††thanks: This work supported by the U.S. Department of Energy, Office of
Science, Grant #DE-SC0016457

# The relation between instant and light-front formulations of quantum field
theory

W. N. Polyzou<EMAIL_ADDRESS>Department of Physics and Astronomy
The University of Iowa
Iowa City, IA 52242, USA

###### 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 Poincaré 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. The underlying field theory
is not initially associated with an instant form or light-front form of the
dynamics. In this construction the existence of a vacuum and one-particle mass
eigenstates is assumed and both the light-front and instant-form
representations are constructed to share the same vacuum and one-particle
states. If there is spontaneous symmetry breaking there will be a 0 mass
particle in the mass spectrum (assuming no Higgs mechanism). 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 Poincaré 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 the most
important unanswered question for applications, this work may provide some
insight into the nature of the expected resolution and identifies the origin
of some of differences between the perturbative and non-perturbative
approaches.

##  I Introduction

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 Dirac (1949). 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 Poincaré group
continuously connected to the identity Wigner (1939). The Poincaré Lie algebra
has 3 independent commutators involving rotationless boost and translation
generators that have the Hamiltonian on the right

$[K^{i},P^{j}]=i\delta_{ij}H.$ (1)

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 Poincaré Lie algebra with the minimum number (3-4) of
interaction-dependent Poincaré 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 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 Sokolov and Shatnyi (1979)Keister
and Polyzou (1997). 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 Haag (1955), 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 sense in perturbative quantum field theory with cutoffs,
so the notion of instant- and light-front formulations Chang and Ma
(1969)Kogut and Soper (1970)Chang et al. (1973)Chang and Yan (1973)Yan
(1973a)Yan (1973b) 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:

  * $\bullet$

Is the light-front vacuum the same as the Fock vacuum?

  * $\bullet$

How to understand $P^{+}=0$ (zero mode) singularities?

  * $\bullet$

How to understand spontaneous symmetry breaking in a light-front dynamics?

  * $\bullet$

How to renormalize the theory consistent with rotational covariance.

  * $\bullet$

What is the relation between light-front and canonical quantization of a
quantum field theory?

  * $\bullet$

Are both approaches equivalent?

While there is a general consensus that the two formulations are equivalent,
the answers to the above questions are not as clean as desired. These issues
have been discussed extensively in the literature Chang and Ma (1969) Kogut
and Soper (1970) Leutwyler et al. (1970) Rohrlich (1971) Chang et al. (1973)
Chang and Yan (1973) Yan (1973a) Yan (1973b) Schlieder and Seiler (1972)
Maskawa and Yamawaki (1976) Nakanishi and Yabuki (1977) Nakanishi and Yamawaki
(1977) Leutwyler and Stern (1978) Coester (1992) Coester and Polyzou (1994)
Bylev (1996) Brodsky et al. (1998) Yamawaki (1998) Choi and Ji (1998)
Tsujimaru and Yamawaki (1998) Lenz (2000) Heinzl (2001) Burkardt et al. (2002)
Heinzl (2003) Ullrich and Werner (2006) Bakker (2011) Choi and Ji (2011) Choi
and Ji (2014) Beane (2013) Chabysheva and Hiller (2014) Herrmann and Polyzou
(2015) Ji (2017) Collins (2018) Mannheim et al. (2021).

Most of these discussions are based on perturbation theory; in particular on
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 a compelling resolution of these questions is 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 that are normally assumed in axiomatic formulations of
quantum field theory Streater and Wightman (1980)Haag and Kastler
(1964)Osterwalder and Schrader (1973)Osterwalder and Schrader (1975). One
departure from both of Dirac’s Hamiltonian approaches is that in axiomatic
formulations of quantum field theory states are constructed by applying fields
integrated against Schwartz test functions Gelfand et al. (1964) in four
space-time variables to the vacuum. These states can be evolved, but the
initial data is not on a sharp time slice or light front. The existence of an
asymptotically complete scattering theory is also assumed, based on the Haag-
Ruelle formulation of scattering theory Haag (1958)Brenig and Haag
(1959)Ruelle (1962)Jost (1965). Haag-Ruelle scattering has the advantage that
it can be formulated in terms of wave operators defined by strong limits. The
strong limits justify using many of the methods that are used in non-
relativistic scattering theory and they are used in the construction that
follows. The difference with the LSZ formulation of scattering is that the
interpolating field operators must create one-particle states rather than just
create states that have some overlap with one-particle states. The advantage
of the Haag-Ruelle formulation of scattering is that the weak limits that are
used to define the scattering asymptotic conditions in the LSZ formulation of
scattering can be replaced by strong limits.

A two-Hilbert space representation, which does not require the existence of a
free dynamics on the Hilbert space of the theory, is used to formulate the
scattering theory. This avoids any concerns about Haag’s theorem Streater and
Wightman (1980)Haag (1955), which implies that the interacting and free field
theories are formulated on inequivalent Hilbert-space representations of the
field algebra. The second Hilbert space, which is called the asymptotic
Hilbert space, is the direct sum of tensor products of single-particle Hilbert
spaces with physical masses. This representation is natural in non-
relativistic many-body scattering. For example, in lead-lead scattering the
lead nuclei are treated as elementary particles in the asymptotic Hilbert
space. Products of the lead state vectors,
$|L_{1},\mathbf{p}_{1},\mu_{1}\rangle\times|L_{2},\mathbf{p}_{2},\mu_{2}\rangle$,
define a mapping from the asymptotic Hilbert space of suitably symmetrized
square integrable functions,
$f(\mathbf{p}_{1},\mu_{1},\mathbf{p}_{2},\mu_{2})$, to the Hilbert space of
the theory. In the field theory case the lead wave functions are replaced by
products of operators, $A_{L}^{\dagger}(\mathbf{p},\mu)$, that create point
spectrum lead mass eigenstates out of the vacuum, applied to the vacuum Reed
and Simon (1979) Araki (1999)Baumgärtel and Wollenberg (1983). In the field
theory case the operators $A_{L}^{\dagger}(\mathbf{p},\mu)$ are not creation
operators. They do not commute or anticommute with each other and do not
satisfy canonical commutation relations. Inner products of states constructed
by applying two or more of these operators to the vacuum can have intermediate
states with different particle content.

In the simplest case the asymptotic Hilbert space 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, including the effects of self-interactions, when the
particles are asymptotically separated.

There is a natural unitary representation of the Poincaré group on the
asymptotic Hilbert space, which treats the particles as free particles with
their observed masses. 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
Poincaré group on the asymptotic space is mapped to the dynamical unitary
representation of the corresponding kinematic subgroup on the field-theory
Hilbert space without changing the scattering operator.

A theorem due to Ekstein 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 asymptotically complete wave operators defined as strong limits. In the
two-Hilbert space formulation Ekstein’s condition is a constraint on the
unitary transformation relating two different Hamiltonians that does not
require modifying the Hamiltonian in the asymptotic Hilbert space. Roughly
speaking it means that the unitary operator is a short-range perturbation of
the identity in a manner that Ekstein makes precise (see eq. (34)). 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 Poincaré group. The proof uses
the unitary of the wave operators which holds provided the vacuum and one-body
channels are included in the asymptotic Hilbert space. Unitary transformations
must leave the one-particle masses and the mass gap unchanged. In this work
these transformations are also constructed to leave the vacuum and one-
particle states unchanged.

When these unitary transformations are applied to the unitary representation
of the Poincaré group on the physical Hilbert space, they result in two new
scattering-equivalent representations with light-front and instant-form
kinematic subgroups. These representations have the property that the
injection operators map the kinematic subgroup on the asymptotic Hilbert space
to the corresponding subgroup of the unitary representation of the Poincaré
group on the physical Hilbert space. Both representations are equivalent to
the original “axiomatic” formulation of the theory in the sense that they are
related by $S$-matrix preserving unitary transformations.

One thing that is missing in this construction is the existence of a non-
interacting representation of the Poincaré group on the Hilbert space of the
field theory. Haag’s theorem implies that free fields, even with experimental
masses, are defined on an inequivalent Hilbert space representation of the
field algebra, so there is no corresponding local free field theory on the
Hilbert space of the field theory. In the event that the mapping from the
asymptotic space to the field theory Hilbert space has dense range, and the
unitarized form of this mapping satisfies Ekstein’s asymptotic condition, eq.
(34), then it can be used to map the unitary representation of the Poincaré
group on the asymptotic Hilbert space to a unitary representation of the
Poincaré group on the field theory Hilbert that plays the role of a free-
particle dynamics. Since the two-Hilbert space injection operators are not
unique and are not local operators, this will not be a local field theory.

The results can be summarized as follows. The construction assumes a unitary
representation of the Poincaré group on the Hilbert space of the field theory
and constructs scattering equivalent representations with light-front and
instant-form kinematic symmetries. The representations are related by a
unitary transformation that satisfies an asymptotic condition that preserves
the scattering matrix elements. Because the one-body states are eigenstates of
the mass operator, there is no mass renormalization. While the form of the
interactions is representation dependent, the absence of mass renormalization
suggests the that simplest types of interactions that change particle number
are short-range $2\leftrightarrow 3$-particle interactions rather than
$1\leftrightarrow 2$ vertex interactions.

This paper has five 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 brief discussion of spontaneous symmetry
breaking. The construction is compared to the corresponding construction in
relativistic formulations of quantum mechanics of a finite number of degrees
of freedom in section 5. Section 6 has a summary of the results and
conclusions. There are two appendices. The first one contains a discussion of
Ekstein’s treatment of scattering equivalences Ekstein (1960) that is used in
section 3. The second appendix discusses the construction of Haag-Ruelle
injection operators with the kinematic symmetry properties that are used in
sections two and three.

##  II 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:

  * $\bullet$

The Hilbert space of the field theory is generated by applying bounded
functions, $F$, of field operators smeared over test functions to a unique
vacuum vector, denoted by $|0\rangle$. The smearing is assumed to be over
Schwartz test functions Gelfand et al. (1964) 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, $|\psi\rangle$, can be taken to
have the form:

$|\psi\rangle=F|0\rangle\qquad F=\sum_{n=1}^{N}c_{n}e^{i\phi_{n}(f_{n})}$ (2)

  * where the $c_{n}$ are complex coefficients, the $f_{n}$ are Schwartz functions, $\phi_{n}$ are field operators and $N$ is finite.

