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\begin{document}
\title{Scattering and reflection positivity in relativistic 
Euclidean quantum mechanics}
\author{W. N. Polyzou}
\affiliation{
%Department of Physics and Astronomy, 
The University of Iowa, Iowa City, IA
52242}

\begin{abstract}

  In this paper I discuss a formulation of relativistic few-particle
  scattering theory where the dynamical input is a collection of
  reflection-positive Euclidean covariant Green functions.  This
  formulation of relativistic quantum mechanics has the advantage that
  all calculations can be performed using Euclidean variables.  The
  construction of the Hilbert space inner product requires that the
  Green functions are reflection positive, which is a non-trivial
  constraint on the dynamics.  In this paper I discuss the structure
  of reflection positive Green functions.  I show that scattering wave
  operators exist when the dynamics is given by a Green function with
  the structure discussed in this work.  Techniques for computing
  scattering observables in this Euclidean framework are discussed.

\end{abstract}

\pacs{03.70.+k, 11.10.-z 11.15.Ha 64.60.ae}

\maketitle

\section{Introduction}

In this paper I discuss a formulation of relativistic few-body quantum
mechanics where the dynamics is introduced through a collection of
Euclidean ``Green'' functions.  The motivation for this approach is
similar to the motivation for using Euclidean methods in quantum field
theory.  The complicated singularities of the Minkowski Green
functions are replaced by well-behaved Euclidean Green functions.  In
addition, while the input can in principle be directly calculated from
a field theory, for the purpose of making realistic models it can also
be treated phenomenologically.  Cluster properties, which are
difficult to realize in direct interaction formulations of
relativistic quantum mechanics, are relatively simple to realize in
the Euclidean formulation.  The surprising features of this approach
are (1) all relevant calculations can be performed entirely in
Euclidean space, (2) there is numerical evidence that GeV-scale
scattering calculations can be performed without analytic continuation
(3) and finite Poincar\'e transformations can be performed without
leaving Euclidean space.

In the next section I discuss the general structure of this Euclidean
formulation of relativistic quantum mechanics.  In section three I
discuss the construction of the model Hilbert space and the
construction of a representation of the Poincar\'e Lie algebra on this
space.  This defines what I mean by a relativistic quantum theory.  In
section four I illustrate how the usual Minkowski description of a
single particle is realized in the Euclidean framework.  In section
five I discuss the important constraint of reflection positivity,
which is a required property of acceptable Euclidean Green functions.
I illustrate some of the difficulties with establishing reflection
positivity and generalize a known one-dimensional result to
demonstrate the existence of a large class of reflection positive
four-point Green functions.  In section six I discuss the formulation
of the scattering problem when the dynamics appears in the kernel of a
Hilbert space inner product.  I show that M{\o}ller wave operators
exist for the class of reflection positive Green function introduced
in the previous section.  In section seven I discuss methods for the
numerical computation of scattering observables in this formalism.
Section eight provides a summary of the results and a discussion of open
problems.  The appendix generalizes the discussion in section four to 
massive particles with any spin.

\section{Input}

The input to a Euclidean relativistic quantum theory is a collection of 
Euclidean invariant distributions, which I refer to as 
quasi-Schwinger functions
\beq
\{ S_{m:n}(\x_m, \cdots , \x_1; \y_1, \cdots , \y_n) \}.
\label{c.1}
\eeq
Here san-serif variables $\x_i = (\x^0_i,\mathbf{x}_i)$ and $\y_i :=
(\y^0_i,\mathbf{y}_i)$ are Euclidean space-time variables.  In
principle the collection of quasi-Schwinger functions could be
infinite, however finite collections are appropriate for making
few-body models.  The $\y$ and $\x$ variables are associated with
initial and final degrees of freedom respectively.  For purpose of
illustration I consider the spinless case.  The treatment of particles
with spin is discussed in the appendix.

