert (m,s,d_n) \mathbf{p},\mu_c \rangle =
\sum_{\mu_f=-s}^s  \vert (m,s,d_n) \tilde{\mathbf{p}}(p),\mu_f \rangle
\sqrt{p^+ \over \omega_m(\mathbf{p})}
D^s_{\mu_f \mu_c}[B_f^{-1}(p/m) B_c(p/m)]
\label{S1:6}
\eeq
which assumes that both bases have delta function normalizations:
\beq
\langle (m,s,d_n) \mathbf{p},\mu_c \vert (m',s',d_k)
\mathbf{p}',\mu'_c \rangle =
\delta (\mathbf{p}-\mathbf{p}')\delta_{ss'}\delta_{\mu_c\mu_c'} \delta_{nk}
\delta [m-m']
\label{S1:7}
\eeq
and
\beq
\langle (m,s,d_n) \tilde{\mathbf{p}},\mu_f \vert (m',s',d_k)
\tilde{\mathbf{p}}',\mu'_f \rangle =\delta (p^+-p^{+\prime})
\delta (\mathbf{p}_{\perp}-\mathbf{p}_{\perp}')\delta_{ss'}\delta_{\mu_f\mu_f'} \delta_{nk}
\delta [m-m']
\label{S1:8}
\eeq
where $\delta [m-m']$ is either a Dirac or Kronecker delta function
depending on whether $m$ is in the point or continuous spectrum of the
mass operator.  $B_c(p/m)$ and $B_f(p/m)$ are $SL(2,C)$ matrices
representing a rotationless boost and a light-front preserving boost
from $(1,0,0,0)$ to $p/m$.  The combination $B_f^{-1}(p/m) B_c(p/m)$
is a $SU(2)$ representation of a Melosh rotation \cite{Melosh:1974cu}
that changes the
light-front spin to the canonical spin.

There are similar transformations relating light-front bases
corresponding to different orientations of the light front
(i.e. replacing $\hat{\mathbf{z}}$ by some other unit vector
$\hat{\mathbf{n}}$).  Like (\ref{S1:6}) they involve a variable
change, the square root of a Jacobian and a momentum-dependent
rotation matrix.
%The explicit form is not important in what follows.
%The unitary
%operator $V_{fc}$ that changes the canonical basis to the light front
%basis is defined by
%\[
%V_{fc}= \vert 0 \rangle \langle 0 \vert +
%\]
%\beq
%\sum \int
%\vert (m,s,d_n) \tilde{\mathbf{p}},\mu_f \rangle d\tilde{\mathbf{p}}
%\langle (m,s,d_n) \tilde{\mathbf{p}},\mu_f \vert (m',s',d'_k),
%{\mathbf{p}}',\mu_c \rangle d\mathbf{p'}
%\langle (m',s',d'_k),
%{\mathbf{p}}',\mu_c \vert
%\label{S1:9a)} 
%\eeq
%where
%\[
%\langle (m,s,d_n) \tilde{\mathbf{p}},\mu_f \vert (m',s',d'_k),
%{\mathbf{p}}',\mu_c \rangle =
%\delta (\tilde{\mathbf{p}}(\mathbf{p}')-\tilde{\mathbf{p}})
%\delta [m-m'] \delta_{ss'}\delta_{d_n d_k'}
%\sqrt{p^+ \over \omega_m(\mathbf{p})}
%D^s_{\mu_f \mu_c}[B_f^{-1}(p/m) B_c(p/m)]
%=
%\]
%\beq
%\delta (\mathbf{p}(\tilde{\mathbf{p}})-{\mathbf{p}}')
%\delta [m-m'] \delta_{ss'}\delta_{d_n d_k'}
%\sqrt{ \omega_m(\mathbf{p})\over p^+ }
%D^s_{\mu_f \mu_c}[B_f^{-1}(p/m) B_c(p/m)] .
%\label{S1:9b}
%\eeq
%An analogous construction can be used to relate bases with different
%orientations of the light front.

The action of $U(\Lambda ,a)$ on each of these irreducible basis
states is a consequence of the transformation properties of the
Poincar\'e generators and the basis choice.
For the ``instant-form'' basis
\beq
U(\Lambda ,a) \vert 0 \rangle = \vert 0 \rangle ,
\label{S1:10}
\eeq
\beq
U(\Lambda ,a) \vert (m,s,d_n) {\mathbf{p}},\mu \rangle
=
e^{ia \cdot \Lambda p}  \sum_{\mu'=-s}^s
\vert (m,s,d_n) {\pmb{\Lambda}{p}},\mu' \rangle
\sqrt{{ \omega_m (\pmb{\Lambda}p) \over \omega_m (\mathbf{p})}}
D^s_{\mu'\mu}[B_c^{-1}(\Lambda ( p/m)) \Lambda B_c(p/m)]
\label{S1:11}
\eeq
and for the ''light-front'' basis.
\beq
U(\Lambda ,a) \vert 0 \rangle = \vert 0 \rangle ,
\label{S1:12}
\eeq
\beq
U(\Lambda ,a) \vert (m,s,d_n) {\tilde{\mathbf{p}}},\mu \rangle
=
e^{ia \cdot \Lambda p} \sum_{\mu'=-s}^s
\vert (m,s,d_n) {\tilde{\pmb{\Lambda}}{p}},\mu' \rangle
\sqrt{{(\Lambda p)^+ \over p^+}}
D^s_{\mu'\mu}[B_f^{-1}(\Lambda (p/m)) \Lambda B_f(p/m)].
\label{S1:13}
\eeq
These equations define the dynamical
unitary representation of the Poincar\'e group on these two
irreducible bases.

The connection of these bases with kinematic subgroups is that the
coefficients of the basis functions on the right hand side of
equations (\ref{S1:11}) and (\ref{S1:13}) do not depend on the mass
eigenvalue $m$ when $(\Lambda,a)$ is an element of the kinematic
subgroup.  For the basis (\ref{S1:11}) the kinematic subgroup is
generated by rotations and spatial translations while for the basis
(\ref{S1:13}) the kinematic subgroup is the subgroup that leaves the
light front hyperplane $x^+=x^0 + x=0$ invariant.

The next step is to discuss the formulation of scattering theory.  A
two-Hilbert space representation \cite{Coester:1982vt}\cite{baumgartl:1983}, will be used for this purpose.  The
starting point is the assumption that the mass operator,
$M=\sqrt{-p^2}$ has one-particle eigenstates.  In this
non-perturbative context one-particle means that the mass operator, $M
= \sqrt{-p^2}$, has a non-empty point spectrum. The discrete mass
eigenvalues are assumed to be strictly positive.
There is no distinction between elementary and composite particles.

Normalizable one-particle states in the ``instant form'' basis
can be constructed by smearing the irreducible basis functions with a
wave packet:
\beq
\vert \psi_{cg} \rangle := \int d\mathbf{p} \sum_{\mu=-s}^s
\vert  (m,s,d_n) \mathbf{p},\mu \rangle g (\mathbf{p},\mu)
\label{S1:14}
\eeq
where $m$ is a discrete mass eigenvalue and $g (\mathbf{p},\mu)$
is a square integrable wave packet.
The corresponding construction in
the ``light-front'' basis has the form
\beq
\vert \psi_{fg} \rangle := \int d^2\mathbf{p}_{\perp} \int_0^\infty dp^+
\sum_{\mu=-s}^s
\vert  (m,s,d_n) \tilde{\mathbf{p}},\mu \rangle \tilde{g} (\tilde{\mathbf{p}},\mu). 
\label{S1:15}
\eeq
These normalizable vectors are linear in the smearing functions and
can be represented as elements of the field algebra applied to the
vacuum. This means that they can be expressed as
\beq
\vert \psi_{cg} \rangle  = \sum_{\mu=-s}^s \int A^{\dagger}_{m,s,d_n}({p},\mu)\vert 0 \rangle d\mathbf{p}
g (\mathbf{p},\mu) = A^{\dagger}(g)\vert 0 \rangle
\label{S1:16}
\eeq
and
\beq
\vert \psi_{fg} \rangle  = \sum_{\mu=-s}^s \int \tilde{A}^{\dagger} _{m,s,d_n}(\tilde{{p}},\mu)\vert 0 \rangle dp^+ d^2\mathbf{p}_{\perp}
\tilde{g} (\tilde{\mathbf{p}},\mu) = \tilde{A}^{\dagger}(\tilde{g})\vert 0 \rangle
\label{S1:17}
\eeq
These states will be equal if $g ({\mathbf{p}},\mu) =
\langle (m,s) \mathbf{p},\mu \vert g \rangle $ and
$\tilde{g} (\tilde{\mathbf{p}},\mu)
= \langle (m,s) \tilde{\mathbf{p}},\mu \vert g \rangle $
are related by the change of basis (\ref{S1:6}).

The operators $A^{\dagger}_{m,s,d_n}({p},\mu)$ and
$\tilde{A}^{\dagger}_{m,s,d_n}(\tilde{{p}},\mu)$ are functions of the
fields that create particles of mass $m$, spin $s$, momentum
$\mathbf{p}$ and magnetic quantum number $\mu$ {\it out of the vacuum}.
The construction of these operators starting from operators that couple
the vacuum to the one-particle states of the theory is discussed in
appendix \ref{inj}.

Since $A^{\dagger}_{m,s,d_n}({p},\mu)$ and
$\tilde{A}^{\dagger}_{m,s,d_n}(\tilde{{p}},\mu)$ are in the field
algebra (after smearing), they can be multiplied.  While the field
theory has one-particle states, {\it it does not have free
  many-particle states}, so $N$ repeated application of these
operators, while defined, cannot be interpreted as creating
$N$-particle states out of the vacuum. 

The following quantity
\beq
A^{\dagger}_{m_1,s_1,d_{n_1}}(\mathbf{p_1},\mu_1) \cdots A^{\dagger}_{m_k,s_k,d_{n_k}}
(\mathbf{p_k},\mu_k) \vert 0 \rangle 
\label{S1:18}
\eeq
can be considered as a mapping from a space of square integrable
functions of $\mathbf{p}_1, \mu_1 \cdots \mathbf{p}_k,\mu_k$ to the
Hilbert space of the field theory.  Note that while
$A^{\dagger}(g)\vert 0 \rangle = \tilde{A}^{\dagger}
(\tilde{g})\vert 0 \rangle$ for $g$ and $\tilde{g}$
related by (\ref{S1:6}), this is not true for products of these
operators applied to the vacuum.
From equation (\ref{A2:17}) in
appendix \ref{inj} it follows that this mapping has the following
properties
\[
\mathbf{P} A^{\dagger}_{m_1,s_1,d_{n_1}}(\mathbf{p}_1,\mu_1) \cdots
A^{\dagger}_{m_N,s_N,d_{n_N}}(\mathbf{p}_N,\mu_N)
\vert 0 \rangle =
\]
\beq
\sum_{n=1}^N\mathbf{p}_n
A^{\dagger}_{m_1,s_1,d_{n_1}}(\mathbf{p}_1,\mu_1) \cdots
A^{\dagger}_{m_N,s_N,d_{n_N}}(\mathbf{p}_N,\mu_N)
\vert 0 \rangle
\label{S1:19}
\eeq
and
\[
U(R,0 )A^{\dagger}_{m_1,s_1,d_{n_1}}(\mathbf{p}_1,\mu_1) \cdots
A^{\dagger}_{m_N,s_N,d_{n_N}}(\mathbf{p}_1,\mu_N)
\vert 0 \rangle
 =
\]
\beq
\sum
A^{\dagger}_{m_1,s_1,d_{n_1}}(R\mathbf{p}_1,\nu_1) \cdots
A^{\dagger}_{m_N,s_N,d_{n_N}}(R\mathbf{p}_N,\nu_N)
\vert 0 \rangle 
\vert 0 \rangle
\prod_{n=1}^N D^{s_n}_{\nu_n \mu_n}(R) .
\label{S1:20}
\eeq
For the mapping in the
``light-front'' basis the corresponding relations are 
\[
\tilde{\mathbf{P}} \tilde{A}^{\dagger}_{m_1,s_1,d_{n_1}}(\tilde{\mathbf{p}}_1,\mu_1) \cdots \tilde{A}^{\dagger}_{m_N,s_N,d_{n_N}}(\tilde{\mathbf{p}}_N,\mu_N) \vert 0 \rangle =
\]
\beq
\sum_{n=1}^N\tilde{\mathbf{p}}_n \tilde{A}^{\dagger}_{m_1,s_1,d_{n_1}}(\tilde{\mathbf{p}}_1,\mu_1) \cdots \tilde{A}^{\dagger}_{m_N,s_N,d_{n_N}}(\tilde{\mathbf{p}}_N,\mu_N) \vert 0 \rangle 
\label{S1:21}
\eeq
\[
U(B_f,0) \tilde{A}^{\dagger}_{m_1,s_1,d_{n_1}}(\tilde{\mathbf{p}}_1,\mu_1) \cdots
\tilde{A}^{\dagger}_{m_N,s_N,d_{n_N}}(\tilde{\mathbf{p}}_N,\mu_N) \vert 0 \rangle =
\]
\beq
\tilde{A}^{\dagger}_{m_1,s_1,d_{n_1}}(\tilde{\mathbf{B}}_f p_1,\mu_1) \cdots
\tilde{A}^{\dagger}_{m_N,s_N,d_{n_N}}(\tilde{\mathbf{B}}_f p_N,\mu_N) \vert 0 \rangle
\prod_{n=1}^N {\sqrt {B_f(p_n)^+\over p_n^+}}.
\label{S1:22}
\eeq
\[
U(R_z(\phi),0) \tilde{A}^{\dagger}_{m_1,s_1,d_{n_1}}(\tilde{\mathbf{p}}_1,\mu_1) \cdots
\tilde{A}^{\dagger}_{m_N,s_N,d_{n_N}}(\tilde{\mathbf{p}}_N,\mu_N) \vert 0 \rangle =
\]
\beq
\tilde{A}^{\dagger}_{m_1,s_1,d_{n_1}}(R_z(\phi)\tilde{\mathbf{p}}_1,\mu_1) \cdots
\tilde{A}^{\dagger}_{m_N,s_N,d_{n_N}}(R(\phi) \tilde{\mathbf{p}}_N,\mu_N) \vert 0 \rangle
\prod_{n=1}^N e^{i \mu_n \phi}.
\label{S1:23}
\eeq
%These vectors, after smearing with test functions, do NOT represent
%$N$-particle states in the field theory Hilbert space.