  * $\bullet$

There is a unitary representation of the Poincaré 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 (2) is

$U(\Lambda,a)|\psi\rangle=\sum_{c}c_{n}e^{i\phi(f_{n}^{\prime})}|0\rangle\qquad
f_{n}^{\prime}(x)=f_{n}(\Lambda^{-1}(x-a)).$ (3)

  * $\bullet$

The vacuum, $|0\rangle$, is Poincaré invariant and normalized to unity:

$U(\Lambda,a)|0\rangle=|0\rangle\qquad\langle 0|0\rangle=1.$ (4)

  * $\bullet$

The unitary representation of the Poincaré 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 Poincaré group Wigner
(1939)Bargmann (1947) on the orthogonal complement of the vacuum subspace. The
mass Casimir operator is assumed to have point-spectrum eigenstates
representing particles of the theory.

  * $\bullet$

There is a complete set of asymptotically complete Haag-Ruelle Haag
(1958)Brenig and Haag (1959)Ruelle (1962)Jost (1965) scattering states.

A characteristic element of Dirac’s forms of dynamics is the notion of a
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 Poincaré group. These manifolds are prominent in Dirac’s work; in this
work the focus will be on the subgroups of the Poincaré 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 Poincaré 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. While free
fields with physical masses have the same mass spectrum, they act on an
inequivalent Hilbert space representation of the field algebra by Haag’s
theorem, so there is no free dynamics on the Hilbert space of the interacting
field 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
Poincaré group. The unitary representation of the Poincaré group has a set of
10 infinitesimal generators that are self-adjoint operators on the Hilbert
space of the field theory satisfying the Poincaré Lie algebra. This follows
because each one is the generator of a unitary one-parameter group. The
unitary representation of the Poincaré 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 Poincaré 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
Poincaré group. There will also be Poincaré 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

$|0\rangle\cup\\{|(m,s,d_{n})\mathbf{p},\mu_{c}\rangle\\}\qquad\mbox{and}\qquad|0\rangle\cup\\{|(m,s,d_{n})\tilde{\mathbf{p}},\mu_{F}\rangle\\}$
(5)

where the light-front components of the four momentum are defined by

$\tilde{\mathbf{p}}=(\mathbf{p}\cdot\hat{\mathbf{x}},\mathbf{p}\cdot\hat{\mathbf{y}},p^{0}+\mathbf{p}\cdot\hat{\mathbf{z}}):=(\mathbf{p}_{\perp},p^{+}).$
(6)

The variables (6) are eigenvalues of the generators of translations tangent to
the light-front hyperplane. The first basis in (5) 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

$|(m,s,d_{n})\mathbf{p},\mu_{I}\rangle=\sum_{\mu_{f}=-s}^{s}|(m,s,d_{n})\tilde{\mathbf{p}}(p),\mu_{F}\rangle\sqrt{p^{+}\over\omega_{m}(\mathbf{p})}D^{s}_{\mu_{F}\mu_{I}}[B_{F}^{-1}(p/m)B_{I}(p/m)]$
(7)

which assumes that both bases have delta function normalizations:

$\langle(m,s,d_{n})\mathbf{p},\mu_{I}|(m^{\prime},s^{\prime},d_{k})\mathbf{p}^{\prime},\mu^{\prime}_{I}\rangle=\delta(\mathbf{p}-\mathbf{p}^{\prime})\delta_{ss^{\prime}}\delta_{\mu_{I}\mu_{I}^{\prime}}\delta_{nk}\delta[m-m^{\prime}]$
(8)

and

$\langle(m,s,d_{n})\tilde{\mathbf{p}},\mu_{F}|(m^{\prime},s^{\prime},d_{k})\tilde{\mathbf{p}}^{\prime},\mu^{\prime}_{F}\rangle=\delta(p^{+}-p^{+\prime})\delta(\mathbf{p}_{\perp}-\mathbf{p}_{\perp}^{\prime})\delta_{ss^{\prime}}\delta_{\mu_{F}\mu_{F}^{\prime}}\delta_{nk}\delta[m-m^{\prime}]$
(9)

where $\delta[m-m^{\prime}]$ is either a Dirac or Kronecker delta function
depending on whether $m$ is in the point or continuous spectrum of the mass
operator. $B_{I}(p/m)$ 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 $B_{F}^{-1}(p/m)B_{I}(p/m)$ is a $SU(2)$ representation
of a Melosh rotation Melosh (1974) that changes the light-front spin to the
canonical spin. $D^{s}_{\mu_{F}\mu_{I}}[R]=\langle
s,\mu_{F}|U(R)|s,\mu_{I}\rangle$ is the $2s+1$ dimensional unitary
representation of $SU(2)$ and
$\omega_{m}(\mathbf{p})=\sqrt{\mathbf{p}^{2}+m^{2}}$.

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 (7) they involve a
variable change, the square root of a Jacobian and a momentum-dependent
rotation matrix.

The action of $U(\Lambda,a)$ on each of these irreducible basis states is a
consequence of the transformation properties of the Poincaré generators and
the basis choice. For the “instant-form” basis

$U(\Lambda,a)|0\rangle=|0\rangle,$ (10)
$U(\Lambda,a)|(m,s,d_{n}){\mathbf{p}},\mu\rangle=e^{ia\cdot\Lambda
p}\sum_{\mu^{\prime}=-s}^{s}|(m,s,d_{n}){\boldsymbol{\Lambda}{p}},\mu^{\prime}\rangle\sqrt{{\omega_{m}(\boldsymbol{\Lambda}p)\over\omega_{m}(\mathbf{p})}}D^{s}_{\mu^{\prime}\mu}[B_{I}^{-1}(\Lambda(p/m))\Lambda
B_{I}(p/m)]$ (11)

and for the ”light-front” basis,

$U(\Lambda,a)|0\rangle=|0\rangle,$ (12)
$U(\Lambda,a)|(m,s,d_{n}){\tilde{\mathbf{p}}},\mu\rangle=e^{ia\cdot\Lambda
p}\sum_{\mu^{\prime}=-s}^{s}|(m,s,d_{n}){\tilde{\boldsymbol{\Lambda}}{p}},\mu^{\prime}\rangle\sqrt{{(\Lambda
p)^{+}\over p^{+}}}D^{s}_{\mu^{\prime}\mu}[B_{F}^{-1}(\Lambda(p/m))\Lambda
B_{F}(p/m)].$ (13)

These equations define the dynamical unitary representation of the Poincaré
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 (11)
and (13) do not depend on the mass eigenvalue $m$ when $(\Lambda,a)$ is an
element of the kinematic subgroup. For the basis (11) the kinematic subgroup
is generated by rotations and spatial translations while for the basis (13)
the kinematic subgroup is the subgroup that leaves the light front hyperplane
$x^{+}=x^{0}+\mathbf{x}\cdot\hat{\mathbf{z}}=0$ invariant.

The next step is to discuss the formulation of scattering theory. A two-
Hilbert space representation Coester and Polyzou (1982)Baumgärtel and
Wollenberg (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 integrating the irreducible basis functions with a wave packet:

$|\psi_{Ig}\rangle:=\int
d\mathbf{p}\sum_{\mu=-s}^{s}|(m,s,d_{n})\mathbf{p},\mu\rangle
g(\mathbf{p},\mu)$ (14)

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

$|\psi_{Fg}\rangle:=\int
d^{2}\mathbf{p}_{\perp}\int_{0}^{\infty}dp^{+}\sum_{\mu=-s}^{s}|(m,s,d_{n})\tilde{\mathbf{p}},\mu\rangle\tilde{g}(\tilde{\mathbf{p}},\mu)$
(15)

where in the light front case $\tilde{\mathbf{p}}$ should vanish as $p^{+}\to
0$ in a manner that results in a square integrable function of $\mathbf{p}$
under the variable change (6) $\tilde{\mathbf{p}}\to{\mathbf{p}}$.

These normalizable vectors are linear in the functions ${g}({\mathbf{p}},\mu)$
and $\tilde{g}(\tilde{\mathbf{p}},\mu)$. They can be represented by elements
of the field algebra applied to the vacuum. This means that they can be
expressed as

$|\psi_{Ig}\rangle=\sum_{\mu=-s}^{s}\int
A^{\dagger}_{m,s,d_{n}}({p},\mu)|0\rangle
d\mathbf{p}g(\mathbf{p},\mu)=A^{\dagger}(g)|0\rangle$ (16)

and

$|\psi_{Fg}\rangle=\sum_{\mu=-s}^{s}\int\tilde{A}^{\dagger}_{m,s,d_{n}}(\tilde{{p}},\mu)|0\rangle
dp^{+}d^{2}\mathbf{p}_{\perp}\tilde{g}(\tilde{\mathbf{p}},\mu)=\tilde{A}^{\dagger}(\tilde{g})|0\rangle.$
(17)

These states will be equal if
$g({\mathbf{p}},\mu)=\langle(m,s,d_{n})\mathbf{p},\mu_{I}|g\rangle$ and
$\tilde{g}(\tilde{\mathbf{p}},\mu)=\langle(m,s,d_{n})\tilde{\mathbf{p}},\mu_{F}|g\rangle$
are related by the change of basis (7).

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 $\mu$ 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 B.

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 with test functions), they can be multiplied.