The quasi-Schwinger functions are assumed to have the following properties:
\begin{itemize}
\item [{\bf E.1}] Euclidean invariance:
\[
S_{m:n}(\x_m, \cdots, \x_1; \y_1, \cdots , \y_n) =
\]
\beq
S_{m:n}(O\x_m+\a, \cdots , O\x_1+\a; O \y_1+\a, \cdots ,  O \y_n+\a)
\label{c.2}
\eeq
\item [] where $\a$ is a constant Euclidean 4-vector and $O$ is a 
four-dimensional orthogonal transformation. 

\item [{\bf E.2}] Reflection positivity:
\item [] for any $\{f_n (\y_1, \cdots, \y_n )\}$ with support for 
$0 < \y_1^0 < \y_{2}^0 <  \cdots < \y_{n-1}^0 < \y_n^0$:
\[
(f,f)_M = 
\]
\beq
\sum_{k,n} \int f_k^* (\x_k, \cdots ,\x_1 ) 
S_{k:n}(\theta \x_k, \cdots, \theta \x_1; \y_1, \cdots , \y_n)
f_n (\y_1, \cdots, \y_n)d^{4k}\x d^{4n}\y \geq 0 
\label{c.3}
\eeq
\item [] where $\theta \y := \theta (\y^0,\pmb{\y}) := (-\y^0,\pmb{\y})$ 
is the Euclidean time reflection operator.
\item [] In addition the quasi-Schwinger functions must satisfy 
\beq
S_{k:n}(\x_k \cdots \x_1; \y_1 \cdots  \y_n) =
S_{n:k}^*(\y_n \cdots \y_1; \x_1 \cdots  \x_k) 
\label{c.4}
\eeq
so $(f,g)_M = (g,f)_M^*$.    
\item [{\bf E.3}] Cluster properties:
\beq
S_{k:n} (\x_k , \cdots , \x_1; \y_1 , \cdots ,  \y_n) =
\sum \prod S^c_{k_i:n_i} (\x_{ki}, \cdots , \x_{1i}; \y_{1i}, \cdots ,  \y_{ni})
\label{c.5}
\eeq
\item [] where the $S^c_{k_i:n_i}$ satisfy [{\bf E.1-E.2}],
$\sum k_i = k$ and $\sum n_i = n$.  The sum runs over
all partitions of the initial and final variables into clusters
containing at least one initial and one final coordinate in each
cluster.  For a given partition the products are over clusters in
the partition.  The $S^c_{k_i:n_i}$ vanish as distributions whenever
any of the coordinates are asymptotically separated.

\end{itemize}

For quasi-Schwinger functions describing identical particles the
initial and final variables are separately symmetric or anti-symmetric
under interchange of the Euclidean space time coordinates.  In quantum
field theory locality identifies the $n$-point Schwinger functions
with different numbers of initial and final degrees of freedom
\cite{Osterwalder:1973dx}.  In this work the locality requirement is
relaxed which allows for the possibility of having different $n$-point
functions for different combinations of initial and final particles.
 
In this formalism the connected $n$-point quasi-Schwinger functions
replace the $n$-body interactions in conventional potential models.
The problem is to construct a set of model quasi-Schwinger functions
that lead to predictions consistent with a large class of experiments.
The virtue of this type of phenomenology is that if the Schwinger
functions of the field theory could be calculated, the model input could
be replaced by the exact Schwinger functions.  


\section{The quantum theory}

In any quantum theory the fundamental observables are probability
amplitudes.  These are represented by Hilbert space inner products of
unit normalized vectors.  The first step in the construction of a
relativistic quantum model is to specify the structure of the model
Hilbert space.  This is a specification of the model degrees of
freedom, however in the Euclidean framework the relation of the
representation of the Hilbert space to the particle content of the
theory is more abstract.