If the Fourier transform of these states is integrated against wave
packets that are localized in regions separated by large space-like
separations and the field theory satisfies cluster properties
(normally a consequence of uniqueness of the vacuum) then the
expectation is that the resulting vectors look like states of $N$
asymptotically separated particles when they are used in
inner products \cite{jost:1966}\cite{araki-bk}.  The interpretation of the operators
${A}^{\dagger}_{m,s,d_{n}}$ and  $\tilde{A}^{\dagger}_{m,s,d_{n}}$
is that they {\it asymptotically} behave like creation operators .

A scattering channel $\alpha$ is associated with a finite collection
of particles asymptotically.  It could correspond to the state of a
target and incoming projectile or a collection of particles that are
detected in a scattering experiment. The set of all scattering
channels of the theory is denoted by ${\cal A}$.  In what follows both
the vacuum and one-particle states are included in the collection of
channels, ${\cal A}$.

Channel injection operators are defined as
products of the operators (\ref{S1:16}) or (\ref{S1:17}),
applied to the vacuum,  one for each
particle in a scattering channel $\alpha$.  In the canonical
basis they are 
\beq
\Phi_{\alpha} ({\mathbf{p}_1},\mu_1 \cdots {\mathbf{p}_N},\mu_N):=
\prod_{k \in \alpha}  A^{\dagger}_{m_k,s_k,d_{n_k}}({\mathbf{p}_k},\mu_k)\vert 0 \rangle
\label{S1:24}
\eeq
and in the light-front basis they are
\beq
\tilde{\Phi}_{\alpha} (\tilde{\mathbf{p}}_1,\mu_1 \cdots \tilde{\mathbf{p}}_N,\mu_N):=
\prod_{k \in \alpha}
\tilde{A}^{\dagger}_{m_k,s_k,d_{n_k}}(\tilde{\mathbf{p}}_k,\mu_k)\vert0 \rangle . 
\label{S1:25}
\eeq
The order of the product does not matter in the asymptotic region.
The asymptotic channel Hilbert space ${\cal H}_{\alpha}$ is the space
of square integrable functions of the variables $\mathbf{p}_k,\mu_k$
or $\tilde{\mathbf{p}}_k,\mu_k$ for $k \in \alpha$.  It is interpreted
as a space of $N$ particles of mass $m_k$ and spin $s_k$.  If any of the
particles in the channel are identical then the functions representing
identical Bosons should be symmetrized and those representing
identical Fermions should anti-symmetrized.  The channel injection
operators (\ref{S1:24}) and (\ref{S1:25}) are interpreted as mappings
from the $k$-particle channel Hilbert space ${\cal H}_{\alpha}$ to the Hilbert
space ${\cal H}$ of the field theory.

The asymptotic unitary representation of the Poincar\'e group
on ${\cal H}_{\alpha}$ is defined by treating ${\cal H}_{\alpha}$ as a space of
$N$ mutually non-interacting particles of mass $m_k$ and spin $s_k$
for $k \in \alpha$.  This representation is defined by the tensor
product of one-particle irreducible representations in terms of
``instant-form'' variables
\beq
U_\alpha(\Lambda ,a)\vert \mathbf{p}_1, \mu_1, \cdots
\mathbf{p}_k, \mu_k \rangle  :=
e^{ia \cdot \Lambda (\sum_n p_n)}
\vert \pmb{\Lambda}p_1, \nu_1 \cdots \pmb{\Lambda}p_k, \nu_k \rangle
\prod_{n=1}^k
\sqrt{{ \omega_{m_n} (\pmb{\Lambda}p_n) \over \omega_{m_n} (\mathbf{p}_n)}}
D^s_{\nu_n\mu_n}[B_c^{-1}(\Lambda p_n/m_n) \Lambda B_c(p_n/m_n)].
\label{S1:26}
\eeq
The corresponding expression in terms of ``light-front'' variables is
\beq
U_\alpha(\Lambda ,a)\vert \tilde{\mathbf{p}}_1, \mu_1, \cdots
\tilde{\mathbf{p}}_k, \mu_k \rangle  :=
e^{ia \cdot \Lambda (\sum_n p_n)}
\vert \tilde{\pmb{\Lambda}}p_1, \nu_1 \cdots \tilde{\pmb{\Lambda}}p_k, \nu_k \rangle
\prod_{n=1}^k
\sqrt{{(\tilde{\Lambda}p_n)^+ \over {p}^+_n}}
D^s_{\nu_n\mu_n}[B_f^{-1}(\Lambda p_n/m_n) \Lambda B_f(p_n/m_n)].
\label{S1:27}
\eeq
The states in (\ref{S1:26}) and (\ref{S1:27}) are basis vectors in
${\cal H}_{\alpha}$, not ${\cal H}$.  Since the Hilbert space vectors
in the range of $\Phi_{\alpha}$ or $\tilde{\Phi}_{\alpha}$ are not
N-particle states it follows that
\beq
U(\Lambda,a)\Phi_{\alpha} \not= \Phi_{\alpha}U_{\alpha}(\Lambda,a).
\qquad
U(\Lambda,a)\tilde{\Phi}_{\alpha} \not= \tilde{\Phi}_{\alpha}U_{\alpha}(\Lambda,a).
\label{S1:28}
\eeq
This is because 
(\ref{A2:6}) does not hold for $P^0$ due to the
$p^0$ integrals in (\ref{A2:9}) and 
(\ref{A2:6}) does not hold for $P^-$ due to the
$p^-$ integrals on (\ref{A2:9}) (see appendix \ref{inj}).

When $\Lambda_K$ and $a_K$ are elements of the
instant-form or light-front kinematic subgroup,
it follows from (\ref{S1:19}-\ref{S1:23}) that 
\beq
U(\Lambda_K,a_K)\Phi_{\alpha} = \Phi_{\alpha}U_{\alpha}(\Lambda_K,a_K)
\qquad
U(\Lambda_K,a_K)\tilde{\Phi}_{\alpha} = \tilde{\Phi}_{\alpha}U_{\alpha}(\Lambda_K,a_K)
%\qquad (\Lambda_K,a_K) \in \mbox{ kinematic subgroup}
\label{S1:29}
\eeq
because the coefficients of the kinematic transformations are mass
independent.  The operators $\Phi_{\alpha}$ and
$\tilde{\Phi}_{\alpha}$ that map the channel $\alpha$ Hilbert space
${\cal H}_{\alpha}$ into the field theory Hilbert space ${\cal H}$ are
Haag-Ruelle \cite{Haag:1958vt}\cite{Brenig:1959}\cite{Ruelle:1962}
injection operators.  The only difference with the
standard choices is that
$\Phi_{\alpha}$ and
$\tilde{\Phi}_{\alpha}$ 
are designed
to have the intertwining properties (\ref{S1:29}).

In the two-Hilbert space 
formulation \cite{simon}\cite{araki-bk} channel wave operators $\Omega_{\alpha \pm} =
\Omega_{\alpha \pm} (H,\Phi_{\alpha},H_{\alpha})$ are defined by the
strong limits
\beq
\lim_{t \to \pm\infty}
\Vert (\Omega_{\alpha \pm} (H,\Phi_{\alpha},H_{\alpha}) -
e^{iHt} \Phi_{\alpha} e^{-i H_\alpha t}) \vert \psi \rangle \Vert =0
\label{S1:30a}
\eeq
or
\beq
\lim_{t \to \pm\infty}
\Vert (\Omega_{\alpha \pm} (H,\tilde{\Phi}_{\alpha},H_{\alpha}) -
e^{iHt} \tilde{\Phi}_{\alpha} e^{-i H_\alpha t}) \vert \psi \rangle \Vert =0 .
\label{S1:30b}
\eeq
The wave operators satisfy the intertwining property
\beq
U(\Lambda ,a) \Omega_{\alpha \pm}(H,\Phi_{\alpha},H_{\alpha}) =
\Omega_{\alpha \pm}(H,\Phi_{\alpha},H_{\alpha}) U_{\alpha} (\Lambda,a)
\qquad
U(\Lambda ,a) \Omega_{\alpha \pm}(H,\tilde{\Phi}_{\alpha},H_{\alpha}) =
\Omega_{\alpha \pm}(H,\tilde{\Phi}_{\alpha},H_{\alpha}) U_{\alpha} (\Lambda,a)
\label{S1:31}
\eeq
where, unlike (\ref{S1:28}), $(\Lambda,a)$ are not restricted to the
kinematic subgroup.

The purpose of the injection operators in Haag-Ruelle scattering is to
define the asymptotic boundary conditions.  They are constructed so
they behave like creation operators in asymptotically separated
regions.  They replace the free-particle asymptotic states of ordinary
scattering theory by states in the Hilbert spaces of the field theory
that only behave like a system of non-interacting particle in the asymptotic region.
The two injection operators defined in appendix \ref{inj} both have this
property.  They differ in which variable enforces the one-particle mass
shell delta function when applied the the vacuum. 
This means that as operators
\beq
\Omega_{\alpha \pm}(H,{\Phi}_{\alpha},H_{\alpha})=
\Omega_{\alpha \pm}(H,\tilde{\Phi}_{\alpha},H_{\alpha})
\label{S1:32}
\eeq
if they are evaluated in the same one-particle basis.   This identity 
is equivalent to the requirement
\beq
0=
\lim_{t \to \pm \infty} \Vert e^{iHt} ({\Phi}_{\alpha} - \tilde{{\Phi}}_{\alpha})
e^{-iH_{\alpha t}} \vert \psi_{\alpha} \rangle  \Vert =
\lim_{t \to \pm \infty} \Vert ({\Phi}_{\alpha} - \tilde{{\Phi}}_{\alpha})
e^{-iH_{\alpha t}} \vert \psi_{\alpha} \rangle  \Vert 
\label{S1:33}
\eeq
which is precisely the condition that the injection operators
agree asymptotically.