The following quantity

$A^{\dagger}_{m_{1},s_{1},d_{n_{1}}}(\mathbf{p_{1}},\mu_{1})\cdots
A^{\dagger}_{m_{k},s_{k},d_{n_{k}}}(\mathbf{p_{k}},\mu_{k})|0\rangle$ (18)

can be considered as a mapping from a Hilbert space of square integrable
suitably symmetrized functions of
$\mathbf{p}_{1},\mu_{1}\cdots\mathbf{p}_{k},\mu_{k}$ to the Hilbert space of
the field theory. Note that while
$A^{\dagger}(g)|0\rangle=\tilde{A}^{\dagger}(\tilde{g})|0\rangle$ for $g$ and
$\tilde{g}$ related by (7), because both operators create the same single-
particle states out of the vacuum, this identity is not true for products of
these operators applied to the vacuum. From equation (117) in appendix B it
follows that this mapping has the following properties

$\mathbf{P}A^{\dagger}_{m_{1},s_{1},d_{n_{1}}}(\mathbf{p}_{1},\mu_{1})\cdots
A^{\dagger}_{m_{N},s_{N},d_{n_{N}}}(\mathbf{p}_{N},\mu_{N})|0\rangle=$
$\sum_{n=1}^{N}\mathbf{p}_{n}A^{\dagger}_{m_{1},s_{1},d_{n_{1}}}(\mathbf{p}_{1},\mu_{1})\cdots
A^{\dagger}_{m_{N},s_{N},d_{n_{N}}}(\mathbf{p}_{N},\mu_{N})|0\rangle$ (19)

and

$U(R,0)A^{\dagger}_{m_{1},s_{1},d_{n_{1}}}(\mathbf{p}_{1},\mu_{1})\cdots
A^{\dagger}_{m_{N},s_{N},d_{n_{N}}}(\mathbf{p}_{1},\mu_{N})|0\rangle=$ $\sum
A^{\dagger}_{m_{1},s_{1},d_{n_{1}}}(R\mathbf{p}_{1},\nu_{1})\cdots
A^{\dagger}_{m_{N},s_{N},d_{n_{N}}}(R\mathbf{p}_{N},\nu_{N})|0\rangle|0\rangle\prod_{n=1}^{N}D^{s_{n}}_{\nu_{n}\mu_{n}}(R).$
(20)

For the mapping in the “light-front” basis the corresponding relations are

$\tilde{\mathbf{P}}\tilde{A}^{\dagger}_{m_{1},s_{1},d_{n_{1}}}(\tilde{\mathbf{p}}_{1},\mu_{1})\cdots\tilde{A}^{\dagger}_{m_{N},s_{N},d_{n_{N}}}(\tilde{\mathbf{p}}_{N},\mu_{N})|0\rangle=$
$\sum_{n=1}^{N}\tilde{\mathbf{p}}_{n}\tilde{A}^{\dagger}_{m_{1},s_{1},d_{n_{1}}}(\tilde{\mathbf{p}}_{1},\mu_{1})\cdots\tilde{A}^{\dagger}_{m_{N},s_{N},d_{n_{N}}}(\tilde{\mathbf{p}}_{N},\mu_{N})|0\rangle$
(21)
$U(B_{f},0)\tilde{A}^{\dagger}_{m_{1},s_{1},d_{n_{1}}}(\tilde{\mathbf{p}}_{1},\mu_{1})\cdots\tilde{A}^{\dagger}_{m_{N},s_{N},d_{n_{N}}}(\tilde{\mathbf{p}}_{N},\mu_{N})|0\rangle=$
$\tilde{A}^{\dagger}_{m_{1},s_{1},d_{n_{1}}}(\tilde{\mathbf{B}}_{f}p_{1},\mu_{1})\cdots\tilde{A}^{\dagger}_{m_{N},s_{N},d_{n_{N}}}(\tilde{\mathbf{B}}_{f}p_{N},\mu_{N})|0\rangle\prod_{n=1}^{N}{\sqrt{B_{f}(p_{n})^{+}\over
p_{n}^{+}}}.$ (22)
$U(R_{z}(\phi),0)\tilde{A}^{\dagger}_{m_{1},s_{1},d_{n_{1}}}(\tilde{\mathbf{p}}_{1},\mu_{1})\cdots\tilde{A}^{\dagger}_{m_{N},s_{N},d_{n_{N}}}(\tilde{\mathbf{p}}_{N},\mu_{N})|0\rangle=$
$\tilde{A}^{\dagger}_{m_{1},s_{1},d_{n_{1}}}(R_{z}(\phi)\tilde{\mathbf{p}}_{1},\mu_{1})\cdots\tilde{A}^{\dagger}_{m_{N},s_{N},d_{n_{N}}}(R(\phi)\tilde{\mathbf{p}}_{N},\mu_{N})|0\rangle\prod_{n=1}^{N}e^{i\mu_{n}\phi}.$
(23)

If the Fourier transform of these states are 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 resulting vectors look like states of $N$
asymptotically separated particles when they are used in inner products. This
property is used Jost (1965)Araki (1999) to prove the existence of the strong
limits that define wave operators in Haag-Ruelle scattering. The
interpretation of the operators ${A}^{\dagger}_{m,s,d_{n}}$ and
$\tilde{A}^{\dagger}_{m,s,d_{n}}$ is that they 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 an
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 (16) or
(17), applied to the vacuum, one for each particle in a scattering channel
$\alpha$. In the canonical basis they are

$\Phi_{\alpha}({\mathbf{p}_{1}},\mu_{1}\cdots{\mathbf{p}_{N}},\mu_{N}):=\prod_{k\in\alpha}A^{\dagger}_{m_{k},s_{k},d_{n_{k}}}({\mathbf{p}_{k}},\mu_{k})|0\rangle$
(24)

and in the light-front basis they are

$\tilde{\Phi}_{\alpha}(\tilde{\mathbf{p}}_{1},\mu_{1}\cdots\tilde{\mathbf{p}}_{N},\mu_{N}):=\prod_{k\in\alpha}\tilde{A}^{\dagger}_{m_{k},s_{k},d_{n_{k}}}(\tilde{\mathbf{p}}_{k},\mu_{k})|0\rangle.$
(25)

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 $\alpha$ are identical then the functions
representing identical Bosons should be symmetrized and those representing
identical Fermions should anti-symmetrized. The order of the product in (25)
does not matter for suitably symmetrized states in the asymptotic Hilbert
space. The channel injection operators (24) and (25) are interpreted as
mappings from the $k$-particle channel subspace of the asymptotic Hilbert
space ${\cal H}_{\alpha}$ to the Hilbert space ${\cal H}$ of the field theory.

The asymptotic unitary representation of the Poincaré 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

$U_{\alpha}(\Lambda,a)|\mathbf{p}_{1},\mu_{1},\cdots\mathbf{p}_{k},\mu_{k}\rangle:=\sum
e^{ia\cdot\Lambda(\sum_{n}p_{n})}|\boldsymbol{\Lambda}p_{1},\nu_{1}\cdots\boldsymbol{\Lambda}p_{k},\nu_{k}\rangle\prod_{n=1}^{k}\sqrt{{\omega_{m_{n}}(\boldsymbol{\Lambda}p_{n})\over\omega_{m_{n}}(\mathbf{p}_{n})}}D^{s}_{\nu_{n}\mu_{n}}[B_{c}^{-1}(\Lambda
p_{n}/m_{n})\Lambda B_{c}(p_{n}/m_{n})].$ (26)

The corresponding expression in terms of “light-front” variables is

$U_{\alpha}(\Lambda,a)|\tilde{\mathbf{p}}_{1},\mu_{1},\cdots\tilde{\mathbf{p}}_{k},\mu_{k}\rangle:=\sum
e^{ia\cdot\Lambda(\sum_{n}p_{n})}|\tilde{\boldsymbol{\Lambda}}p_{1},\nu_{1}\cdots\tilde{\boldsymbol{\Lambda}}p_{k},\nu_{k}\rangle\prod_{n=1}^{k}\sqrt{{(\tilde{\Lambda}p_{n})^{+}\over{p}^{+}_{n}}}D^{s}_{\nu_{n}\mu_{n}}[B_{f}^{-1}(\Lambda
p_{n}/m_{n})\Lambda B_{f}(p_{n}/m_{n})].$ (27)

The states in (26) and (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 necessarily N-particle states it follows that
in general

$U(\Lambda,a)\Phi_{\alpha}\not=\Phi_{\alpha}U_{\alpha}(\Lambda,a).\qquad
U(\Lambda,a)\tilde{\Phi}_{\alpha}\not=\tilde{\Phi}_{\alpha}U_{\alpha}(\Lambda,a).$
(28)

This is because (106) does not hold for $P^{0}$ due to the $p^{0}$ integrals
in (109) and (106) does not hold for $P^{-}$ due to the $p^{-}$ integrals on
(109) (see appendix B).

When $\Lambda_{K}$ and $a_{K}$ are elements of the instant-form or light-front
kinematic subgroup, it follows from (19-23) that

$U(\Lambda_{K},a_{K})\Phi_{\alpha}=\Phi_{\alpha}U_{\alpha}(\Lambda_{K},a_{K})\qquad
U(\Lambda_{K},a_{K})\tilde{\Phi}_{\alpha}=\tilde{\Phi}_{\alpha}U_{\alpha}(\Lambda_{K},a_{K})$
(29)

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 Haag (1958)Brenig and Haag
(1959)Ruelle (1962) injection operators. The only difference with the standard
choices of Haag Ruelle injection operators is that $\Phi_{\alpha}$ and
$\tilde{\Phi}_{\alpha}$ are designed to preserve the kinematic subgroup (29).

In the two-Hilbert space formulation Reed and Simon (1979)Araki (1999) channel
wave operators
$\Omega_{\alpha\pm}=\Omega_{\alpha\pm}(H,\Phi_{\alpha},H_{\alpha})$ are
defined by the strong limits

$\lim_{t\to\pm\infty}\|(\Omega_{\alpha\pm}(H,\Phi_{\alpha},H_{\alpha})-e^{iHt}\Phi_{\alpha}e^{-iH_{\alpha}t})|\psi\rangle\|=0$
(30)

or

$\lim_{t\to\pm\infty}\|(\Omega_{\alpha\pm}(H,\tilde{\Phi}_{\alpha},H_{\alpha})-e^{iHt}\tilde{\Phi}_{\alpha}e^{-iH_{\alpha}t})|\psi\rangle\|=0.$
(31)

The two Hilbert space wave operators satisfy the Poincaré covariance condition

$U(\Lambda,a)\Omega_{\alpha\pm}(H,\Phi_{\alpha},H_{\alpha})=\Omega_{\alpha\pm}(H,\Phi_{\alpha},H_{\alpha})U_{\alpha}(\Lambda,a)\qquad
U(\Lambda,a)\Omega_{\alpha\pm}(H,\tilde{\Phi}_{\alpha},H_{\alpha})=\Omega_{\alpha\pm}(H,\tilde{\Phi}_{\alpha},H_{\alpha})U_{\alpha}(\Lambda,a)$
(32)

where, unlike (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 particles in the asymptotic region. The two injection operators
defined in appendix B both have this property. They differ in which variable
enforces the one-particle mass-shell delta function when applied the vacuum.
This means that as operators

$\Omega_{\alpha\pm}(H,{\Phi}_{\alpha},H_{\alpha})=\Omega_{\alpha\pm}(H,\tilde{\Phi}_{\alpha},H_{\alpha})$
(33)

if they are evaluated in the same one-particle basis. This identity is
equivalent to the requirement

$0=\lim_{t\to\pm\infty}\|e^{iHt}({\Phi}_{\alpha}-\tilde{{\Phi}}_{\alpha})e^{-iH_{\alpha
t}}|\psi_{\alpha}\rangle\|=\lim_{t\to\pm\infty}\|({\Phi}_{\alpha}-\tilde{{\Phi}}_{\alpha})e^{-iH_{\alpha
t}}|\psi_{\alpha}\rangle\|$ (34)

which is precisely the condition that the injection operators agree
asymptotically.