Given the representation of the Hilbert space, the invariance of the
probability amplitudes under change of inertial coordinate system
requires a unitary representation of the Poincar\'e group
\cite{Wigner:1939cj}.  An equivalent requirement is a set of ten
self-adjoint operators that satisfy the commutation relations of the
Poincar\'e group.  These operators generate unitary one-parameter
groups corresponding to time translation, space translation, rotations
and rotationless Lorentz transformations. Finite Poincar\'e
transformations can be expressed as products of these one-parameter
unitary groups.

In this section I define the model Hilbert space of Euclidean
relativistic quantum mechanics and construct a representation of the
Poincar\'e Lie algebra on this space.

I assume that a collection of quasi-Schwinger functions satisfying
{\bf E.1-E.3} is given.  The quasi-Schwinger functions can be used to
construct two quadratic forms.  The first one is a pure Euclidean form
where the $S_{k:n}$ serve as kernels of the quadratic form:
\beq
(f,g)_E = 
\sum_{m,n} \int f_m^* (\x_1, \cdots ,\x_m ) 
S_{m:n}(\x_m, \cdots, \x_1; \y_1, \cdots , \y_n)
g_n (\y_1, \cdots, \y_n)d^{4m}\x d^{4n}\y .
\label{d.1}
\eeq
Here vectors are represented by collections of test functions     
$\{ g_n (\y_n, \cdots, \y_1) \} $ of Euclidean space-time variables. 

Following Osterwalder and Schrader \cite{Osterwalder:1973dx} 
I use the Euclidean 
time-reflection operator to define a second quadratic form  
\beq
(f,g)_M = 
\sum_{m,n} \int f_m^* (\x_1, \cdots ,\x_m ) 
S_{m:n}(\theta \x_m, \cdots, \theta \x_1; \y_1, \cdots , \y_n)
g_n (\y_1, \cdots, \y_n)d^{4m}\x d^{4n}\y ,
\label{d.2}
\eeq
where $\theta \x = \theta (\tau,\mathbf{x}) = (-\tau,\mathbf{x})$
is the Euclidean time reflection operator.

In both of these forms the integrals are over Euclidean space-time
variables.  The two quadratic forms differ by the presence of the
Euclidean-time reflections, $\theta$, in all of the final-state
variables.  The second quadratic form can be expressed abstractly in
terms of the first form
\beq
(f,g)_M = (\Theta f,g)_E 
\label{d.3}
\eeq
where $\Theta$ represents the  operator that reflects all of the final
Euclidean times.  I distinguish these forms by the subscripts $E$ for
Euclidean and $M$ for Minkowski.  The Minkowski designation of the
second form is not obvious at this point.

To understand the motivation for introducing the second form note that
the Euclidean invariance property,  {\bf E.1}, ensures that the 
representation $U_E(O,a)$ of the four-dimensional Euclidean group, 
defined by
\beq
((U_E(O,a) f_m) (\y_1, \cdots ,\y_m ):=
f_m (O\y_1+a, \cdots , O\y_m+a), 
\label{d.4}
\eeq
satisfies the unitarity condition 
\beq
( U_E(O,a)f, U_E(O,a) g)_E = ( f, g)_E 
\label{d.5}
\eeq
in the sense that it leaves the first quadratic form invariant.

Each of the one-parameter subgroups of $U_E(O,a)$ can be formally
expressed as the exponential of $i$ multiplied by a Hermitian
infinitesimal generator in the $(,)_E$ quadratic form.  The Euclidean
time-reversal operator $\Theta$ changes the sign of the infinitesimal
generators of $U_E(O,a)$ that are linear in Euclidean time or
derivatives with respect to Euclidean time.  There are four generators
with one of these properties.  They are the generator of Euclidean
time translation and the three generators of rotations in Euclidean
space-time planes, $(\x^0,x,0,0), (\x^0,0,y,0), (\x^0,0,0,z)$.  The
additional sign change makes these four generators anti-Hermitian in
the $(,)_M$ quadratic form.  These can be made Hermitian with respect
to the $(,)_M$ quadratic form by multiplying each one by $-i$.  With
this change these four generators become Hermitian with respect to the
quadratic form $(,)_M$.  In addition, a direct calculation shows that
when combined with generators of ordinary rotations and space
translations, the resulting set of 10 operators satisfy the Poincar\'e
commutation relations.