The scattering operator for scattering from channel
$\alpha$ to channel $\beta$  is
\beq
S_{\beta,\alpha}:{\cal H}_{\alpha} \to {\cal H}_{\beta}: =
\Omega^{\dagger}_{\beta +}(H,\Phi_\beta,H_\beta)
\Omega_{\alpha -}(H,\Phi_\alpha,H_\alpha).
\label{S1:34}
\eeq
It follows from the identity (\ref{S1:33}) that
$S_{\alpha\beta}$ is independent of the choice of injection operator.
If follows from (\ref{S1:31}) that  
\beq
U_{\beta}(\Lambda,a)S_{\beta,\alpha} =
S_{\beta,\alpha}U_{\alpha}(\Lambda,a) .
\label{S1:35}
\eeq

This can be extended to all channels, ${\cal A}$, by defining
the asymptotic Hilbert space as the direct sum of all
channel Hilbert spaces ${\cal H}_{\alpha}$ for $\alpha \in {\cal A}$,
where ${\cal A}$ is defined to include the vacuum and one-particle channels
as well as the scattering channels:
\beq
{\cal H}_{\cal A}:= \oplus_{\alpha \in {\cal A}} {\cal H}_{\alpha}. 
\label{S1:36}
\eeq
The asymptotic unitary representation of the Poincar\'e group is defined
on ${\cal H}_{\cal A}$ by
\beq
U_{\cal A}(\Lambda, a) = \oplus_{\alpha \in {\cal A}} U_{\alpha}(\Lambda, a).
\label{S1:37}
\eeq

Multi-channel injection operators are defined as the sum of all channel
injection operators
\beq
\Phi_{\cal A}:= \sum_{\alpha \in {\cal A}} \Phi_{\alpha}
\qquad \mbox{or} \qquad 
\tilde{\Phi}_{\cal A}:= \sum_{\alpha \in {\cal A}} \tilde{\Phi}_{\alpha}
\label{S1:38}
\eeq
where $\Phi_{\alpha}:{\cal H}_{\alpha} \to {\cal H}$.
These can be used to define multi-channel wave operators
\beq
\Omega_{{\cal A}  \pm} (H,\Phi_{\cal A},H_{\cal A})
=
\tilde{\Omega}_{{\cal A}  \pm} (H,\tilde{\Phi}_{\cal A},H_{\cal A})
\label{S1:39}
\eeq
by the strong limits
\beq
\lim_{t \to \pm\infty}
\Vert (\Omega_{{\cal A} \pm} (H,\Phi_{\cal A},H_{\cal A}) -
e^{iHt} \Phi_{\cal A} e^{-i H_{\cal A} t}) \vert \psi \rangle \Vert =0
\label{S1:40}
\eeq
or
\beq
\lim_{t \to \pm\infty}
\Vert (\Omega_{{\cal A} \pm} (H,\tilde{\Phi}_{\cal A},H_{\cal A}) -
e^{iHt} \tilde{\Phi}_{\cal A} e^{-i H_{\cal A} t}) \vert \psi \rangle \Vert =0
\label{S1:41}
\eeq
where $H_{\cal A}:= \sum_{\alpha \in {\cal A}} H_{\alpha}\Pi_{\alpha}$ and
$\Pi_{\alpha}$ is the projection on the subspace ${\cal H}_{\alpha}$
of ${\cal H}_{\cal A}$.  The
assumed asymptotic completeness of the scattering theory means that
the wave operators (including the vacuum and one-body channels) are
unitary mappings from the asymptotic Hilbert space, ${\cal H}_{\cal
  A}$, to the Hilbert, ${\cal H}$, space of the quantum field theory.

With this definition (\ref{S1:31}) becomes
\beq
U(\Lambda ,a) \Omega_{{\cal A} \pm} (H,{\Phi}_{\cal A},H_{\cal A})=
\Omega_{{\cal A}\pm} (H,{\Phi}_{\cal A},H_{\cal A}) U_{{\cal A}} (\Lambda,a)
\label{S1:42}
\eeq
which also holds with $\Phi_{\cal A}$ replaced by $\tilde{\Phi}_{\cal A}$. 

The multi-channel scattering operator is
\beq
S(H,\Phi_{\cal A},{\cal H}_{\cal A})=
\Omega_+^{\dagger}(H,\Phi_{\cal A},{\cal H}_{\cal A})
\Omega_-(H,\Phi_{\cal A},{\cal H}_{\cal A})
\label{S1:43}
\eeq
where it is defined to be the identity on the vacuum and one-particle
subspaces.  


Poincar\'e invariance of the multi-channel
scattering operator on ${\cal H}_{\cal A}$ follows from the intertwining
property of the wave operators 
(\ref{S1:42}) \cite{simon}
\[
U_{\cal A}(\Lambda,a) S(H,\Phi_{\cal A},{\cal H}_{\cal A})=
U_{\cal A}(\Lambda,a) \Omega_+^{\dagger}(H,\Phi_{\cal A},{\cal H}_{\cal A})
\Omega_-(H,\Phi_{\cal A},{\cal H}_{\cal A})=
\]
\[
\Omega_+^{\dagger}(H,\Phi_{\cal A},{\cal H}_{\cal A})
U(\Lambda ,a)\Omega_-(H,\Phi_{\cal A},{\cal H}_{\cal A})=
\]
\beq
\Omega_+^{\dagger}(H,\Phi_{\cal A},{\cal H}_{\cal A})
\Omega_-(H,\Phi_{\cal A},{\cal H}_{\cal A})
U_{\cal A}(\Lambda,a) =
S(H,\Phi_{\cal A},{\cal H}_{\cal A})
U_{\cal A}(\Lambda,a).
\label{S1:44}
\eeq
At this point all that has been demonstrated is that the same
scattering theory is obtained by using injection operators that agree
asymptotically but intertwine different kinematic subgroups.  This
will be used in the next section to construct unitarily equivalent
light-front and instant-form representations of the field theory that
preserve the scattering matrix.

\section{Construction}

The next step is to use the kinematic symmetries of the injection operators
to show that the
Hamiltonian $H$ in the expressions for the wave operators
can be replaced by the light front Hamiltonian, $P^-$.  The relations
\beq
H= P^-+P^3 \qquad H = {1 \over 2}(P^++P^-)
\label{S2:1}
\eeq
will be used in what follows.
In the first case note because $P^3$ intertwines with
$\Phi_{{\cal A}}$ it follows that
\beq
e^{iHt}\Phi_{{\cal A}}e^{-iH_{\cal A}} = 
e^{i(P^-+P^3)t}\Phi_{{\cal A}}e^{-i(P^-_{\cal A}+ P^3_{\cal A})t}=
e^{iP^-t}\Phi_{{\cal A}}e^{-iP^-_{\cal A}t}.
\label{S2:2}
\eeq
Similarly since $P^+$ intertwines with $\tilde{\Phi}_{\cal A}$ 
\beq
e^{iHt}\tilde{\Phi}_{{\cal A}}e^{-iH_{\cal A}} = 
e^{i(P^-+P^+)t/2}\tilde{\Phi}_{{\cal A}}e^{-i(P^-_{\cal A}+ P^+_{\cal A})t/2}=
e^{iP^-t/2}\tilde{\Phi}_{{\cal A}}e^{-iP^-_{\cal A}t/2}.
\label{S2:3}
\eeq
This means
the wave operators constructed using $H$ are identical
to the ones using $P^-$ with both injection operators.
Combining these results with (\ref{S1:39}) gives the following
identifications
\beq
\Omega_{\pm}(H,\Phi_{\cal A},H_{\cal A}) =
\Omega_{\pm}(H,\tilde{\Phi}_{\cal A},H_{\cal A}) =
\Omega_{\pm}(P^-,\Phi_{\cal A},P^-_{\cal A}) =
\Omega_{\pm}(P^-,\tilde{\Phi}_{\cal A},P^-_{\cal A}). 
\label{S2:4}
\eeq
The final step in the construction is to introduce two
unitary operators $V_F$ and $V_I$ that act on 
field theory Hilbert space ${\cal H}$ with the following
properties:
\beq
V_F \vert 0 \rangle = V_I \vert 0 \rangle = \vert 0 \rangle
\label{S2:5}
\eeq
\beq
[U(\Lambda_{K_I},a_{K_I}),V_I=0] \qquad [U(\Lambda_{K_F},a_{K_F}),V_F]=0   
\label{S2:6}
\eeq
\beq
\lim_{t \to \pm \infty}
\Vert (\Phi_{\cal A} - V_I\Phi_{\cal A}) e^{-i H_{\cal A}t}\vert \psi_{\alpha} \rangle \Vert =0
\label{S2:7}
\eeq
\beq
\lim_{t \to \pm \infty}
\Vert (\tilde{\Phi}_{\cal A} - V_F\tilde{\Phi}_{\cal A}) e^{-i H_{\cal A}t}\vert \psi_{\alpha} \rangle \Vert =0 
\label{S2:8}
\eeq
where $(\Lambda_{K_I},a_{K_I})$ and $(\Lambda_{K_F},a_{K_F})$ are in the
instant-form and light-front kinematic subgroups respectively. 
Note that for equations (\ref{S2:7} and \ref{S2:8}) to hold the
asymptotic Hamiltonian must have a non-trivial absolutely continuous
spectrum.  While this is true on the scattering channel subspaces,
it is not true for the vacuum or one-particle states.  This
implies that in addition to (\ref{S2:5}) that the unitary
operators $V_I$ and $V_f$ also act like the identity on the one-particle subspaces.

The next step is to define two new unitary representations of the Poincar\'e
group on the Hilbert space ${\cal H}$ of the quantum field theory by:
\beq
U_F(\Lambda ,a):=V_F U(\Lambda,a) V_F^{\dagger}
\qquad
U_I(\Lambda ,a):=V_I U(\Lambda,a) V_I^{\dagger}
\label{S2:9}
\eeq
with the corresponding dynamical generators:
\beq
H_F:=V_F H V_F^{\dagger} \qquad  H_I:=V_I H V_I^{\dagger}
\label{S2:10}
\eeq
\beq
P^-_F := V_F P^- V_F^{\dagger}  \qquad P^-_I := V_I P^- V_I^{\dagger}.  
\label{S2:11}
\eeq
It follows from (\ref{S2:6}) that

\beq
U_I(\Lambda_{K_I} ,a_{K_I}) =U(\Lambda_{K_I},a_{K_I}) 
\qquad
U_F(\Lambda_{K_F} ,a_{K_F}):= U(\Lambda_{K_F},a_{K_F})
\label{S2:12}
\eeq
where $(\Lambda_{K_I} ,a_{K_I})$ is an instant-form kinematic  Poincar\'e
transformation and $(\Lambda_{K_F} ,a_{K_F})$ is a front-form kinematic
transformation.

In Appendix \ref{seq} it is  shown \cite{Ekstein:1960} that
as a consequence of
(\ref{S2:7}) and (\ref{S2:8}) that  
\beq
\Omega_{\pm}(H_I,\Phi_{\cal A},H_{\cal A}) =
V_I \Omega_{\pm}(H,\Phi_{\cal A},H_{\cal A})
\label{S2:13}
\eeq
and
\beq
\Omega_{\pm}(H_F,\tilde{\Phi}_{\cal A},H_{\cal A}) = V_F \Omega_{\pm}(H_F,\tilde{\Phi}_{\cal A},H_{\cal A}).
\label{S2:14}
\eeq
  
Taken together with (\ref{S2:4}) gives:  
\beq
\Omega_{\pm}(H_I,\Phi_{\cal A},H_{\cal A}) =
V_I \Omega_{\pm}(H,\Phi_{\cal A},H_{\cal A}) =
V_I \Omega_{\pm}(H,\tilde{\Phi}_{\cal A},H_{\cal A})=
V_I \Omega_{\pm}(P^-,\tilde{\Phi}_{\cal A},P^-_{\cal A})=
V_IV_F^{\dagger} \Omega_{\pm}(P_F^-,\tilde{\Phi}_{\cal A},P^-_{\cal A}).
\label{S2:15}
\eeq
Since (\ref{S2:15}) holds for the same $V_IV_f^{\dagger}$ for both time limits
it follows that 
\[
S(H_I,\Phi{\cal A},H_{\alpha}) =
\Omega^{\dagger}_{+}(H_I,\Phi_{\cal A},H_{\cal A})
\Omega_{-}(H_I,\Phi_{\cal A},H_{\cal A}) =
\]
\[
\Omega^{\dagger}_{+}(H_F,\tilde{\Phi}_{\cal A},H_{\cal A})
V_FV_I^{\dagger} V_IV_F^{\dagger}
\Omega_{-}(H_F,\tilde{\Phi}_{\cal A},H_{\cal A}) =
\]
\[
\Omega^{\dagger}_{+}(P^-_F,\tilde{\Phi}_{\cal A},P^-_{\cal A})
\Omega_{-}(P^-_F,\tilde{\Phi}_{\cal A},P^-_{\cal A}) =
\]
\beq
S(P^-_F,\tilde{\Phi}_{\cal A},P^-_{\cal A}).
\label{S2:17}
\eeq
This means both representations of the dynamics result in the same scattering
operators on the asymptotic Hilbert space.
Next consider the two unitary representations of
the Poincar\'e group, $U_I(\Lambda,a)$ and $U_F(\Lambda,a)$
defined above.
For the first one the operators $\{\mathbf{P},s,s_z\}$
are mutually commuting self-adjoint functions of the instant-form
kinematic generators of the representation $U_I(\Lambda,a)$.  For the
second one the operators $\{\tilde{\mathbf{P}},s_{z_f}\}$
are mutually commuting self-adjoint functions of the
front-form kinematic generators of the representation
$U_F(\Lambda,a)$.

It is possible to construct a basis for $U_I(\Lambda,a)$
consisting of the eigenvalues of $\{\mathbf{P},s,s_z\}$ and
some additional kinematically invariant commuting observables $x_I$.
Similarly it is possible to construct a basis for
$U_F(\Lambda,a)$ consisting of the eigenvalues of $\{\tilde{\mathbf{P}},s_{zF}\}$ and some additional kinematically invariant commuting observables $x_F$.