The scattering operator for scattering from channel $\alpha$ to channel
$\beta$ is

$S_{\beta,\alpha}:{\cal H}_{\alpha}\to{\cal
H}_{\beta}:=\Omega^{\dagger}_{\beta+}(H,\Phi_{\beta},H_{\beta})\Omega_{\alpha-}(H,\Phi_{\alpha},H_{\alpha}).$
(35)

It follows from the identity (34) that $S_{\alpha\beta}$ is independent of the
choice of injection operator. It follows from (32) that

$U_{\beta}(\Lambda,a)S_{\beta,\alpha}=S_{\beta,\alpha}U_{\alpha}(\Lambda,a).$
(36)

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:

${\cal H}_{\cal A}:=\oplus_{\alpha\in{\cal A}}{\cal H}_{\alpha}.$ (37)

The asymptotic unitary representation of the Poincaré group is defined on
${\cal H}_{\cal A}$ by

$U_{\cal A}(\Lambda,a)=\oplus_{\alpha\in{\cal A}}U_{\alpha}(\Lambda,a).$ (38)

Multi-channel injection operators are defined as the sum of all channel
injection operators

$\Phi_{\cal A}:=\sum_{\alpha\in{\cal
A}}\Phi_{\alpha}\qquad\mbox{or}\qquad\tilde{\Phi}_{\cal
A}:=\sum_{\alpha\in{\cal A}}\tilde{\Phi}_{\alpha}$ (39)

where $\Phi_{\alpha}:{\cal H}_{\alpha}\to{\cal H}$. These can be used to
define multi-channel wave operators

$\Omega_{{\cal A}\pm}(H,\Phi_{\cal A},H_{\cal A})=\tilde{\Omega}_{{\cal
A}\pm}(H,\tilde{\Phi}_{\cal A},H_{\cal A})$ (40)

by the strong limits

$\lim_{t\to\pm\infty}\|(\Omega_{{\cal A}\pm}(H,\Phi_{\cal A},H_{\cal
A})-e^{iHt}\Phi_{\cal A}e^{-iH_{\cal A}t}|\psi\rangle\|=0$ (41)

or

$\lim_{t\to\pm\infty}\|(\Omega_{{\cal A}\pm}(H,\tilde{\Phi}_{\cal A},H_{\cal
A})-e^{iHt}\tilde{\Phi}_{\cal A}e^{-iH_{\cal A}t}|\psi\rangle\|=0$ (42)

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 (32) becomes

$U(\Lambda,a)\Omega_{{\cal A}\pm}(H,{\Phi}_{\cal A},H_{\cal A})=\Omega_{{\cal
A}\pm}(H,{\Phi}_{\cal A},H_{\cal A})U_{{\cal A}}(\Lambda,a)$ (43)

which also holds with $\Phi_{\cal A}$ replaced by $\tilde{\Phi}_{\cal A}$.

The multi-channel scattering operator is

$S(H,\Phi_{\cal A},{\cal H}_{\cal A})=\Omega_{+}^{\dagger}(H,\Phi_{\cal
A},{\cal H}_{\cal A})\Omega_{-}(H,\Phi_{\cal A},{\cal H}_{\cal A})$ (44)

where it was constructed to be the identity on the vacuum and one particle
states.

Poincaré invariance of the multi-channel scattering operator on ${\cal
H}_{\cal A}$ follows from the intertwining property of the wave operators (43)
Reed and Simon (1979)

$U_{\cal A}(\Lambda,a)S(H,\Phi_{\cal A},{\cal H}_{\cal A})=U_{\cal
A}(\Lambda,a)\Omega_{+}^{\dagger}(H,\Phi_{\cal A},{\cal H}_{\cal
A})\Omega_{-}(H,\Phi_{\cal A},{\cal H}_{\cal A})=$
$\Omega_{+}^{\dagger}(H,\Phi_{\cal A},{\cal H}_{\cal
A})U(\Lambda,a)\Omega_{-}(H,\Phi_{\cal A},{\cal H}_{\cal A})=$
$\Omega_{+}^{\dagger}(H,\Phi_{\cal A},{\cal H}_{\cal
A})\Omega_{-}(H,\Phi_{\cal A},{\cal H}_{\cal A})U_{\cal
A}(\Lambda,a)=S(H,\Phi_{\cal A},{\cal H}_{\cal A})U_{\cal A}(\Lambda,a).$ (45)

At this point all that has been demonstrated is that the same scattering
theory is obtained by using injection operators that agree asymptotically but
commute with 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.

##  III 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

$H=P^{-}+P^{3}\qquad H={1\over 2}(P^{+}+P^{-})$ (46)

will be used in what follows. In the first case note because
$P^{3}{\Phi}_{\cal A}={\Phi}_{\cal A}P^{3}_{\cal A}$ it follows that

$e^{iHt}\Phi_{{\cal A}}e^{-iH_{\cal A}}=e^{i(P^{-}+P^{3})t}\Phi_{{\cal
A}}e^{-i(P^{-}_{\cal A}+P^{3}_{\cal A})t}=e^{iP^{-}t}\Phi_{{\cal
A}}e^{-iP^{-}_{\cal A}t}.$ (47)

Similarly since $P^{+}\tilde{\Phi}_{\cal A}=\tilde{\Phi}_{\cal A}P^{+}_{\cal
A}$

$e^{iHt}\tilde{\Phi}_{{\cal A}}e^{-iH_{\cal
A}}=e^{i(P^{-}+P^{+})t/2}\tilde{\Phi}_{{\cal A}}e^{-i(P^{-}_{\cal
A}+P^{+}_{\cal A})t/2}=e^{iP^{-}t/2}\tilde{\Phi}_{{\cal A}}e^{-iP^{-}_{\cal
A}t/2}.$ (48)

This means that the wave operators constructed using $H$ are identical to the
ones using $P^{-}$ with both injection operators. Combining these results with
(40) gives the following identifications

$\Omega_{\pm}(H,\Phi_{\cal A},H_{\cal A})=\Omega_{\pm}(H,\tilde{\Phi}_{\cal
A},H_{\cal A})=\Omega_{\pm}(P^{-},\Phi_{\cal A},P^{-}_{\cal
A})=\Omega_{\pm}(P^{-},\tilde{\Phi}_{\cal A},P^{-}_{\cal A}).$ (49)

The final step in the construction is to introduce two unitary operators
$V_{F}$ and $V_{I}$ that act on the field theory Hilbert space ${\cal H}$ with
the following properties:

$V_{F}|0\rangle=V_{I}|0\rangle=|0\rangle$ (50)
$[U(\Lambda_{K_{I}},a_{K_{I}}),V_{I}]=0\qquad[U(\Lambda_{K_{F}},a_{K_{F}}),V_{F}]=0$
(51) $\lim_{t\to\pm\infty}\|(\Phi_{\cal A}-V_{I}\Phi_{\cal A})e^{-iH_{\cal
A}t}|\psi_{\alpha}\rangle\|=0$ (52) $\lim_{t\to\pm\infty}\|(\tilde{\Phi}_{\cal
A}-V_{F}\tilde{\Phi}_{\cal A})e^{-iH_{\cal A}t}|\psi_{\alpha}\rangle\|=0$ (53)

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 (52 and 53) to hold the asymptotic Hamiltonian must have a non-
trivial absolutely continuous mass spectrum. While this is true on the
scattering channel subspaces, it is not true for the vacuum or one-particle
states, where $e^{-iH_{\cal A}t}$ either factors out because the energy
depends on the mass and kinematic variables or is $1$ in the case of the
vacuum. In these cases (53) becomes

$\|(\tilde{\Phi}_{\cal A}-V_{F}\tilde{\Phi}_{\cal
A})|\psi_{\alpha}\rangle\|=0.$ (54)

In the construction of this section the operators $V_{I}$ and $V_{F}$ were
chosen to leave the vacuum unchanged. While this is not necessary, this choice
gives S-matrix equivalent light-front and instant-form field theories with the
same vacuum vectors.

The next step is to define two new unitary representations of the Poincaré
group on the Hilbert space ${\cal H}$ of the quantum field theory by:

$U_{F}(\Lambda,a):=V_{F}U(\Lambda,a)V_{F}^{\dagger}\qquad
U_{I}(\Lambda,a):=V_{I}U(\Lambda,a)V_{I}^{\dagger}$ (55)

with the corresponding dynamical generators:

$H_{F}:=V_{F}HV_{F}^{\dagger}\qquad H_{I}:=V_{I}HV_{I}^{\dagger}$ (56)
$P^{-}_{F}:=V_{F}P^{-}V_{F}^{\dagger}\qquad
P^{-}_{I}:=V_{I}P^{-}V_{I}^{\dagger}.$ (57)

It follows from (51) that

$U_{I}(\Lambda_{K_{I}},a_{K_{I}})=U(\Lambda_{K_{I}},a_{K_{I}})\qquad
U_{F}(\Lambda_{K_{F}},a_{K_{F}}):=U(\Lambda_{K_{F}},a_{K_{F}})$ (58)

where $(\Lambda_{K_{I}},a_{K_{I}})$ is an instant-form kinematic Poincaré
transformation and $(\Lambda_{K_{F}},a_{K_{F}})$ is a front-form kinematic
transformation.

In Appendix A it is shown Ekstein (1960) that as a consequence of (52) and
(53) that

$\Omega_{\pm}(H_{I},\Phi_{\cal A},H_{\cal A})=V_{I}\Omega_{\pm}(H,\Phi_{\cal
A},H_{\cal A})$ (59)

and

$\Omega_{\pm}(H_{F},\tilde{\Phi}_{\cal A},H_{\cal
A})=V_{F}\Omega_{\pm}(H,\tilde{\Phi}_{\cal A},H_{\cal A}).$ (60)

Taken together with (49) gives:

$\Omega_{\pm}(H_{I},\Phi_{\cal A},H_{\cal A})=V_{I}\Omega_{\pm}(H,\Phi_{\cal
A},H_{\cal A})=V_{I}\Omega_{\pm}(H,\tilde{\Phi}_{\cal A},H_{\cal
A})=V_{I}\Omega_{\pm}(P^{-},\tilde{\Phi}_{\cal A},P^{-}_{\cal
A})=V_{I}V_{F}^{\dagger}\Omega_{\pm}(P_{F}^{-},\tilde{\Phi}_{\cal
A},P^{-}_{\cal A}).$ (61)

Since (61) holds for the same $V_{I}V_{F}^{\dagger}$ for both time limits it
follows that

$S(H_{I},\Phi{\cal A},H_{\alpha})=\Omega^{\dagger}_{+}(H_{I},\Phi_{\cal
A},H_{\cal A})\Omega_{-}(H_{I},\Phi_{\cal A},H_{\cal A})=$
$\Omega^{\dagger}_{+}(H_{F},\tilde{\Phi}_{\cal A},H_{\cal
A})V_{F}V_{I}^{\dagger}V_{I}V_{F}^{\dagger}\Omega_{-}(H_{F},\tilde{\Phi}_{\cal
A},H_{\cal A})=$ $\Omega^{\dagger}_{+}(P^{-}_{F},\tilde{\Phi}_{\cal
A},P^{-}_{\cal A})\Omega_{-}(P^{-}_{F},\tilde{\Phi}_{\cal A},P^{-}_{\cal A})=$
$S(P^{-}_{F},\tilde{\Phi}_{\cal A},P^{-}_{\cal A}).$ (62)

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 Poincaré 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 $\\{\mathbf{P},s,s_{zI}\\}$ 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

$\langle\mathbf{p},s,\mu_{I},x_{I}|\psi\rangle\qquad\mbox{or}\qquad\langle\tilde{\mathbf{p}},\mu_{F},x_{F}|\psi\rangle.$
(63)

Matrix elements of the dynamical operators are non-trivial matrices in the
$``x^{\prime\prime}$ variables:

$\langle\mathbf{p},s,\mu_{I},x_{I}|H_{I}|\mathbf{p}^{\prime},s^{\prime},\mu_{I}^{\prime},x_{I}^{\prime}\rangle=\delta(\mathbf{p}-\mathbf{p}^{\prime})\delta_{ss^{\prime}}\delta_{\mu_{I}\mu_{I}^{\prime}}\langle
x_{I}\|H_{I}(\mathbf{P},s)\|x_{I}^{\prime}\rangle$ (64)

and

$\langle\tilde{\mathbf{p}},\mu_{F},x_{F}|P^{-}_{F}|\tilde{\mathbf{p}}^{\prime},\mu_{F}^{\prime},x_{F}^{\prime}\rangle=\delta(\tilde{\mathbf{p}}-\tilde{\mathbf{p}}^{\prime})\delta_{\mu_{F}\mu_{F}^{\prime}}\langle
x_{F}\|P^{-}_{F}(\tilde{\mathbf{P}},\mu_{F})\|x_{F}^{\prime}\rangle.$ (65)

The eigenvalue problems for the dynamical operators have the forms

$\sum\int\langle x_{I}\|H_{I}(\mathbf{P},s)\|x_{I}^{\prime}\rangle
dx_{I}^{\prime}\langle\mathbf{p},s,\mu_{I},x_{I}|\psi\rangle=E(\mathbf{P},s)\langle\mathbf{p},s,\mu_{I},x_{I}|\psi\rangle$
(66)

and

$\sum\int\langle
x_{F}\|P^{-}_{F}(\tilde{\mathbf{P}},\mu_{F})\|x_{F}^{\prime}\rangle
dx_{F}^{\prime}\langle\tilde{\mathbf{p}},\mu,x_{F}^{\prime}|\psi\rangle=P^{-}(\tilde{\mathbf{P}},\mu_{F})\langle\tilde{\mathbf{p}},\mu_{F},x_{F}|\psi\rangle.$
(67)

The matrices $\langle x_{I}\|H_{I}(\mathbf{P},s)\|x_{I}^{\prime}\rangle$ or
$\langle x_{F}\|P_{F}^{-}(\tilde{\mathbf{P}},\mu)\|x_{F}^{\prime}\rangle$ must
be diagonalized in order to compute dynamical Poincaré transformations. On the
other hand kinematic transformations can be computed by applying the inverse
kinematic transformation to basis vectors:

$\langle\mathbf{p},s,\mu,x_{I}|U_{I}(R,\mathbf{a})|\psi\rangle=\langle\psi|U_{I}^{\dagger}(R,\mathbf{a})|\mathbf{p},s,\mu,x_{I}\rangle^{*}=$
$e^{i\mathbf{a}\cdot\mathbf{p}}\langle
R^{-1}\mathbf{p},s,\nu,x_{I}|\psi\rangle D^{s}_{\mu\nu}(R).$ (68)

Similarly in the light-front case for light-front preserving boosts and
translations:

$\langle\tilde{\mathbf{p}},\mu,x_{F}|U_{F}(\Lambda_{KF},a_{KF})|\psi\rangle=\langle\psi|U_{F}^{\dagger}(\Lambda_{KF},a_{KF})|\tilde{\mathbf{p}},\mu,x_{F}\rangle^{*}=$
$e^{i\tilde{\mathbf{a}}\cdot\tilde{\mathbf{p}}}\langle\tilde{\boldsymbol{\Lambda}}^{-1}_{k}p,\mu,x_{F}|\psi\rangle\sqrt{{(\Lambda^{-1}_{k}p)^{+}\over
p^{+}}}$ (69)

and for rotations about the $z$ axis

$\langle\tilde{\mathbf{p}},\mu,x_{F}|U_{F}(R_{z}(\phi),0)\psi\rangle=\langle\psi|U_{F}^{\dagger}(R_{z}(\phi))|\tilde{\mathbf{p}},\mu,x_{F}\rangle^{*}=$
$\langle\tilde{\mathbf{R}}_{z}(\phi)^{-1}p,\mu,x_{F}|\psi\rangle
s^{i\mu\phi}.$ (70)

Equation (68) shows that $U_{I}(\Lambda,a)$ has an instant-form kinematic
subgroup while equations (69) and (70) show that $U_{F}(\Lambda,a)$ has a
light-front kinematic subgroup. In addition the two representations are
related by a unitary transformation:

$U_{F}(\Lambda,a)=V_{F}V_{I}^{\dagger}U_{I}(\Lambda,a)V_{I}V_{F}^{\dagger}$
(71) $|0\rangle_{F}=V_{F}V_{I}^{\dagger}|0\rangle_{I}=|0\rangle_{I}.$ (72)

In this construction it was chosen so that it preserves the vacuum.

If follows from (62) 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 (51), the
injection operators satisfy

$U_{I}(\Lambda_{KI},a_{KI})\Phi_{\cal A}=\Phi_{\cal A}U_{\cal
A}(\Lambda_{KI},a_{KI})$ (73)

for the instant-form kinematic subgroup and

$U_{F}(\Lambda_{KF},a_{KF})\tilde{\Phi}_{\cal A}=\tilde{\Phi}_{\cal A}U_{\cal
A}(\Lambda_{KF},a_{KF})$ (74)

for the light-front kinematic subgroup.

The results can be summarized by noting that it is possible to choose bases in
the field theory Hilbert space that transform 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 in these representations.

This construction did not assume the existence of a free dynamics on the
Hilbert space of the field theory. If the range of $\tilde{\Phi}$ is all of
${\cal H}$ and

$W:=(\tilde{\Phi}_{\cal A}\tilde{\Phi}_{\cal
A}^{\dagger})^{-1/2}\tilde{\Phi}_{\cal A}$ (75)

satisfies

$\lim_{t\to\pm\infty}\|(W-\tilde{\Phi}_{\cal A})e^{-iH_{\cal
A}t}|\psi_{\alpha}\rangle\|=0$ (76)

then $W$ is a kinematically invariant unitary mapping from ${\cal H}_{\cal A}$
to ${\cal H}$ and $U_{0}(\Lambda,a)=WU_{\cal A}(\Lambda,a)W^{\dagger}$ defines
a consistent “free dynamics” on ${\cal H}$ that satisfies

$U_{I}(\Lambda_{KI},a_{KI})=U_{0}(\Lambda_{KI},a_{KI})$ (77)

with a similar relation in the light-front case. If these representation
exist, they are not unique since they depend on the choice of injection
operator.

##  IV Spontaneous Symmetry Breaking

One of the puzzling aspects of light-front quantum field theory is how
spontaneous symmetry breaking is realized. The signal for spontaneous symmetry
breaking is a conserved local current, $j^{\mu}(x)$, that arises from the
symmetry and the presence of a 0 mass Goldstone boson in the spectrum. For a
local scalar field theory the following condition Coleman (2019)

$\lim_{R\to\infty}\langle 0|[Q_{R},\phi(y)]|0\rangle\not=0$ (78)

implies the existence of a 0 mass particle. Here

$\langle 0|[Q_{R},\phi(y)]|0\rangle:=\langle 0|[\int
d\mathbf{x}\chi_{R}(|\mathbf{x}|)j^{0}(\mathbf{x},t),\phi(y)]|0\rangle$ (79)

where $\chi_{R}(|\mathbf{x}|)$ is a smooth function that that is 1 for
$|\mathbf{x}|<R$ and $0$ for $|\mathbf{x}|>R+\epsilon$ for some finite
positive $\epsilon$. The cutoff function $\chi_{R}$ ensures that the integral
converges for large $|\mathbf{x}|$. The commutator vanishes for $x-y$ space-
like. For fixed $y$ and $t$ and sufficiently large $R$, if $|\mathbf{x}|>R$
then $x-y$ is space-like and the commutator vanishes. In this case

$\langle 0|[Q_{R},\phi(y)]|0\rangle:=\langle 0|[\int
d\mathbf{x}\chi_{R}(|\mathbf{x}|)j^{0}(\mathbf{x},t),\phi(y)]|0\rangle=\int
d\mathbf{x}\langle 0|[j^{0}(\mathbf{x},t),\phi(y)]|0\rangle.$ (80)

This does not require that the charge operator to exists.