It follows that in these bases wave functions have the form
\beq
\langle \mathbf{p}, s,\mu ,x_I \vert \psi \rangle 
%\label{S2:18}
%\eeq
\qquad \mbox{or} \qquad
%or
%\beq
\langle \tilde{\mathbf{p}} ,\mu ,x_F \vert \psi \rangle .
\label{S2:19}
\eeq
Matrix elements of the dynamical operators
are non-trivial matrices in the $''x''$ variables: 
\beq
\langle \mathbf{p}, s,\mu ,x_I \vert H_I \vert
\mathbf{p}', s',\mu' ,x_I'  \rangle =
\delta (\mathbf{p}-\mathbf{p}') \delta_{ss'}\delta_{\mu \mu'} 
\langle x_I \Vert H_I(\mathbf{P},s) \Vert x_I' \rangle
\label{S2:20}
\eeq
and
\beq
\langle \tilde{\mathbf{p}},\mu ,x_F \vert P^-_F \vert
\tilde{\mathbf{p}}',\mu' ,x_F'  \rangle =
\delta (\tilde{\mathbf{p}}-\tilde{\mathbf{p}}') \delta_{\mu \mu'} 
\langle x_F \Vert P^-_F(\tilde{\mathbf{P}},\mu) \Vert x_F' \rangle .
\label{S2:21}
\eeq

The eigenvalue problems for the dynamical operators have the forms
\beq
\sum\int \langle x_I \Vert H_I(\mathbf{P},s) \Vert x_I' \rangle
dx_I' \langle \mathbf{p}, s,\mu ,x_I \vert \psi \rangle =
E(\mathbf{P},s) \langle \mathbf{p}, s,\mu ,x_I \vert \psi \rangle
\label{S2:22}
\eeq
and
\beq
\sum\int 
\langle x_F \Vert P^-_F(\tilde{\mathbf{P}},\mu) \Vert x_F' \rangle
dx_F' \langle \tilde{\mathbf{p}},\mu ,x_F' \vert \psi \rangle =
P^-(\tilde{\mathbf{P}},\mu) \langle \tilde{\mathbf{p}},\mu ,x_F
\vert \psi \rangle .
\label{S2:23}
\eeq
The matrices
$\langle x_I \Vert H_I(\mathbf{P},s) \Vert x_I' \rangle$ or
$\langle x_F \Vert P_F^-(\tilde{\mathbf{P}},\mu) \Vert x_F' \rangle$
must be diagonalized in this basis in order to compute
dynamical Poincar\'e transformations.  On the other hand
kinematic transformations can be computed by applying the inverse
transformation to basis vectors:
\[
\langle \mathbf{p}, s,\mu ,x_I \vert U_I(R,\mathbf{a})\vert  \psi \rangle =
\langle \psi \vert  U_I^{\dagger}(R,\mathbf{a}) \vert
\mathbf{p}, s,\mu ,x_I  \rangle^* =
\]
\beq
e^{i \mathbf{a} \cdot \mathbf{p}} \langle 
R^{-1}\mathbf{p}, s,\nu ,x_I \vert \psi  \rangle
D^s_{\mu \nu}(R).
\label{S2:24}
\eeq
Similarly in the light-front case for light-front preserving
boosts and translations:
\[
\langle \tilde{\mathbf{p}},\mu ,x_F \vert U_F(\Lambda_{KF},a_{KF})\vert  \psi \rangle =
\langle \psi \vert  U_F^{\dagger}(\Lambda_{KF},a_{KF}) \vert
\tilde{\mathbf{p}},\mu ,x_F \rangle^* = 
\]
\beq
e^{i \tilde{\mathbf{a}} \cdot \tilde{\mathbf{p}}}
\langle \tilde{\pmb{\Lambda}}^{-1}_k p,\mu ,x_F \vert  \psi \rangle
\sqrt{{ (\Lambda^{-1}_k p)^+ \over p^+}}
\label{S2:25}
\eeq
and for rotations about the $z$ axis
\[
\langle \tilde{\mathbf{p}},\mu ,x_F \vert U_F(R_z(\phi),0) \psi \rangle =
\langle \psi \vert  U_F^{\dagger}(R_z(\phi)) \vert
\tilde{\mathbf{p}},\mu ,x_F \rangle^* = 
\]
\beq
\langle \tilde{\mathbf{R}}_z(\phi)^{-1}p,\mu ,x_F \vert  \psi \rangle
s^{i \mu \phi}.
\label{S2:26}
\eeq
Equation (\ref{S2:24}) shows that $U_I(\Lambda,a)$ has an instant form kinematic subgroup
while equations (\ref{S2:25}) and (\ref{S2:26}) show that
$U_F(\Lambda,a)$
has a light-front
kinematic subgroup.  In addition the two representation are related
by a unitary transformation that preserves the vacuum and one particle
states:
\beq
U_F(\Lambda,a)= V_FV_I^{\dagger}U_I(\Lambda,a)V_IV_f^{\dagger}
\label{S2:27}
\eeq
\beq
\vert 0 \rangle_F =  V_FV_I^{\dagger} \vert 0 \rangle_I = \vert 0 \rangle_I .
\label{S2:28}
\eeq

If follows from (\ref{S2:17}) that both representations give the same unitary scattering
operators on the asymptotic Hilbert space ${\cal H}_{\cal A}$.  In addition
because $V_I$ and $V_f$ are kinematically invariant (\ref{S2:6}),
the injection operators satisfy
\beq
U_I(\Lambda_{KI} ,a_{KI}) \Phi_{\cal A} =
\Phi_{\cal A}U_{\cal A}(\Lambda_{KI} ,a_{KI})
\label{S2:29}
\eeq
for the instant form kinematic subgroup and 
\beq
U_F(\Lambda_{KF} ,a_{KF}) \tilde{\Phi}_{\cal A} =
\tilde{\Phi}_{\cal A}U_{\cal A}(\Lambda_{KF} ,a_{KF})
\label{S2:30}
\eeq
for the light-front kinematic subgroup.

The result can be summarized by noting that it is possible to
choose bases in the field theory Hilbert space that transforms
covariantly under either kinematic subgroup.  In both cases a
free dynamics is not assumed. The two representations are related
by $S$-matrix preserving unitary transformations.  Both representation have
the same vacuum and one particle subspaces.  There are no bare particles
or production vertices in these representations.

Finally note that if the range of $\tilde{\Phi}$ is all of
${\cal H}$ and
\beq
W:=  (\tilde{\Phi}_{\cal A} \tilde{\Phi}_{\cal A}^{\dagger})^{-1/2} \tilde{\Phi}_{\cal A} 
\label{S2:31}
\eeq
satisfies
\beq
\lim_{t \to \pm \infty}
\Vert (W -\tilde{\Phi}_{\cal A}) e^{-i H_{\cal A}t}\vert \psi_{\alpha} \rangle \Vert =0
\label{S2:32}
\eeq
then it 
is a kinematically invariant unitary mapping from
${\cal H}_{\cal A}$ to ${\cal H}$ and
$U_0(\Lambda,a) = W U_{\cal A}(\Lambda,a) W^{\dagger}$
defines a consistent ``free dynamics'' on ${\cal H}$
that satisfies
\beq
U_I(\Lambda_{KI},a_{KI}) = U_0(\Lambda_{KI},a_{KI})
\label{S2:33}
\eeq
with a similar relation in the light-front case.

\section{Analysis and Conclusions}

This work demonstrated the equivalence of formulations of quantum
field theories with light-front and instant-form kinematic symmetries.
It differs from most approaches to this problem because it is
completely non-perturbative.  This results in a number of differences
with approaches that assume the existence of both a free and
interacting unitary representation of the Poincar\'e group on the
Hilbert space representation of the field theory.  The problem with
this assumption is that in a local field theory the free and dynamical
unitary representations of the Poincar\'e group act on inequivalent
representations of the Hilbert space.  This requires re-thinking about
the meaning of instant and front-form dynamics.  In this approach,
kinematic Poincar\'e transformations can be performed without
diagonalizing the mass operator.  Both the light front and instant
representations are unitarily equivalent and give the same
multichannel scattering operators.  Some of the differences are (1)
there are no true multi-particle states in the field theory; that
notion only makes sense asymptotically.  This also means that there is
no Fock space and no ``quantization'' (2) the fields are not
restricted to an instant or light front hyperplane (3) there is no
free-field dynamics (4) there are no zero modes (5) there is one
vacuum, and all of the particles are physical.  The advantage of
starting with the Hilbert space representation of the interacting theory
is that all of the issues with renormalization are defined away.

This work provides one way to understand Dirac's forms of dynamics in a
non-perturbative setting.  This approach assumed the
existence of local quantum field theory with all of the expected
properties.  Using this assumption unitarily equivalent representations with the
same scattering operators were constructed.  Bases that transformed
covariantly with respect to light-front and instant-form kinematic
subgroups where constructed for each of these equivalent
representations of the dynamics.  Kinematic Poincar\'e transformations
on these wave functions could be computed without diagonalizing the
dynamical operators.  One aspect of the these representations of the
dynamics is that they both act irreducibly on the vacuum and {\it
  one-particle} states.  This means that the 0 and 1 body solutions
are basis vectors in both bases.  Both representations give the same
Poincar\'e invariant scattering operator.  The Haag-Ruelle injection
operators that are used to formulate the scattering asymptotic
conditions in both cases intertwine with corresponding kinematic
subgroups.  If the injection operators have full range on the Hilbert
space of the field theory there is the analog of a free dynamics on
the Hilbert of the field theory, it has the same vacuum as the full
theory and the particles all have physical masses (no mass
renormalization).  The construction of equivalent instant-form and
light-front unitary representations of the Poincar\'e group is
essentially identical to the corresponding construction in the finite
number of degree of freedom case
\cite{Sokolov:1977im}\cite{Keister:1996bd}\cite{Polyzou:19892b}\cite{Polyzou:2002cp}.
In that case, when there is production, the particles all have their
physical masses and there are no vertex interactions that can change
masses \cite{Polyzou:2003dt}. The simplest interactions that change particle
number are $2 \leftrightarrow 3$ interactions.

While the representations discussed in this analysis are not useful
for applications, the results support the conclusions of most of the
quoted references that the two formulations of the theory are
equivalent.  The vacuum is the same in both representations.  While
the vacuum is kinematically invariant, it is also rotationally invariant,
which is an additional constraint on the vacuum in the perturbative case. 

\appendix
\section{Scattering equivalences}
\label{seq}
The two-Hilbert space multi-channel wave operators are
defined by the strong limits where the $\Phi_{\cal A}$ are
two-Hilbert space injection operators that map the
asymptotic Hilbert space ${\cal H}_{\cal A}$ to the Hilbert space
${\cal H}$ of the quantum field theory:
\beq
\Omega_\pm(H,\Phi_{\cal A},H_{\cal A}) =
s-\lim_{t \to \pm \infty}
e^{iHt} \Phi_{\cal A} e^{-iH_{\cal A}t}. 
\label{A1:1}
\eeq
The wave operators are assumed to exist and be Poincar\'e
invariant in the sense
\beq
U(\Lambda ,a)  \Omega_\pm^{\dagger}(H,\Phi_{\cal A},H_{\cal A})=
\Omega_\pm^{\dagger}(H,\Phi_{\cal A},H_{\cal A})U_{\cal A} (\Lambda ,a).
\label{A1:2}
\eeq
where $U_{\cal A} (\Lambda ,a)$ is the natural representation of the
Poincar\'e group on ${\cal H}_{\cal A}$ that has the form of a direct
sum of tensor products of irreducible representations.  The set of
channels ${\cal A}$ is assumed to include the vacuum channel, and
one-particle channels in addition to multi-particle scattering
channels.  The wave operators are assumed to be asymptotically complete
unitary mappings from asymptotic Hilbert space ${\cal H}_{\cal A}$ to
the Hilbert space ${\cal H}$ of the field theory.  

The scattering operator is defined by 
\beq
S(H,\Phi,H_{\cal A}) =
\Omega_+^{\dagger}(H,\Phi_{\cal A},H_{\cal A})
\Omega_-(H,\Phi_{\cal A},H_{\cal A}).
\label{A1:3}
\eeq
Consider two identical scattering operators based on different
Hamiltonian's
\beq
S(H,\Phi_{\cal A},H_{\cal A}) =S(H',\Phi_{\cal A}',H_{\cal A}) 
\label{A1:4}
\eeq
It follows from the definitions and unitarity of the
wave operators that
\beq
W:= \Omega_+(H',\Phi'_{\cal A},H_{\cal A})
\Omega_+^{\dagger}(H,\Phi_{\cal A},H_{\cal A})=
\Omega_-(H',\Phi'_{\cal A},H_{\cal A})
\Omega_-^{\dagger}(H,\Phi_{\cal A},H_{\cal A})
\label{A1:5}
\eeq
is a unitary operator on ${\cal H}$.
The identification of the scattering operators (\ref{A1:4}) means
that $W$ is the same for the incoming ($t \to +\infty$) or outgoing
($t \to -\infty$) multi-channel wave operator.
It follows that
\beq
W \Omega_\pm(H,\Phi_{\cal A},H_{\cal A})
=
\Omega_\pm(H',\Phi_{\cal A}',H_{\cal A}).
\label{A1:6}
\eeq
In addition, the intertwining property of the wave operators \cite{simon},
\[
WH'=
\Omega_\pm(H,\Phi_{\cal A},H_{\cal A})
\Omega_\pm^{\dagger}(H',\Phi_{\cal A}',H_{\cal A})H'
=
\Omega_\pm(H,\Phi_{\cal A},H_{\cal A})
H_{\cal A} \Omega_\pm^{\dagger}(H',\Phi'_{\cal A},H_{\cal A})
\]
\beq
=
H\Omega_\pm(H,\Phi_{\cal A},H_{\cal A})
\Omega_\pm^{\dagger}(H',\Phi'_{\cal A},H_{\cal A})=
HW
\label{A1:7}
\eeq
means that the two Hamiltonians are related by the unitary
transformation $W$.