Current conservation implies

$\langle 0|[\int
d\mathbf{x}\partial_{\mu}j^{\mu}(\mathbf{x},t),\phi(y)]|0\rangle=0.$ (81)

Inserting a complete set of intermediate states gives

$0=\sum\int d\mathbf{x}\left(\langle
0|\partial_{\mu}j^{\mu}(\mathbf{x},t)|p,n\rangle{d\mathbf{p}\over
2p_{n}^{0}}\langle p,n|\phi(y)|0\rangle-\langle
0|\phi(y)|p,n\rangle{d\mathbf{p}\over 2p_{n}^{0}}\langle
p,n|\partial_{\mu}j^{\mu}(\mathbf{x},t)|0\rangle\right)=$ $\sum\int
d\mathbf{x}\left(\langle
0|\partial_{\mu}j^{\mu}(\mathbf{0},0)|p_{r},n\rangle{d\mathbf{p}\over
2p_{n}^{0}}\langle p_{r},n|\phi(0)|0\rangle e^{ip\cdot(x-y)}-\langle
0|\phi(0)|p_{r},n\rangle{d\mathbf{p}\over 2p_{n}^{0}}\langle
p_{r},n|\partial_{\mu}j^{\mu}(\mathbf{0},0)|0\rangle e^{ip\cdot(y-x)}\right)$
(82)

where Poincaré covariance has been used to remove the non-trivial $x,y$ and
$p$ dependence from the matrix elements. $p_{r}$ is the constant rest four
momentum for massive states and a constant light-like vector for massless
states. For a scalar field theory the vacuum expectation value of the current
vanishes so the vacuum does not appear as an intermediate state. The Lehmann
weights that appear in this matrix element

$\sigma(m_{n})m_{n}^{2}=\langle
0|\partial_{\mu}j^{\mu}(\mathbf{0},0)|p_{r},n\rangle\langle
p_{nr},n|\phi(0)|0\rangle$ (83)

and

$\sigma^{*}(m_{n})m_{n}^{2}=\langle 0|\phi(0)|p_{nr},n\rangle\langle
p_{nr},n|\partial_{\mu}j^{\mu}(\mathbf{0},0)|0\rangle$ (84)

are functions of the invariant mass of the intermediate states. The factor
$m_{n}^{2}=-p_{n}^{2}$ is due to the $\partial_{\mu}j^{\mu}(\mathbf{0},0)$
assuming that the current is a local function of the scalar field.

Current conservation requires that this quantity vanishes which follows if
either $\sigma(m_{n})=0$ or $m_{n}^{2}=0$. Inserting intermediate states in
(79) gives a similar result with $\sigma(m_{n})m_{n}^{2}$ replaced by
$p_{nr}^{0}\sigma(m_{n})$ where $p_{nr}^{0}$ is a non-zero constant.

Since this does not include the $m^{2}$ factor, it vanishes by current
conservation unless $\sigma(m_{n})$ contains a $\delta(m_{n})$. Thus for (78)
to be non-vanishing the sum over intermediate states must have a contribution
from a 0 mass particle. This is the non-perturbative form of Goldstone’s
theorem.

In Coleman (2019) Coleman avoids directly computing the charge operator. The
current is an operator valued distribution, so there is no reason to expect
that integrating against the constant $1$ will result in a well defined
operator. There is another reason for this. If the integrals that define the
charge operator converge and there is no contribution to the current on the
surface at infinity, then the charge operator will be translationally
invariant and time independent. This means that it would commute with the four
momentum operator. It follows that even if $Q_{I}|0\rangle$ is not $0$, it is
eigenstate of the four momentum, with 0 eigenvalue. While the Goldstone boson
has 0 mass, its momentum cannot be 0, which means that a translationally
invariant time-independent charge operator cannot couple the vacuum to the
Goldstone boson.

In the light-front representation it is still possible to define $Q_{IR}(t)$
and the analysis above still applies. The assumption that the current is a
local function of the fields means that the light-front dynamics must be used
to get its value on the fixed-time surface where the locality of the theory
can be utilized to control the volume integral.

While the non-vanishing of (78) is a sign of spontaneous symmetry breaking,
the argument used above does not work when it is applied to the light-front
charge. The problem is that there is no compact region on the light front
where outside of that region $(x-y)$ is always space-like for fixed $y$ and
$x^{+}=0$. This is true even if the light-front charge is computed in the
instant-form representation. If the integrals that define the light-front
charge exist and there are no contributions from the divergence theorem, the
situation is similar to the instant-form case - the charge operator will
commute with the four momentum which means that the light-front charge applied
to the vacuum is an eigenstate of the four momentum with 0 eigenvalue, which
by the argument used above means that the charge operator cannot couple the
vacuum to the Goldstone boson.

##  V Miscellaneous remarks

This section provides a brief discussion comparing the structure of the light-
front and instant-form representations discussed in this work to the
corresponding relation in relativistic quantum mechanics of a finite number of
degrees of freedom.

The starting point of this work is the assumed existence of a local field
theory with the expected properties. Normally in an instant or front-form
representation of a quantum field theory “solving the field theory” means
constructing a unitary transformation that decomposes the Hilbert space into a
direct integral of irreducible representations. This is the relativistic
analog of diagonalizing the Hamiltonian. This is also true in relativistic
quantum theories of a finite number of degrees of freedom, except in that case
the interacting and non-interacting theories are formulated on the same
representation of the Hilbert space. In the finite number of degree of freedom
case different choices of independent variables in each irreducible subspace
are related by a unitary change of variables. This defines a unitary, S-matrix
preserving transformation relating the two forms of dynamics. It is non-
trivial because there is a different (mass-dependent) variable change on each
irreducible subspace. The light-front and instant-form bases have the property
that matrix elements of the unitary representation of the kinematic subgroups
in each case are mass independent.

In this work this process is reversed; the direct integral is assumed to be a
property of a quantum field theory with all of the expected properties. The
general class of S-matrix preserving unitary transformations is used to
construct equivalent representations of the dynamics. The freedom to choose
different $S-$matrix preserving unitary transformations was exploited to
choose S-matrix preserving unitary transformations that commute with the
instant form or light-front kinematic subgroups of the theory, which resulted
in equivalent light-front or instant-form representations of the field theory.

The unitarity of these transformations means that they must preserve the
spectrum of the mass operator. This means that point-spectrum eigenvalues of
the mass operator must be the same in all three representations. The unitary
transformation must also leave the vacuum energy or the mass gap unchanged.
Preserving the $S$-matrix means that acceptable unitary transformations must
be short-range perturbations of the identity in the sense that they do not
require a modification of scattering asymptotic conditions. The unitary
transformations that satisfy all of these conditions are not unique and are
discussed in appendix (A).

The construction of equivalent instant-form and light-front unitary
representations of the Poincaré group in this paper is essentially identical
to the corresponding construction in the finite number of degree of freedom
case Sokolov and Shatnyi (1979)Keister and Polyzou (1997)Polyzou (1989)Polyzou
(2002). In that case, when there is physical particle production, the
particles all have their physical masses and there are no vertex interactions
that can change masses Polyzou (2003). In that case the simplest interactions
that change particle number are $2\leftrightarrow 3$ interactions.

Given the spectrum of one-particle states, the mass spectrum of the field
theory Hamiltonian includes the one-particle masses and continuous eigenvalues
starting at the two-particle thresholds. This is identical to the mass
spectrum of a sum of free-field Hamiltonians with physical particle masses.
The conclusion of Haag’s theorem Streater and Wightman (1980) is that a
spectrally-equivalent free field theories are defined on inequivalent
representation of the Hilbert space. This why there are no free multiparticle
states in the field theory and is the motivation for using the two-Hilbert
space representation in order to formulate the scattering asymptotic
conditions.

In the absence of free multi-particle states in the theory, the notion of
many-particle interactions needs to be defined. One possibility is to define
interactions using matrix elements in states constructed by applying products
of the operators that create one-particle states out of vacuum to the vacuum.
Matrix elements of the Hamiltonian in these states have two properties, the
first is that they become products of one-particle states as the particles are
asymptotically separated. In this representation, because the one-particle
states are point spectrum eigenstates of the Hamiltonian, the Hamiltonian
cannot couple them to the continuum. This means that in this representation a
“production vertex” cannot couple to a multiparticle scattering state. This is
one way of saying that stable particle cannot decay, however particle
production is possible since interactions that couple “two-particle states” to
“three-particle states”, etc. are possible in this representation.

A final comment concerns vacuum polarization in this representation. The
physical vacuum is orthogonal to all bound and scattering eigenstates of the
Hamiltonian, but operators that couple the vacuum to other states in the
Hilbert space are possible and are needed to describe local vacuum
excitations.

##  VI Analysis and Conclusions

This work provides one way to understand Dirac’s forms of dynamics in a non-
perturbative setting. This approach assumed the existence of a 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 transform covariantly with respect to light-front and
instant-form kinematic subgroups where constructed for each of these
equivalent representations of the dynamics. Kinematic Poincaré transformations
on these wave functions could be computed without diagonalizing the dynamical
operators. One aspect of the construction used in this work is that both
representations of the dynamics leave the vacuum and one-particle states
unchanged and both representations give the same Poincaré invariant scattering
operator. While preserving the vacuum states is not required, the construction
demonstrates that it is a consistent assumption.

A key difference with the perturbative approach is the non-existence of an
underlying free-field dynamics on the Hilbert space of the field theory. If
the injection operators have full range on the Hilbert space of the field
theory and satisfy (76) there is the analog of a free dynamics on the Hilbert
of the field theory. If such a representation exists, it cannot be a local
free field theory with physical masses.

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 up to a unitary transformation and can be taken as the same in
both representations. While the light-front vacuum is kinematically invariant,
it is also rotationally invariant, which is an additional dynamical constraint
on the light-front vacuum that separates it from the free-field Fock vacuum.
By starting with the assumed solution to the field theory there are no issues
with renormalization, rotational covariance or 0 modes. These issues need to
be addressed in applications. If there is spontaneous symmetry breaking, the
non-perturbative condition for the presence of a Goldstone boson in the mass
spectrum is the same in both representations, but it does not couple to the
vacuum with a space-time translationally invariant charge operator, whether it
is on the fixed-time surface or the light-front hyperplane.

The author would like to thank the referee for some useful suggestions.

##  Appendix A Scattering equivalences

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:

$\Omega_{\pm}(H,\Phi_{\cal A},H_{\cal
A})=s-\lim_{t\to\pm\infty}e^{iHt}\Phi_{\cal A}e^{-iH_{\cal A}t}.$ (85)

The wave operators are assumed to exist and be Poincaré invariant in the sense

$U(\Lambda,a)\Omega_{\pm}^{\dagger}(H,\Phi_{\cal A},H_{\cal
A})=\Omega_{\pm}^{\dagger}(H,\Phi_{\cal A},H_{\cal A})U_{\cal A}(\Lambda,a).$
(86)

where $U_{\cal A}(\Lambda,a)$ is the natural representation of the Poincaré
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

$S(H,\Phi,H_{\cal A})=\Omega_{+}^{\dagger}(H,\Phi_{\cal A},H_{\cal
A})\Omega_{-}(H,\Phi_{\cal A},H_{\cal A}).$ (87)

Consider two identical scattering operators based on different Hamiltonians

$S(H,\Phi_{\cal A},H_{\cal A})=S(H^{\prime},\Phi_{\cal A}^{\prime},H_{\cal
A})$ (88)

It follows from the definitions and unitarity of the wave operators that

$W:=\Omega_{+}(H^{\prime},\Phi^{\prime}_{\cal A},H_{\cal
A})\Omega_{+}^{\dagger}(H,\Phi_{\cal A},H_{\cal
A})=\Omega_{-}(H^{\prime},\Phi^{\prime}_{\cal A},H_{\cal
A})\Omega_{-}^{\dagger}(H,\Phi_{\cal A},H_{\cal A})$ (89)

is a unitary operator on ${\cal H}$. The identification of the scattering
operators (88) means that $W$ is the same for the incoming ($t\to+\infty$) or
outgoing ($t\to-\infty$) multi-channel wave operator. It follows that

$W\Omega_{\pm}(H,\Phi_{\cal A},H_{\cal A})=\Omega_{\pm}(H^{\prime},\Phi_{\cal
A}^{\prime},H_{\cal A}).$ (90)

In addition, the intertwining property of the wave operators Reed and Simon
(1979),

$WH^{\prime}=\Omega_{\pm}(H,\Phi_{\cal A},H_{\cal
A})\Omega_{\pm}^{\dagger}(H^{\prime},\Phi_{\cal A}^{\prime},H_{\cal
A})H^{\prime}=\Omega_{\pm}(H,\Phi_{\cal A},H_{\cal A})H_{\cal
A}\Omega_{\pm}^{\dagger}(H^{\prime},\Phi^{\prime}_{\cal A},H_{\cal A})$
$=H\Omega_{\pm}(H,\Phi_{\cal A},H_{\cal
A})\Omega_{\pm}^{\dagger}(H^{\prime},\Phi^{\prime}_{\cal A},H_{\cal A})=HW$
(91)

means that the two Hamiltonians are related by the unitary transformation $W$.

It follows that

$\Omega_{-}(H^{\prime},\Phi_{\cal A}^{\prime},H_{\cal
A})=W\Omega_{-}(H,\Phi_{\cal A},H_{\cal A})=$
$\Omega_{-}(WHW^{\dagger},W\Phi_{\cal A},H_{\cal
A})=\Omega_{-}(H^{\prime},W\Phi_{\cal A},H_{\cal A}).$ (92)

Taking the difference of the right and left side of equation (92) gives

$0=\lim_{t\to\pm\infty}\|e^{iH^{\prime}t}(\Phi_{\cal A}^{\prime}-W\Phi_{\cal
A})e^{-iH_{\cal A}t}|\psi\rangle\|.$ (93)

The unitarity of $e^{iH^{\prime}t}$ gives

$0=\lim_{t\to\pm\infty}\|(\Phi_{\cal A}^{\prime}-W\Phi_{\cal A})e^{-iH_{\cal
A}t}|\psi\rangle\|.$ (94)

That this condition holds for both time limits is important. For the vacuum
and one-particle channels this condition requires that

$\Phi_{\cal A}^{\prime}-W\Phi_{\cal A}=0.$ (95)

Next consider the converse. Assume that $W$ is a unitary operator satisfying
$H^{\prime}=WHW^{\dagger}$, and the scattering operators

$S(H,\Phi,H_{\cal A})=\Omega_{+}^{\dagger}(H,\Phi_{\cal A},H_{\cal
A})\Omega_{-}(H,\Phi_{\cal A},H_{\cal A})$ (96)
$S(H^{\prime},\Phi^{\prime}_{\cal A},H_{\cal
A})=\Omega_{+}^{\dagger}(H^{\prime},\Phi^{\prime}_{\cal A},H_{\cal
A})\Omega_{-}(H^{\prime},\Phi^{\prime}_{\cal A},H_{\cal A})$ (97)

both exist. If $W$ satisfies (94) for both time limits then the $S$ matrices
are identical

$S(H^{\prime},\Phi^{\prime}_{\cal A},H_{\cal A})=S(H,\Phi_{\cal A},H_{\cal
A}).$ (98)

The proof follows from

$\Omega_{\pm}(H^{\prime},\Phi^{\prime}_{\cal A},H_{\cal
A})=\Omega_{\pm}(WHW^{\dagger},\Phi^{\prime}_{\cal A},H_{\cal
A})=W\Omega_{\pm}(H,W^{\dagger}\Phi^{\prime}_{\cal A},H_{\cal A})=$
$W\Omega_{\pm}(H,W^{\dagger}(\underbrace{\Phi^{\prime}_{\cal A}-W\Phi{\cal
A}}_{\to 0}+W\Phi_{\cal A}),H_{\cal A})=W\Omega_{\pm}(H,\Phi_{\cal A}),H_{\cal
A}).$ (99)

It then follows that

$S(H^{\prime},\Phi^{\prime}_{\cal A},H_{\cal
A})=\Omega_{+}^{\dagger}(H^{\prime},\Phi^{\prime}_{\cal A},H_{\cal
A})\Omega_{-}(H^{\prime},\Phi^{\prime}_{\cal A},H_{\cal A})=$
$\Omega_{+}^{\dagger}(H,\Phi_{\cal A},H_{\cal
A})W^{\dagger}W\Omega_{-}(H,\Phi_{\cal A},H_{\cal A})=S(H,\Phi_{\cal
A},H_{\cal A}).$ (100)

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 Hamiltonians 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 (94) 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 they have asymptotically complete scattering operators. The
field theory generalization of multi-particle scattering is the Haag-Ruelle
formulation of scattering which involves the strong limits used in this
appendix.

##  Appendix B Two Hilbert space injection operators

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

$\langle 0|B^{\dagger}B|0\rangle=\sum_{n}\langle 0|B^{\dagger}|n\rangle\langle
n|B|0\rangle$ (101)

has one-body intermediate states (here one-body means states with discrete
positive-mass eigenvalues, so 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 that 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

$B(x):=e^{-iP\cdot x}Be^{-iP\cdot x}=U^{\dagger}(I,x)BU(I,x)$ (102)

where $P^{\mu}$ is the four momentum operator. It follows from (102) that

${\partial B(x)\over\partial x_{\mu}}=-i[P^{\mu},B(x)].$ (103)

The next step is to compute the Fourier transform of the operator density
$B(x)$:

$\hat{B}(q):=\int{d^{4}x\over(2\pi)^{2}}e^{iq\cdot x}B(x).$ (104)

Multiplying both sides of (103) by ${e^{iq\cdot x}\over(2\pi)^{2}}$ and
integrating over $d^{4}x$ by parts, assuming that ${B}(x)$ is an operator
valued distribution, gives

$-i[P^{\mu},\hat{B}(q)]=-iq^{\mu}\hat{B}(q)$ (105)

or

$[P^{\mu},\hat{B}(q)]=q^{\mu}\hat{B}(q).$ (106)

It follows from (106) that $\hat{B}(q)|0\rangle$ is either $0$ or an
eigenstate of the four momentum with eigenvalue $q^{\mu}$:

$P^{\mu}\hat{B}(q)|0\rangle=\hat{B}(q)P^{\mu}|0\rangle=q^{\mu}\hat{B}(q)|0\rangle=q^{\mu}\hat{B}(q)|0\rangle.$
(107)

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 $q^{0}>0$, and identically $0$ on the rest of the spectrum of intermediate
states in (101) and define

${A}^{\dagger}(q):=\hat{B}(q)h(q).$ (108)

Additional integrations are used to define the operators:

${A}^{\dagger}(\mathbf{q}):=\int{A}^{\dagger}(q)dq^{0}\qquad\mbox{and}\qquad\tilde{A}^{\dagger}(\tilde{\mathbf{q}}):=\int{A}^{\dagger}(q)dq^{-}/2.$
(109)

Because $h(q)$ is localized, these operators are distributions in the three
momentum or light-front components of the four momentum. When these operators
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 $q^{-}$ it follows that (106) must be
replaced by

$[\mathbf{P},{A}^{\dagger}(\mathbf{q})]=\mathbf{q}{A}^{\dagger}(\mathbf{q}).\qquad[\tilde{\mathbf{P}},\tilde{A}^{\dagger}(\tilde{\mathbf{q}})]=\tilde{\mathbf{q}}\tilde{A}^{\dagger}(\tilde{\mathbf{q}})$
(110)

which when applied to the vacuum becomes

$\mathbf{P}A^{\dagger}(\mathbf{q})|0\rangle=\mathbf{q}{A}^{\dagger}(\mathbf{q})|0\rangle\qquad\tilde{\mathbf{P}}\tilde{A}^{\dagger}(\tilde{\mathbf{q}})|0\rangle=\tilde{\mathbf{q}}\tilde{A}^{\dagger}(\tilde{\mathbf{q}})|0\rangle.$
(111)

This means that ${A}^{\dagger}(\mathbf{q})$ creates a one-particle state with
momentum $\mathbf{q}$, mass $m$ and spin $s$ out of the vacuum. Similarly