It follows that
\[
\Omega_-(H',\Phi_{\cal A}',H_{\cal A}) = 
W \Omega_-(H,\Phi_{\cal A},H_{\cal A}) =
\]
\beq
\Omega_-(WHW^{\dagger} ,W\Phi_{\cal A},H_{\cal A})=
\Omega_-(H',W\Phi_{\cal A},H_{\cal A}).
\label{A1:8}
\eeq
Taking the difference of the right and left side of equation 
(\ref{A1:8})
gives
\beq
0=\lim_{t \to \pm \infty}
\Vert e^{iH't} (\Phi_{\cal A}' - W\Phi_{\cal A}) e^{-iH_{\cal A}t}\vert \psi \rangle \Vert .
\label{A1:9}
\eeq
The unitarity of gives $e^{iH't}$
\beq
0=\lim_{t \to \pm \infty} \Vert (\Phi_{\cal A}' - W\Phi_{\cal A})
e^{-iH_{\cal A}t}\vert \psi \rangle \Vert .
\label{A1:10}
\eeq
That this condition holds for both time limits is important.

Next consider the converse.  Assume that $W$ is a unitary operator
satisfying $H'=WHW^{\dagger}$, and the scattering operators
\beq
S(H,\Phi,H_{\cal A}) =
\Omega_+^{\dagger}(H,\Phi_{\cal A},H_{\cal A})
\Omega_-(H,\Phi_{\cal A},H_{\cal A})
\label{A1:11}
\eeq
\beq
S(H',\Phi'_{\cal A},H_{\cal A}) =
\Omega_+^{\dagger}(H',\Phi'_{\cal A},H_{\cal A})
\Omega_-(H',\Phi'_{\cal A},H_{\cal A})
\label{A1:12}
\eeq
both exist.
If $W$ satisfies (\ref{A1:10}) for both time limits then the $S$
matrices are identical
\beq
S(H',\Phi'_{\cal A},H_{\cal A}) = S(H,\Phi_{\cal A},H_{\cal A}).
\label{A1:13}
\eeq
The proof follows from
\[
\Omega_\pm(H',\Phi'_{\cal A},H_{\cal A}) =
\Omega_\pm(WHW^{\dagger} ,\Phi'_{\cal A},H_{\cal A})=
W\Omega_\pm(H,W^{\dagger}\Phi'_{\cal A},H_{\cal A})=
\]
\beq
W\Omega_\pm(H,W^{\dagger}(\underbrace{\Phi'_{\cal A}- W\Phi{\cal A}}_{\to 0} + W\Phi_{\cal A}),H_{\cal A})=
W\Omega_\pm(H,\Phi_{\cal A}),H_{\cal A}).
\label{A1:14}
\eeq
It then follows that
\[
S(H',\Phi'_{\cal A},H_{\cal A})=
\Omega_+^{\dagger}(H',\Phi'_{\cal A},H_{\cal A})
\Omega_-(H',\Phi'_{\cal A},H_{\cal A})=
\]
\beq
\Omega_+^{\dagger}(H,\Phi_{\cal A},H_{\cal A})W^{\dagger}
W
\Omega_-(H,\Phi_{\cal A},H_{\cal A})=
S(H,\Phi_{\cal A},H_{\cal A}).
\label{A1:15}
\eeq
Note that unitary equivalence is not a sufficient condition for
$S$-matrix equivalence.  This is not hard to understand in the case of ordinary
quantum mechanics, where two Hamiltonian's with short-range
repulsive interactions have
the same spectrum (so they are unitarily equivalent) but generally have
different $S$-matrix elements or phase shifts.

The conclusion of this section is that unitary operators $W$ that
satisfy the asymptotic condition (\ref{A1:10}) can be used to relate
Hamiltonians that have the same scattering matrix.  This asymptotic
condition means that $W$ does not
disturb the asymptotic structure of the states that define the
asymptotic condition.

The above condition applies to both quantum mechanics and quantum
field theory assuming that the have asymptotically complete scattering
operators.  Equation (\ref{A1:10}) requires that $W$ leaves the
one-particle states unchanged.  The field theory generalization of
multi-particle scattering is the Haag-Ruelle formulation of scattering
which involves strong limits used in this appendix.


\section{Two Hilbert space injection operators}
\label{inj}
This appendix discusses the construction of injection operators from operators
in the field algebra.  The starting point is to let $B$ be a function of the
fields that creates a one-body state out of the vacuum.  This means
that the completeness sum 
\beq
\langle 0 \vert B^{\dagger}B \vert 0 \rangle =
\sum_n \langle 0 \vert B^{\dagger}\vert n \rangle
\langle n\vert  B \vert 0 \rangle 
\label{A2:1}
\eeq
has one-body intermediate states (here one-body means states with
discrete positive mass eigenvalues - there is no distinction between
elementary and composite one-body states).  For simplicity it is
assumed that the one-body spectrum is non-degenerate, which means each
one-particle state has a different mass, and each one-body mass
eigenstate has a given spin.

To isolate the operators that create the one-body states use
space-time translations to define the operator valued distributions
\beq
B(x) := e^{-i P\cdot x} B e^{-i P\cdot x} = U^{\dagger}(I,x) B U(I,x)
\label{A2:2}
\eeq
where $P^{\mu}$ is the four momentum operator.  It follows from
(\ref{A2:2}) that
\beq
{\partial B(x)\over \partial x_{\mu}}= -i [P^{\mu},B(x)].
\label{A2:3}
\eeq
The next step is to compute the Fourier transform of the operator density
$B(x)$: 
\beq
\hat{B}(q) := \int {d^4x \over (2 \pi)^2}
e^{iq\cdot x} B(x) .
\label{A2:4}
\eeq
Multiplying both sides of (\ref{A2:3}) by ${e^{iq\cdot x}\over (2 \pi)^2}$
and integrating over $d^4x$ by parts, assuming that ${B}(x)$ is an operator
valued distribution, gives
\beq
-i [P^{\mu},\hat{B}(q)]= -i q^{\mu} \hat{B}(q)
\label{A2:5}
\eeq
or 
\beq
[P^{\mu},\hat{B}(q)] = q^{\mu} \hat{B}(q).
\label{A2:6}
\eeq
It follows from (\ref{A2:6}) that $\hat{B}(q)\vert 0 \rangle$ is
either $0$ or an
eigenstate of the four momentum with eigenvalue $q^{\mu}$:
\beq
P^{\mu}\hat{B}(q) \vert 0 \rangle =
\hat{B}(q) P^{\mu}  \vert 0 \rangle +
q^{\mu} \hat{B}(q)\vert 0 \rangle=
q^{\mu} \hat{B}(q)\vert 0 \rangle .
\label{A2:7}
\eeq

Let $m$ be a discrete mass eigenvalue of $M=\sqrt{-P^2}$ and let $s$
denote the spin of the particle of mass $m$.  Next let $h(q)$ be a
smooth Lorentz invariant function of compact support in $q^2$ that is 1
when $q^2=-m^2$ and $m>0$, and
identically $0$
on the rest of the spectrum of intermediate states in (\ref{A2:1}) and
define
\beq
{A}^{\dagger}(q):= \hat{B}(q)h(q).
\label{A2:8}
\eeq
Also define the operators:
\beq
{A}^{\dagger}(\mathbf{q}):= \int {A}^{\dagger}(q)  dq^0
%\label{A2:8a}
%\eeq
%\beq
\qquad
\mbox{and}
\qquad
\tilde{A}^{\dagger}(\tilde{\mathbf{q}}):= \int {A}(q) dq^-/2 .
\label{A2:9}
\eeq
Because $h(q)$ is localized these operators are distributions in the
three momentum or light-front components of the four momentum.  When
these are applied to the vacuum they create one-particle states of
mass $m$.  Since by assumption, there is a unique spin $s$ associated
with each discrete mass eigenvalue, the particle created out of the
vacuum also has spin $s$.

Because of the integrals over $q^0$ or $q^-$ 
%affect the remaining components of $q^{\mu}$ 
it follows that (\ref{A2:6}) must be replaced by 
\beq
[\mathbf{P},{A}^{\dagger}(\mathbf{q})] = \mathbf{q} {A}^{\dagger}(\mathbf{q}) .
\qquad
[\tilde{\mathbf{P}},\tilde{A}^{\dagger}(\tilde{\mathbf{q}})] =
\tilde{\mathbf{q}} \tilde{A}^{\dagger}(\tilde{\mathbf{q}}) 
\label{A2:10}
\eeq
which when applied to the vacuum becomes
\beq
\mathbf{P}A^{\dagger}(\mathbf{q}) \vert 0 \rangle  = \mathbf{q}{A}^{\dagger} (\mathbf{q})\vert 0 \rangle
\qquad
\tilde{\mathbf{P}}\tilde{A}^{\dagger}(\tilde{\mathbf{q}}) \vert 0 \rangle  =
\tilde{\mathbf{q}}\tilde{A}^{\dagger} (\tilde{\mathbf{q}})\vert 0 \rangle .
\label{A2:11}
\eeq
This means that ${A}^{\dagger}(\mathbf{q})$ creates a one-particle
state with momentum $\mathbf{q}$ and mass $m$ and spin $s$ out of the vacuum.
Similarly
\beq
\tilde{\mathbf{P}}
\tilde{A}^{\dagger}( \tilde{\mathbf{q}}) \vert 0 \rangle  =
\tilde{\mathbf{q}}{A}^{\dagger} (\tilde{\mathbf{q}})\vert 0 \rangle .
\label{A2:12}
\eeq
creates a one-particle state with light front momentum $\mathbf{q}$
and mass $m$ and spin $s$ out of the vacuum.

The next step is to construct operators that create simultaneous
eigenstates of mass and spin and magnetic quantum numbers.

Define the operator 
\beq
{A}^{\dagger}_{m,s}( \mathbf{p},\mu) := \int_{SU(2)}
U(R,0){A}^{\dagger}(R\mathbf{p} )U^{\dagger}(R,0)
D^{s*}_{\mu s}(R)
%D^{s*}_{\mu s}(R^{\prime-1}R^{\prime \prime})
dR 
\label{A2:13}
\eeq
where the integral is over the $SU(2)$ Haar measure and
$D^{s*}_{\mu s}(R)$ is the spin-$s$ $SU(2)$ Wigner function
\beq
D^{s}_{\mu s}(R) = \langle s, \mu \vert U(R)\vert s,s \rangle .
\label{A2:14}
\eeq

It follows from (\ref{A2:13}) that
\[
U(R',0){A}^{\dagger}_{m,s}(\mathbf{p},\mu)U^{\dagger}(R',0) =
\]
\beq
\int_{SU(2)}
U(R'R,0){A}^{\dagger}(R^{-1}\mathbf{p} )U^{\dagger}(R'R,0) D^{s*}_{\mu s}(R) dR .
\label{A2:15}
\eeq
Changing variables,  
$R''=R'R$ using the invariance of the Haar measure $dR = dR''$
for fixed $R'$ gives
\[
(\ref{A2:15})=\int_{SU(2)}
U(R'',0){A}^{\dagger}(R^{\prime \prime-1}R'\mathbf{p} )U^{\dagger}(R'',0) dR''
D^{s*}_{\mu s}(R^{\prime -1}R^{\prime \prime})=
\]
\[
\int_{SU(2)}\sum_{\nu=-s}^s
U(R'',0)A^{\dagger}(R^{\prime \prime-1}R'\mathbf{p} )U^{\dagger}(R'',0) dR''
D^{s*}_{\mu\nu} (R^{\prime -1})D^{s*}_{\nu s}(R^{\prime \prime})=
\]
\[
\int_{SU(2)}\sum_{\nu=-s}^s
U(R'',0){A}^{\dagger}(R^{\prime \prime-1}R'\mathbf{p} )U^{\dagger}(R'',0) dR''
D^{s*}_{\nu s}(R^{\prime -1}R^{\prime \prime})
D^{s}_{\nu\mu} (R^{\prime}) =
\]
\beq
\sum_{\nu=-s}^s
{A}^{\dagger}_{(m,j)}(R' \mathbf{p},\nu) D^{s}_{\nu\mu} (R^{\prime}).
\label{A2:16}
\eeq
When applied to the vacuum (\ref{A2:16}) gives  
\beq
U(R,0){A}^{\dagger}_{(m,s)}(\mathbf{p},\mu)\vert 0 \rangle  =
\sum_{\nu=-s}^s {A}^{\dagger}_{m,s}( R\mathbf{p},\nu)\vert 0 \rangle D^{s}_{\nu\mu} (R)
\label{A2:17}
\eeq
which means that either this vanishes or it transforms like a particle
of mass $m$, spin $s$, momentum $\mathbf{p}$ and magnetic quantum
number $\mu$.  The spin in these states created out of the vacuum is
the canonical spin.  This will vanish if there are no one-particle
intermediate states with mass $m$ and spin $s$ in (\ref{A2:1}).  While
the notation is purposely suggestive,
$A^{\dagger}_{m,s}(\mathbf{p},\mu)$ is {\it not a creation operator}.
In addition, $p^0$ is only equal to $\sqrt{\mathbf{p}^2+m^2}$ when
${A}^{\dagger}_{m,s}( \mathbf{p},\mu)$ is applied to the vacuum.

Because rotations do not change the time component, these operators
also satisfy
\[
[ \mathbf{P},{A}_{(m,s)}^{\dagger}(\mathbf{q},\mu)]   =
\int_{SU(2)} dR [\mathbf{P}, 
U(R,0){A}^{\dagger}(R^{-1}\mathbf{q} )U^{\dagger}(R,0)]
D^{s*}_{\mu s}(R) =
\]
\[
\int_{SU(2)} dR
U(R,0)[ R \mathbf{P},  {A}^{\dagger}(R^{-1}\mathbf{q} )]U^{\dagger}(R,0)
D^{s*}_{\mu s}(R) =
\]
\beq
 RR^{-1}\mathbf{q} \int_{SU(2)} dR
U(R,0)  {A}^{\dagger}(R^{-1}\mathbf{q} )U^{\dagger}(R,0)
D^{s*}_{\mu s}(R) =
\mathbf{q}
{A}^{\dagger}_{(m,s)} (\mathbf{q},\mu).
\label{A2:18}
\eeq
The normalization can be chosen so the states created out of the
vacuum have the normalization (\ref{S1:8}). 

The construction above cannot be used in the ``light-front'' case due to the
integral over $p^-$ in (\ref{A2:9}). However in that case it is enough
to project out the magnetic quantum number using a rotation
$R_z(\phi)$ which leaves $p^-$ in (\ref{A2:9}) unchanged.  The spin
is identified with the highest non-zero weight, $s=\mu_{max}$. In this case
equation(\ref{A2:13}) is replaced by
\beq
\tilde{{A}}^{\dagger}_{m,s}( \tilde{\mathbf{p}},\mu) := \int_{0}^{2\pi}
U(R_z(\phi),0)\tilde{{A}}^{\dagger}(R_z(\phi)^{-1}\mathbf{p} )U^{\dagger}(R_z(\phi),0)
e^{-i \mu \phi}
%D^{s*}_{\mu s}(R^{\prime-1}R^{\prime \prime})
{d\phi \over 2 \pi},
\label{A2:19}
\eeq
(\ref{A2:17}) and (\ref{A2:18}) are replaced by
\beq
[ \tilde{\mathbf{P}},\tilde{{A}}_{(m,s)}^{\dagger}(\tilde{\mathbf{q}},\mu)]   =
\tilde{\mathbf{q}}
\tilde{{A}}^{\dagger}_{(m,s)} (\tilde{\mathbf{q}},\mu),
\label{A2:20}
\eeq
\beq
U(\Lambda_K,0) \tilde{{A}}^{\dagger}_{(m,s)} (\tilde{\mathbf{q}},\mu)
U^{\dagger}(\Lambda_K,0) = \tilde{{A}}^{\dagger}_{(m,s)} (\tilde{\Lambda}_k q,\mu),
\label{A2:21}
\eeq
and
\beq
U(R_z(\phi),0) \tilde{{A}}^{\dagger}_{(m,s)}
(\tilde{\mathbf{q}},\mu) U^{\dagger} (R_z(\phi),0) =
\tilde{{A}}^{\dagger}_{(m,s)} (R_z(\phi)\tilde{\mathbf{q}},\mu)
e^{i\mu\phi}.
\label{A2:22}
\eeq
The proof (\ref{A2:20}) is essentially the same as the proof of (\ref{A2:18}).
To show (\ref{A2:22}) note that for rotations about the $z$ axis 
\beq
B_f(p/m) R_z = R_z B_f (R_z^{-1}p/m)
\label{A2:23}
\eeq
where $B_f(p/m)$ is a light front boost.  It follows that
\[
U(B_f(p/m),0)\tilde{{A}}^{\dagger}_{m,s}( \tilde{\mathbf{q}},\mu)
U(B_f(p/m),0)^{\dagger} =
\]
\[
\int_{0}^{2\pi}
U(B_f(p/m) R_z(\phi),0)\tilde{{A}}^{\dagger}(R_z(\phi)^{-1}\mathbf{q} )U^{\dagger}(
B_f(p/m)R_z(\phi),0)
e^{-i \mu \phi} =
\]
\[
\int_{0}^{2\pi}
U(R_z(\phi) B_f(R_z^{-1}(\phi)p/m),0)\tilde{{A}}^{\dagger}(R_z(\phi)^{-1}\mathbf{q} )U^{\dagger}(R_z(\phi) B_f(R_z^{-1}(\phi)p/m),0)
e^{-i \mu \phi}=
\]
\[
\int_{0}^{2\pi}
U(R_z(\phi),0)U( B_f(R_z^{-1}(\phi)p/m),0)\tilde{{A}}^{\dagger}(R_z(\phi)^{-1}\mathbf{q} )U(B_f(R_z^{-1}(\phi)p/m),0))^{\dagger}
U^{\dagger}(R_z(\phi),0)^{\dagger} 
e^{-i \mu \phi}=
\]
\[
\int_{0}^{2\pi}
U(R_z(\phi),0)U(,0)\tilde{{A}}^{\dagger}( B_f(R_z^{-1}(\phi)p/m)R_z(\phi)^{-1}\mathbf{q} )
U^{\dagger}(R_z(\phi),0)^{\dagger} 
e^{-i \mu \phi}=
\]
\[
\int_{0}^{2\pi}
U(R_z(\phi),0)U(,0)\tilde{{A}}^{\dagger}( R_z(\phi)^{-1} B_f(p/m)\mathbf{q} )
U^{\dagger}(R_z(\phi),0)^{\dagger} 
e^{-i \mu \phi}=
\]
\beq
\tilde{{A}}^{\dagger}_{m,s}( \tilde{\mathbf{
B}}_f(p/m)q ,\mu)
\label{A2:24}
\eeq
which proves (\ref{A2:21}). For (\ref{A2:22}) note that 
\[
U(R_z(\phi'),0) \tilde{{A}}^{\dagger}_{(m,s)}
(\tilde{\mathbf{q}},\mu) U^{\dagger} (R_z(\phi'),0) =
\]
\beq
\int_{0}^{2\pi}
U(R_z(\phi+\phi'),0)\tilde{{A}}^{\dagger}(R_z(\phi)^{-1}\mathbf{p} )U^{\dagger}(R_z(\phi+\phi'),0)
e^{-i \mu \phi}
%D^{s*}_{\mu s}(R^{\prime-1}R^{\prime \prime})
{d\phi \over 2 \pi}.
\label{A2:25}
\eeq
Let $ \phi''= \phi + \phi'$ so (\ref{A2:25})) becomes 
\[
\int_{0}^{2\pi}
U(R_z(\phi''),0)\tilde{{A}}^{\dagger}(R_z(\phi''-\phi')^{-1}\mathbf{p} )U^{\dagger}(R_z(\phi''),0)
e^{-i \mu (\phi''-\phi')}
%D^{s*}_{\mu s}(R^{\prime-1}R^{\prime \prime})
{d\phi'' \over 2 \pi}
\]
\[
\int_{0}^{2\pi}
U(R_z(\phi''),0)\tilde{{A}}^{\dagger}(R_z(\phi'')^{-1}R_z(\phi')\mathbf{p} )U^{\dagger}(R_z(\phi''),0)
e^{-i \mu (\phi''}e^{i\phi' \mu)} =
\]
\beq
\tilde{{A}}^{\dagger}_{(m,s)} (R_z(\phi')\tilde{\mathbf{q}},\mu)
e^{i\mu\phi'}.
\label{A2:26}
\eeq
The normalization can be chosen so these states created out of the
vacuum have the normalization (\ref{S1:8}). 


\bibliography{collins.bib}
\end{document}

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\cite{Werner:2006}
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\bibitem{collins} J. Collins,
%``The non-triviality of the vacuum in light-front quantization: An elementary treatment'',
arXiv:1801.03960v1(2018).

\bibitem{marc} M. Herrmann and W. N. Polyzou,
%$(D+$��(JLight-front vacuum,$(D+$��(J%
Phys. Rev., D{\bf}91,085043(2015).



When two one-particle states have identical masses then it is
necessary to find an observable that can be use to distinguish the
particles.

To perform the corresponding construction in the light-front case
we cannot use the constrution above the integral over the light front 

Note that the normalization can be adjusted so the one particle states
created out of the vacuum have ordinary or light-front delta function
normalizations.
${A}^{\dagger}_{m,s}( p,\mu)$  can be used to define
\beq
\tilde{A}^{\dagger}_{ms}(p, \mu)=
\sum_{\nu=-s}^s {A}^{\dagger}_{ms}(p, \nu)\sqrt{P^+ \over \omega_m(\mathbf{p})}
D^{s}_{\nu\mu} (B_c^{-1}(p_m/m) B_f(p_m/m))
\label{A2:15}
\eeq
where $p$ can represent $p^\mu$ or $p^-,p^+,\mathbf{p}_{\perp}$ and
$p_m$ represents the four momentum restricted to the mass shell.
This operator has the property that it creates a one-particle state
of mass $m$ and light-front spin magnetic quantum number
$\mu$ out of the vacuum light front spin.

Note that there is considerable freedom in choosing the operator
$B$ that was used to start this construction.    If these operators
are used to construct wave operators in the Haag-Ruelle
formulation, the resulting scattering operator is independent of the
choice of $B$. 
This is the origin of the field redefinition invariance
of the scattering matrix in quantum field theory.
 The choices used in this appendix have properties that
 are used in the body this paper.



Before constructing
the dynamical models of interest

The space time translational invariance of the multicahnnel
scattering operator means
\[
[P^{\mu}_{\cal A}, S] =0
\]
This means that matrix elements of $S$ in eigenstates of total linear momentum
have the form
\[
\delta (\mathbf{p}-\mathbf{p}') \hat{S}(\mathbf{p})
\]
and in terms of light front momentum 
\[
\delta (\tilde{\mathbf{p}}-\tilde{\mathbf{p}}') \hat{S}(\mathbf{p})
\]

This is because while the individual
$\Phi_{m_k,s_k,d_{kn_k}}(\tilde{\mathbf{p}_k},\mu_k)$ create
one-particle states out of the vacuum, it does not follow that they
create $k+1$ particle states out of $k$ particle states.  However, if
the product states are smeared with wave packets whose supports have
large space-like separations, and the field satisfy a cluster
separability conditions, then they behave like $N$-particle states.
In addition at large spacelike separations the order for the operators
does not matter.

The $\alpha$ asymptotic Hilbert space is defined as the
space of square integrable functions of the variables
${\mathbf{p}_k},\mu_k$ for $k \in \alpha$
or
$\tilde{\mathbf{p}}_k,\mu_k$ for $k \in \alpha$.
The asymptotic channel Hilbert space will be denoted by
${\cal H}_{\alpha}$.  The operator $\Phi_{\alpha}\vert 0 \rangle$
is a mapping from ${\cal H}_{\alpha}$ to the Hilbert space of
the field theory.

It is also useful to define the asymptotic Hilbert space ${\cal H}_{\cal A}$
as the direct sum of the channel Hilbert spaces over all channels -
including the vacuum and 1 body channels:
\[
{\cal H}_{\cal A} = \oplus_{\alpha \in {\cal A}} {\cal H}_{\alpha}
\]
An injection operator from ${\cal H}_{\cal A}$ to the field theoretic
Hilbert space is defined by 
\[
\Phi_{\cal A}  =\sum_{\alpha \in {\cal A}}
%{\cal H}_{\alpha}
\Phi_{\alpha} \vert 0 \rangle
\]
where each $\Phi_{\alpha}$ acts on the subspace ${\cal H}_{\alpha}$ of
${\cal H}_{\cal A}$,

The next step is to define an asymptotic representation of the
Poincar'e group on ${\cal H}_{\alpha}$ by
\[
U_\alpha(\Lambda ,a)\vert \mathbf{p}_1, \mu_1, \cdots
\mathbf{p}_n, \mu_n \rangle  =
e^{ia \cdot \Lambda (\sum_k p_k)}
\vert \pmb{\Lambda}p_1, \nu_1 \cdots \pmb{\Lambda}p_n, \nu_n \rangle
\prod
\sqrt{{ \omega_{m_k} (\pmb{\Lambda}p_k) \over \omega_{m_k} (\mathbf{p}_k)}}
D^s_{\nu\mu}[B_c^{-1}(\Lambda p_k/m_k) \Lambda B_c(p_k/m_k)]
\]
where $m_k$ is the mass of the corresponding on particle state.  This
is the transformation property of $N$ free particles of mass
$m_k$ and spin $s_k$, $k\in \alpha$.