$\tilde{\mathbf{P}}\tilde{A}^{\dagger}(\tilde{\mathbf{q}})|0\rangle=\tilde{\mathbf{q}}{A}^{\dagger}(\tilde{\mathbf{q}})|0\rangle.$
(112)

creates a one-particle state with light-front momentum $\tilde{\mathbf{q}}$
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

${A}^{\dagger}_{m,s}(\mathbf{p},\mu):=\int_{SU(2)}U(R,0){A}^{\dagger}(R\mathbf{p})U^{\dagger}(R,0)D^{s*}_{\mu
s}(R)dR$ (113)

where the integral is over the $SU(2)$ Haar measure and $D^{s*}_{\mu s}(R)$ is
the spin-$s$ $SU(2)$ Wigner function

$D^{s}_{\mu s}(R)=\langle s,\mu|U(R)|s,s\rangle.$ (114)

It follows from (113) that

$U(R^{\prime},0){A}^{\dagger}_{m,s}(\mathbf{p},\mu)U^{\dagger}(R^{\prime},0)=$
$\int_{SU(2)}U(R^{\prime}R,0){A}^{\dagger}(R^{-1}\mathbf{p})U^{\dagger}(R^{\prime}R,0)D^{s*}_{\mu
s}(R)dR.$ (115)

Changing variables, $R^{\prime\prime}=R^{\prime}R$ using the invariance of the
Haar measure $dR=dR^{\prime\prime}$ for fixed $R^{\prime}$ gives

$(\ref{A2:15})=\int_{SU(2)}U(R^{\prime\prime},0){A}^{\dagger}(R^{\prime\prime-1}R^{\prime}\mathbf{p})U^{\dagger}(R^{\prime\prime},0)dR^{\prime\prime}D^{s*}_{\mu
s}(R^{\prime-1}R^{\prime\prime})=$
$\int_{SU(2)}\sum_{\nu=-s}^{s}U(R^{\prime\prime},0)A^{\dagger}(R^{\prime\prime-1}R^{\prime}\mathbf{p})U^{\dagger}(R^{\prime\prime},0)dR^{\prime\prime}D^{s*}_{\mu\nu}(R^{\prime-1})D^{s*}_{\nu
s}(R^{\prime\prime})=$
$\int_{SU(2)}\sum_{\nu=-s}^{s}U(R^{\prime\prime},0){A}^{\dagger}(R^{\prime\prime-1}R^{\prime}\mathbf{p})U^{\dagger}(R^{\prime\prime},0)dR^{\prime\prime}D^{s*}_{\nu
s}(R^{\prime-1}R^{\prime\prime})D^{s}_{\nu\mu}(R^{\prime})=$
$\sum_{\nu=-s}^{s}{A}^{\dagger}_{(m,j)}(R^{\prime}\mathbf{p},\nu)D^{s}_{\nu\mu}(R^{\prime}).$
(116)

When applied to the vacuum (116) gives

$U(R,0){A}^{\dagger}_{(m,s)}(\mathbf{p},\mu)|0\rangle=\sum_{\nu=-s}^{s}{A}^{\dagger}_{m,s}(R\mathbf{p},\nu)|0\rangle
D^{s}_{\nu\mu}(R)$ (117)

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 (101). While the notation is purposely suggestive,
$A^{\dagger}_{m,s}(\mathbf{p},\mu)$ is 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

$[\mathbf{P},{A}_{(m,s)}^{\dagger}(\mathbf{q},\mu)]=\int_{SU(2)}dR[\mathbf{P},U(R,0){A}^{\dagger}(R^{-1}\mathbf{q})U^{\dagger}(R,0)]D^{s*}_{\mu
s}(R)=$
$\int_{SU(2)}dRU(R,0)[R\mathbf{P},{A}^{\dagger}(R^{-1}\mathbf{q})]U^{\dagger}(R,0)D^{s*}_{\mu
s}(R)=$
$RR^{-1}\mathbf{q}\int_{SU(2)}dRU(R,0){A}^{\dagger}(R^{-1}\mathbf{q})U^{\dagger}(R,0)D^{s*}_{\mu
s}(R)=\mathbf{q}{A}^{\dagger}_{(m,s)}(\mathbf{q},\mu).$ (118)

The normalization can be chosen so the states created out of the vacuum have
the normalization (9).

The construction above cannot be used in the “light-front” case due to the
integral over $p^{-}$ in (109). However in that case it is enough to project
out the magnetic quantum number using a rotation $R_{z}(\phi)$ which leaves
$p^{-}$ in (109) unchanged. The spin is identified with the highest non-zero
weight, $s=\mu_{max}$. In this case equation(113) is replaced by

$\tilde{{A}}^{\dagger}_{m,s}(\tilde{\mathbf{p}},\mu):=\int_{0}^{2\pi}U(R_{z}(\phi),0)\tilde{{A}}^{\dagger}(R_{z}(\phi)^{-1}\mathbf{p})U^{\dagger}(R_{z}(\phi),0)e^{-i\mu\phi}{d\phi\over
2\pi},$ (119)

(117) and (118) are replaced by

$[\tilde{\mathbf{P}},\tilde{{A}}_{(m,s)}^{\dagger}(\tilde{\mathbf{q}},\mu)]=\tilde{\mathbf{q}}\tilde{{A}}^{\dagger}_{(m,s)}(\tilde{\mathbf{q}},\mu),$
(120)
$U(\Lambda_{K},0)\tilde{{A}}^{\dagger}_{(m,s)}(\tilde{\mathbf{q}},\mu)U^{\dagger}(\Lambda_{K},0)=\tilde{{A}}^{\dagger}_{(m,s)}(\tilde{\Lambda}_{k}q,\mu),$
(121)

and

$U(R_{z}(\phi),0)\tilde{{A}}^{\dagger}_{(m,s)}(\tilde{\mathbf{q}},\mu)U^{\dagger}(R_{z}(\phi),0)=\tilde{{A}}^{\dagger}_{(m,s)}(R_{z}(\phi)\tilde{\mathbf{q}},\mu)e^{i\mu\phi}.$
(122)

The proof (120) is essentially the same as the proof of (118). To show (122)
note that for rotations about the $z$ axis

$B_{F}(p/m)R_{z}=R_{z}B_{F}(R_{z}^{-1}p/m)$ (123)

where $B_{F}(p/m)$ is a light-front boost. It follows that

$U(B_{F}(p/m),0)\tilde{{A}}^{\dagger}_{m,s}(\tilde{\mathbf{q}},\mu)U(B_{F}(p/m),0)^{\dagger}=$
$\int_{0}^{2\pi}U(B_{F}(p/m)R_{z}(\phi),0)\tilde{{A}}^{\dagger}(R_{z}(\phi)^{-1}\mathbf{q})U^{\dagger}(B_{F}(p/m)R_{z}(\phi),0)e^{-i\mu\phi}=$
$\int_{0}^{2\pi}U(R_{z}(\phi)B_{F}(R_{z}^{-1}(\phi)p/m),0)\tilde{{A}}^{\dagger}(R_{z}(\phi)^{-1}\mathbf{q})U^{\dagger}(R_{z}(\phi)B_{F}(R_{z}^{-1}(\phi)p/m),0)e^{-i\mu\phi}=$
$\int_{0}^{2\pi}U(R_{z}(\phi),0)U(B_{F}(R_{z}^{-1}(\phi)p/m),0)\tilde{{A}}^{\dagger}(R_{z}(\phi)^{-1}\mathbf{q})U(B_{F}(R_{z}^{-1}(\phi)p/m),0))^{\dagger}U^{\dagger}(R_{z}(\phi),0)^{\dagger}e^{-i\mu\phi}=$
$\int_{0}^{2\pi}U(R_{z}(\phi),0)U(,0)\tilde{{A}}^{\dagger}(B_{F}(R_{z}^{-1}(\phi)p/m)R_{z}(\phi)^{-1}\mathbf{q})U^{\dagger}(R_{z}(\phi),0)^{\dagger}e^{-i\mu\phi}=$
$\int_{0}^{2\pi}U(R_{z}(\phi),0)U(,0)\tilde{{A}}^{\dagger}(R_{z}(\phi)^{-1}B_{F}(p/m)\mathbf{q})U^{\dagger}(R_{z}(\phi),0)^{\dagger}e^{-i\mu\phi}=$
$\tilde{{A}}^{\dagger}_{m,s}(\tilde{\mathbf{B}}_{F}(p/m)q,\mu)$ (124)

which proves (121). For (122) note that

$U(R_{z}(\phi^{\prime}),0)\tilde{{A}}^{\dagger}_{(m,s)}(\tilde{\mathbf{q}},\mu)U^{\dagger}(R_{z}(\phi^{\prime}),0)=$
$\int_{0}^{2\pi}U(R_{z}(\phi+\phi^{\prime}),0)\tilde{{A}}^{\dagger}(R_{z}(\phi)^{-1}\mathbf{p})U^{\dagger}(R_{z}(\phi+\phi^{\prime}),0)e^{-i\mu\phi}{d\phi\over
2\pi}.$ (125)

Let $\phi^{\prime\prime}=\phi+\phi^{\prime}$ so (125)) becomes

$\int_{0}^{2\pi}U(R_{z}(\phi^{\prime\prime}),0)\tilde{{A}}^{\dagger}(R_{z}(\phi^{\prime\prime}-\phi^{\prime})^{-1}\mathbf{p})U^{\dagger}(R_{z}(\phi^{\prime\prime}),0)e^{-i\mu(\phi^{\prime\prime}-\phi^{\prime})}{d\phi^{\prime\prime}\over
2\pi}$
$\int_{0}^{2\pi}U(R_{z}(\phi^{\prime\prime}),0)\tilde{{A}}^{\dagger}(R_{z}(\phi^{\prime\prime})^{-1}R_{z}(\phi^{\prime})\mathbf{p})U^{\dagger}(R_{z}(\phi^{\prime\prime}),0)e^{-i\mu(\phi^{\prime\prime}}e^{i\phi^{\prime}\mu)}=$
$\tilde{{A}}^{\dagger}_{(m,s)}(R_{z}(\phi^{\prime})\tilde{\mathbf{q}},\mu)e^{i\mu\phi^{\prime}}.$
(126)

The normalization can be chosen so these states created out of the vacuum have
the normalization (9).

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