There is a similar representation in the light front basis.

The asymptotic representation of the Poincar\'e group on the asymptotic
Hilbert space is defined as
\[
U_{\cal A}(\Lambda,a) := \sum_{\alpha \in {\cal A}} U_\alpha (\Lambda,a)
\]
where each $U_\alpha (\Lambda,a)$ acts on the mutually orthogonal subspace
${\cal H}_\alpha$.  

Next note that there is an additional freedom in terms of how to define the operators $\Phi((m,j,d_n)\mathbf{p},\mu)$.  The requirement that they create one
particle states out of the vacuum does not fix how they operate on non-vacuum
states.  In what follows it is assumed that they
can be chosen so they satisfy
\[
U(I ,\mathbf{a}) \Phi_{\alpha} ({\mathbf{p}_1},\mu_1 \cdots {\mathbf{p}_N},\mu_n) =
\Phi_{\alpha} (\mathbf{p}_1,\mu_1 \cdots {\mathbf{p}_N},\mu_n)
e^{i \mathbf{a} \sum_{k=1}^n \mathbf{p}_k}
\]
\[
U(R ,0) \Phi_{\alpha} ({\mathbf{p}_1},\mu_1 \cdots {\mathbf{p}_N},\mu_n) =
\Phi_{\alpha} (R\mathbf{p}_1,\nu_1 \cdots {R\mathbf{p}_N},\nu_n)
\prod D^{s_k}_{\nu_k \mu_k}[R]
\]
for the canonical basis and
\[
U(I ,\tilde{\mathbf{a}}) \Phi_{\alpha}(\Phi_{\alpha} (\tilde{\mathbf{p}_1},\mu_1 \cdots \tilde{\mathbf{p}_N},\mu_n) =
\Phi_{\alpha} (\tilde{\mathbf{p}_1},\mu_1 \cdots \tilde{\mathbf{p}_N},\mu_n)
e^{i \tilde{\mathbf{a}} \sum_{k=1}^n \tilde{\mathbf{p}}_k}
\]
\[
U(\Lambda_k,0) \Phi_{\alpha} (\tilde{\mathbf{p}}_1,\mu_1 \cdots
\tilde{\mathbf{p}}_N,\mu_n) =
\Phi_{\alpha} (\tilde{\pmb{\Lambda}}{p}_1,\mu_1 \cdots
\tilde{\pmb{\Lambda}_k}{p}_N,\mu_N )
\]
\[
U(R_z(\phi),0) \Phi_{\alpha} (\tilde{\mathbf{p}}_1,\mu_1 \cdots
\tilde{\mathbf{p}}_N,\mu_n) =
\Phi_{\alpha} (R_z(\phi)_k\tilde{\mathbf{p}}_1,\mu_1 \cdots
R_z(\phi)_k\tilde{\mathbf{p}}_N,\mu_N )
\prod D^{s_k}_{\nu_k \mu_k}[R_z(\phi)]
\]
These equations imply
\[
U(\Lambda_K ,a_K) \Phi_{\cal A}= \Phi_{\cal A} U_{\cal A} (\Lambda_K ,a_K) 
\]
for $(\Lambda_K,a_K)$ is the kinematic subgroup of the instant form
dynamics or the front form dynamics.  These equations imply that
while repeated application of these operators do not create particles
they do increase linear and angular momentum.    

Likewise in the case of the light front basis $\Phi_{\alpha}$ creates
a state out of the vacuum that asumptotically looks like an $N$
particle state and in general has the same kinematic transformations
as the associated asymptotic state.

These equation do not hold for full Poincar\'e group.
The injection operators $\Phi_{\cal A}$ in these two cases are different
and are not uniquely defined.  

Channel $\alpha$ scattering state vectors in the Hilbert space and be
defined by
\[
\lim_{t \to \pm \infty} \Vert e^{-iHt}  \vert \Psi_{\pm}(0) \rangle -
\Phi_{\alpha} e^{-iH_{\alpha}t} \vert f_{\alpha} \rangle \Vert  =0
\]
which an be written using unitarity of the time translation operator as
\[
\lim_{t \to \pm \infty} \Vert \vert \Psi_{\pm}(0) \rangle -
e^{-iHt} \Phi_{\alpha} e^{-iH_{\alpha}t} \vert f_{\alpha} \rangle \Vert  =0.
\]
The term on the right 
\[
s-lim_{t \to \pm \infty} e^{-iHt} \Phi_{\alpha} e^{-iH_{\alpha}t}:=
\Omega_{\pm \alpha}
\]
is the two-Hilbert space representation of the
Haag-Ruelle channel wave operator.  More generally we can
define
\[
\[
s-lim_{t \to \pm \infty} e^{-iHt} \Phi_{\cal A} e^{-iH_{\cal A}t}:=
\Omega_{\pm {\cal A}}
\]
If the vacuum and one particle states are included in the set of channels
then $\Omega_{\pm {\cal A}}$ is a unitary operator from ${\cal H}_{\cal A}$
to ${\cal H}$ and the scattering operator
\[
S:= \Omega_{+{\cal A}}^{\dagger}\Omega_{-{|cal A}}
\]
is a unitary operator on the asymptotic Hilbert space.
The useful quantity is channel scattering operator
\[
S_{\alpha,\beta} = \Omega_{+{\alpha}}^{\dagger}\Omega_{-{\beta}}
\]
matrix elements of which give the probability amplitude for scattering from
channel $\beta$ to channel $\alpha$.
The Poincar\'e invaraince of the scattering operator is a consequence of
the intertwinng property of the Haag Ruelle wave operators: 
\[
U(\Lambda ,a) \Omega_{\pm {\cal A}} = 
\Omega_{\pm {\cal A}} U_{ {\cal A}} (\Lambda ,a)
\]
A similar construction and be done with the injection operator
with the light-front kinematic symmetry.  In that case note that
\[
s-\lim_{t \to \pm \infty} e^{-iP^-t} \Phi_{\alpha} e^{-iP^-_{\alpha}t}=
s-\lim_{t \to \pm \infty} e^{-i(M^2/P^+) t} \Phi_{\alpha} e^{-i(M^2_{\alpha}/P^+)t}=
\]
\[
s-\lim_{t \to \pm \infty} e^{-iM^2t} \Phi_{\alpha} e^{-iM^2_{\alpha}t}=
s-\lim_{t \to \pm \infty} e^{-iMt} \Phi_{\alpha} e^{-iM_{\alpha}t}
\]
where we used the kinematic nature of $P^+$ and $\mathbf{P}_{\perp}$
, $P^+>0$ and the invariance principle to and invariance to
get the identfication The case of interest are single channel 

This point there is one vacuum and one scattereing operator, although
we have used different representations in the asymptotic space

W_{fc} S_{c} W^{\dagger}_{fc} = S_f

W_{fc} \Omega_{+c}^{\dagger} \Omega_{-c} W_{fc}^{\dagger} = \Omega_{+f}^{\dagger} \Omega_{-f}

using unitarity gives

\Omega_{+f} W_{fc} \Omega_{+c}^{\dagger} = \Omega_{-f} W_{fc} \Omega_{=c}^{\dagger}
V_{fc}

while W_{fc} is unitary on the asymptotic Hilbert space $V_{fc}$ is
unitary on the field theory Hilbert space.

V_{fc} \Omega_{+c} = \Omega_{+f} W_{fc} = \Omega_{-f} W_{fc}

e^{i Ht} \Phi_{f{\cal A}} e^{-iH_{f{\cal A}} W{fc} 

e^{i Ht} \Phi_{c{\cal A}} e^{-iH_{c{\cal A}}  
that when they are applied
to an eigenstate of linear momentum they increase linear momentum by an
amount $\mathbf{p}$ and when they are applied to an eigenstate of
spin 
\[
[\mathbf{P} ,\Phi(m,j,d_n)\mathbf{p},\mu)] =
\mathbf{p} \Phi(m,j,d_n)\mathbf{p},\mu) 
\]
where $\mathbf{P}$ is an operator and $\mathbf{p}$ is a number and
and
\[
[\mathbf{P} ,\Phi(m,j,d_n)\mathbf{p},\mu)] =
\mathbf{p} \Phi(m,j,d_n)\mathbf{p},\mu) 

they can be chosen
to satisfy
\[
[\mathbf{P} ,\Phi(m,j,d_n)\mathbf{p},\mu)] =
\mathbf{p} \Phi(m,j,d_n)\mathbf{p},\mu) 
\]
where $\mathbf{P}$ is an operator and $\mathbf{p}$ is a number and
\[
U(R) \Phi((m,j,d_n)\mathbf{p},\mu) U^{\dagger}(R)  =
\Phi((m,j,d_n)R \mathbf{p},\nu)D^j_{\nu\mu}[R] 
\]
why?

bases corresponding to
different light fronts
\[
\vert (m,s,d_n) \tilde{\mathbf{p}}_{\hat{\mathbf{z}}},\mu_{f\hat{\mathbf{z}}} \rangle =
\]
\[
\sum_{\mu_{\hat{\mathbf{n}}}=-s}^s\vert 0 \rangle \cup \{ \vert (m,s,d_n)
\tilde{\mathbf{p}}_{\hat{\mathbf{n}}}(\tilde{\mathbf{p}}_{\hat{\mathbf{z}}}),
\mu_{\hat{\mathbf{n}}} \rangle
\sqrt{{p^+_{\hat{\mathbf{n}}} \over p^+_{\hat{\mathbf{z}}}}}
D^s_{\mu_{\hat{\mathbf{n}}} \mu_{\hat{\mathbf{z}}}}[B_{f\hat{\mathbf{z}}}^{-1}(p/m)
B_{f\hat{\mathbf{z}}}(p/m)]
\]
We use the notation the define these change of bases
\[
W_{f_{\hat{\mathbf{z}}}c} \langle (m,s,d_n) \mathbf{p},\mu_c \vert (m',s',d_m)
\mathbf{p}',\mu'_f \rangle =
\langle (m,s,d_n) \tilde{\mathbf{p}},\mu_f \vert (m',s',d_m)
\tilde{\mathbf{p}},\mu_c \rangle
\]
and
\[
W_{f_{\hat{\mathbf{z}}f_{\hat{\mathbf{n}}}}} \langle (m,s,d_n) \tilde{\mathbf{p},\mu_c \vert (m',s',d_m)
\mathbf{p}',\mu'_f \rangle =
\]
The operator $U(\Lambda ,a)$ acts irreduciblly on these states.



3-components of the four momentum and
a projection of the spin on a given axis.  There will be additional
invariant degeneracy quantum numbers.  



On each irreducible subspace we choose a maximal set of commuting functions of
the Poincar\'e generators.  For the purpose of this illustration
they would be the three components of the linear momentum and the
projection of the canonical spin on the $z$ axis or the three
generators of translations tangent to the light-front and the $z$
component of the light front spin.  In addition to these quantum numbers
there will generally be invariant degeneracy quantum numbers.

Basis states have the form

\[
\vert (m,s,d) \mathbf{p},\mu \rangle
\qquad
\vert (m,s,d) \tilde{\mathbf{p}},\mu \rangle 
\]
Normalizable states have the form
\[
\sum_d \sum_{\mu=-s}^s \int
\vert (m,s,d) {\mathbf{p}},\mu \rangle d{\mathbf{p}}dm dd f((m,s,d){\mathbf{p}'\mu})
=
\sum_d \sum_{\mu=-s}^s \int
\Phi (m,s,d)\mathbf{p},\mu) \vert 0 \rangle
d{\mathbf{p}}dm dd f((m,s,d){\mathbf{p}'\mu})
\]
or 
\[
\sum_d \sum_{\mu=-s}^s \int
\vert (m,s,d) \tilde{\mathbf{p}},\mu \rangle d\tilde{\mathbf{p}}
d\tilde{\mathbf{p}}dm dd f((m,s,d)\tilde{\mathbf{p}},\mu) =
\sum_d \sum_{\mu=-s}^s \int
\Phi ((m,s,d)\tilde{\mathbf{p}},\mu) \vert 0 \rangle
d\tilde{\mathbf{p}}dm  f((m,s,d)\tilde{\mathbf{p}},\mu)
\]
where we have used (1) that these states are linear in the functions
$f(\tilde{\mathbf{p}},\mu)$,  $f(\tilde{\mathbf{p}},\mu)$ and
normalizable vectors can be expressed as operator valued functions of
field operators applied to the vacuum of the theory.

One particles states correspond to point spectrum eignetates of
the mass operator 
\[
m^2 = - p^2
\]
The operator
\[
\Phi ((m,s,d),\tilde{\mathbf{p}},\mu)
\]
when $m$ is in the point spectrum of the mass operator creates a one
particle state of mass $m$ spin $s$ momentum $\mathbf{p}$ and
spin projection $\mu$ out the vacuum.

Consider a product of these operators
\[
\prod_{n} \Phi ((m_n,s_n,d_n),\tilde{\mathbf{p}}_n,\mu_n)
\]
This can be considered as a mapping from Hilbert space of square intergrable
functions of $\mathbf{p}_1, \mu_1, \cdots ,\mathbf{p}_n, \mu_n $
to the Hilbert space of the quantum field theory.  It is only when
the supports of the Fourier transforms of the these operators have
large spacelike separations that the  above product looks like a state of
$N$ asymptotically separated particles.

While $\Phi ((m,s,d),\tilde{\mathbf{p}},\mu)$ creates a single particle state when it is applied to the vacuum, products of the $N$ of these operators
do not create $N$-particle states.

Because the kinematic subgroups is independent of the mass in this
representation, $\Phi ((m_n,s_n,d_n),{\mathbf{p}}_n,\mu_n)$
$\Phi ((m_n,s_n,d_n),\tilde{\mathbf{p}}_n,\mu_n)$ transform covariantly
under the kinematic subgroup. [what about the degeneracy parameters]

HF

rep of U on HF

Phi intertwines

kinematic subgroup

Wave operators

V kinematc scattering equiv (mix d,j,s)

Inertwining




Note that we do not assume a Lagrangian, or that the Generators come
from Noether's theorm.  All we have used is that they generate unitary one
parameter groups in the Hilbert space.

The next step is to discuss the scattering theory.  In what follows
we use a two-Hilbert space formulation of Haag-Ruelle scattering.
This assumes that the direct integral has one-particle states.

Start with an operator $\Phi (\mathbf{p},\mu)$ with the property that 
\[
\sum_{\mu} \int d\mathbf{p}  \Phi (\mathbf{p},\mu) \vert 0 \rangle f(\mathbf{p},\mu)  
\]
creates one particle states out of the vacuum with momentum and spin
distribution $f(\mathbf{p},\mu)$.  The construction of 
$\Phi (\mathbf{p},\mu)$ is discussed in the appendix.

For and $N$ particle scattering channel 
\[
\prod \Phi_{m_i,s_i} (\mathbf{p}_i ) 
\]
defines a mapping from  

To construct a projection on the one particle subspace
first define
Define
\[
A[x] = e^{-iP\cdot x}A e^{iP \cdot x},
\]
multiply by
\[
h(x) = \int e^{-i p x} h(p^2) f(\mathbf{p})d^4 p
\]
adn defined
\[
A_h = \int A[x] h(x) d^4 x = \int \delta^4 (p-P) h(p^2)   
\]

\[
\langle 0 \vert A^{\dagger} A \vert 0 \rangle
=\]
\[
\langle 0 \vert A^{\dagger}\vert p \mu d\rangle
\langle p \mu d \vert A \vert 0 \rangle 
\]
Define
\[
A[x] = e^{-iP\cdot x}A e^{iP \cdot x}
\]
and
\[
A[\mathbf{p}t,]= \int A[x] e^{-i p x} h(p^2)
\]
\[
\langle 0 \vert A^{\dagger}[x] A[y] \vert 0 \rangle =
\langle 0 \vert A^{\dagger}[0] \vert p \langle e^{i p (x-y)} dp 
\langle p \vert A[0] \vert 0 \rangle =
\]


\[
A[p]= \int A[x] e^{-ipx} = e^{iP\cdot x }A e^{-iP \cdot x}
\]
Assume that we have a quantum field theory.  In this case there is a vacuum
state $\vert 0 \rangle$ and an algebra of fields, which are operator valued distributions.

The algebra of field operators applied to the vacuum generate the
Hilbert space of the theory.
\[
\vert \psi \rangle = \sum_{i=1}^N c_i e^{i \phi(f_i)} \vert 0 \rangle
\]
\[
\langle \phi \vert \psi \rangle = \sum_{i=1}^N\aum_{j=1}^M
d_j^*c_i \langle 0 \vert e^{-i \phi^{\dagger}(g_j)}e^{i \phi(f_i)} \vert 0 \rangle
\]
We also assume the existence of a set of self adjoint operators satisfying
the commutation relations of the Poincare group.  These are formally defined
as generators of finite Poincare transformations.

Here these are defined as generators of one parameter unitary groups.
At this stage we have not assumed the existence of a Lagrangian or
used Noethers theorem.

We can define light front or instant form generators.  These are just
specific linear combinations of generators.

What do we mean by kinematic subgroups.  Since we have a unitary
representation of the Poincare group we can decompose it into a direct
integral of irreducible representations.  We assume that these are
positive mass representations positive energy representations.

We can construct a basis of generalized eigentates of linear momentum and
canonical spin or light front components of momenta and light front spin.

Given a ``kinematic subgroup'' consider an arbitrary unitary 
transformation $V_k$ that preserves only the kinematic subgroup.

In this case the remaining generators become
\[
G_i = V G_i V^{\dagger}
\]
while the generators of the kinematic subgroup have the form
\[
K_i = V K_i V^{\dagger} = K_i
\]

The transformed representation is no longer a  direct integral of irreducible
representations.  There is a kinematic subgroup and a unitary representation of the Poincar'e group - but there is not a kinematic representation of the
of the Poincar\'e group.

This has the property that kinematic transformations can still be made
in the transformed basis states, without diagonalizing the dynamcis.

We also want to demand that the unitary transformation $V$ is a scattering
equivalence.

Some consequences:

1 There are 2 vacuua on the Hilbert space - \vert 0 \rangle and
$V \vert 0 \rangle$.  Both are kinematically invariant, but each one
in fully Poincare invariant with respect to only one representation.
What we do not have is a Fock representation with the usual constraints.




We also assume the existence of an asymptotically complete scattering
operator.



We define the vacuum functional 
\[
S[f] :=
\langle 0 \vert e^{i \phi (f) } \vert 0 \rangle=
1 + \sum_{n=1}^\infty {(i)^n \over n!} \langle 0 \vert \phi (f)^n
\vert 0 \rangle
\]
\[
\vert \psi \rangle := \sum_{i=1}^n c_i e^{i \phi(f_i)}
\]


The first question that we consider is what is meant by a light front
dynamics of an instant form dynamics.

Dear John:

I really enjoyed your talk today.  I appreciate the approach that you are
taking using operators and worrying about what happens non-perturbatively.

I had a number of comments related to your talk - most were too numerous
and/or technical to bring up in the duscussion.

I have been interested in this subject on and off for some time and
enjoyed your talk on the subject at the recent jlab light cone meeting.
The subject  seems to persist because there the answers are not completely
satisfying.  I put myself in the camp where I still think that there
are some missing things.   My background is more in relativistic quantum
mechanics which is somewhat complementary to the approach in perturbative
field theory, but in some ways it is closer 

Comment 1:  There is a nice paper by Araki which supports some of the
things that you mentioned.  He argues that the Hamiltonian is fixed
(weakly) if (1) there is an invariant vacuum (2) the $\pi$ and $\phi$ fields
satisfy canonical commutation relations (3) the Hamiltonian is quadratic in
the generalized momentum.  (3) Ensures convexity which implies the existence
of a Legendre transformation to a Lagrangian formulation.  Fritz Coester and
I tried to see if this could be generalized in the light front case, but
we did not make the hoped for progress (this was my introduction to the
subject).

Comment 2: Unlike perturbative quantum field theory, in an interacting
theory the notion of kinematic subgroup does not make sense.  This is
because the free fields and Heisenberg fields act on different
inequivalent representations of the Hilbert space.  If one puts in
cutoffs this problem goes away because the free and interacting fields
are operator valued distributions on the same representation of the
Hilbert space, but I think that one has to think about what happens to
the kinematic subgroup in the non-perturbative case.  This seems like
it needs some more thought and a more precise characterization.

Comment 3: From my perspective many of the issues related to
equivalence are tied to the question of how to get self adjoint
Poincare generators acting on a representation of the Hilbert space.
On one hand we normally assume that the generators are self adjoint,
but in quantum field theory this is what non-perturbative
renormalization is all about.  One has self adjointness in
relativistic quantum mechancis of systems of a finite number of
degrees of freedom and in axiomatic treatments of quantum field
theory.

Comment 4: The notion of scattering equivalence seems to play an
interesting role.  In quantum field I have read that this is tied to
Borcher's classes, but I have not followed that up.  My understanding
is through wave operators which are expressed as strong limits.  In
axiomatic field theory Haag-Ruelle scattering results in wave
operators constructed as strong limits.  It is the natural generalization
of the non-relativistic methods.   These operators have the form
\[
\Omega_{\pm}  := s-lim_{t \to \pm \infty} e^{-iHt} \Phi e^{iH_At}
\]
In an asymptotically complete theory these are unitary maps
from an asymptotic Hilbert space (that included one-body states) to
the Hilbert space of the field theory or quantum theory.

The scattering operator is a unitary operator on the asymptotoic
Hilbert space 
\[
S(H,\Phi,H_A) :=
\Omega_^{\dagger} (H,\Phi,H_A)\Omega_- (H,\Phi,H_A)  \qquad    (H,\Phi,H_A) 
\]
A necessary and sufficient contition for two Hamiltonain's related by
a unitary transformations to give the same multichannel $S$ matrix:
\[
S(H,\Phi,H_A) = S(H',\Phi',H_A)
\]
if and only if 
\[
\lim_{t \to \pm \infty} \Vert (\Phi'- V\Phi ) e^{iH_At} \vert \psi_A \rangle \Vert
=0
\]
where
\[
V:= \Omega_- (H',\Phi',H_A) \Omega_-^{\dagger} (H,\Phi,H_A) =
\Omega_+ (H',\Phi',H_A) \Omega_+^{\dagger} (H,\Phi,H_A) 
\]
satisfies
\[
H'=VH  \qquad V^{\dagger}V=I
\]
This can be shown using simple properties of strong limits.  In the
Haag-Ruelle case the injection operator $\Phi$ can be based on
different functions of the fields, which is how one normally justified
the invariance with respect to field defefinition.

Both time limits are required.  Unitarity only ensures that the two
Hamiltonain's have the same spectrum, but that does not mean that
they have the same $S$ matrix (consider a two-body problem
with two different repulsive potentials)

In the case that there is a kinematic subgroup consider the operators
\[
V= U(R) U_0^{\dagger}(R)
\]
where $U(R)$ is the dynamical rotation operator and $U_0(R)$ is the free
rotation operator.  It can be shown that this operator changes the orientation
of the light front.  This is a scattering equivalence if it satisfied the
asymptotic condition [].




The connection with the light front dynamics

In the quantum mechanical case



Appendix 

then it follows that 

\[
\Omega_{\pm} (H',\Phi',H_A) 
V \Omega_{\pm} (H,\Phi,H_A)  =
\Omega_{\pm} (VHV^{\dagger},V\Phi,H_A) =
\Omega_{\pm} (H',V\Phi,H_A)
\]
\[
0 = \Omega_{\pm} (H', \Phi',H_A) - \Omega_{\pm} (H',V\Phi,H_A) =
\lim_{t \to \pm \infty} \Vert e^{\pm -iH't} (\Phi'- V\Phi ) e^{iH_At} \vert \psi_A \rangle \Vert =
\lim \Vert (\Phi'- V\Phi ) e^{iH_At} \vert \psi_A \rangle \Vert 
\]



\Omega_{\pm}^{\dagger} (H',\Phi,H_A) =
\Omega_{\pm}^{\dagger} (VHV^{\dagger},\Phi,H_A)
= V \Omega_{\pm}^{\dagger} (H, V^{\dagger}\Phi,H_A)


e^{iH_At} \Phi^{\dagger} e^{-iHt} 

\[
S(H,\Phi,H_A) = S(H',\Phi',H_A)
\]
if and only if





H'=VH



\Omega_- (H,\Phi,H_A)
=\Omega_- (H',\Phi',H_A)



P^{\mu}\hat{A}(q) \vert 0 \rangle =
\hat{A}(q) P^{\mu}  \vert 0 \rangle +
q^{\mu} \hat{A}(q)\vert 0 \rangle=
q^{\mu} \hat{A}(q)\vert 0 \rangle
\]
\[
P^{\mu}\hat{A}(q) \vert 0 \rangle =
\hat{A}(q) P^{\mu}  \vert 0 \rangle +
q^{\mu} \hat{A}(q)\vert 0 \rangle=
q^{\mu} \hat{A}(q)\vert 0 \rangle
\]

\end{document}