The reason for this connection is because the covering groups of both the 
complex orthogonal groups and complex Lorentz groups are identical.  
Because of this relationship the group of real orthogonal transformations 
can be considered to be a subgroup of the complex Lorentz group.

To understand this relation note that the determinants of the
matrices 
\beq
X:= \left (
\begin{array}{cc} 
t+z & x-iy \\
x+iy & t-z 
\end{array}
\right )
\qquad \mbox{and} \qquad
\X:= \left (
\begin{array}{cc} 
i\tau +z & x-iy \\
x+iy & i\tau -z 
\end{array}
\right )
\label{d.6}
\eeq
are (up to sign) the Minkowski and Euclidean line elements respectively.
The determinant of both  $X$ and $\X$ are preserved by
the same linear $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$ transformation 
\beq
X' = AXB^T \qquad \X' = A\X B^T 
\label{d.7}
\eeq
where $A$ and $B$ are $SL(2,\mathbb{C})$ matrices.  While these transformations
preserve the Euclidean and Minkowski displacements, they do not 
generally preserve the reality of the corresponding coordinates.
Thus, the resulting transformations can be considered representations of 
either complex orthogonal or complex Lorentz  transformations. 

The Hermiticity of the modified generators on $(,)_M$ implies that 
they can be exponentiated to construct a unitary representation of 
the Poincar\'e group with respect to the form $(,)_M$.  The only problem 
is that when the test functions, $\{ f_m (\x_m, \cdots , \x_1 ) \}$, are
unrestricted, this quadratic form has negative norm vectors.  This is
easily seen by choosing $f(\cdots)$ to be odd with respect to
Euclidean time reflection.  An indefinite inner product is not
suitable for a quantum-mechanical interpretation. 
%
%Abstractly, because
%of the Poincar\'e invariance of $(,)_M$, there is a Poincar\'e invariant
%subspace of positive norm vectors, but is does not follow that is
%space in preserved under transformations that separate
%clusters of particles.

Osterwalder and Schrader \cite{Osterwalder:1973dx} solve this problem
by restricting the test functions to have support for positive
relative Euclidean times $0 < \x_1^0 < \x_{2}^0 < \cdots <\x_{m-1}^0 <
\x_m^0 $ and limiting the types of acceptable Schwinger functions to
what they refer to as reflection-positive Schwinger functions.

Following Osterwalder and Schrader we call a collection of
quasi-Schwinger functions $\{S_{k:n}\}$ satisfying
\beq
(f , f)_M \geq 0 
\label{d.8}
\eeq
for functions with positive time and positive relative time support
{\it reflection positive}.  Reflection positivity is a property of the
collection of quasi-Schwinger functions.  The quadratic form $(,)_M$
can be completed to a Hilbert space inner product if the quasi Schwinger
functions are reflection positive.  In this case vectors are
equivalence classes of collections of positive relative time test
functions, where $f \sim g$ if and only if $(f-g,f-g)_M =0$.  This
space is completed by including Cauchy sequences of these equivalence
classes.  The resulting Hilbert space is denoted by ${\cal H}_M$.  For
identical particles the symmetry of the quasi-Schwinger functions
means that test functions with non-overlapping Euclidean time support
can always be reordered to have positive relative time support.

One problem with this realization of the Hilbert space is that the
Euclidean time translations and rotations in Euclidean space-time
planes do not generally preserve the positive Euclidean time-support
constraint.  Nevertheless, for suitable quasi-Schwinger functions,
semigroup methods can be used to establish that the generators,
extracted from infinitesimal versions of these transformations, are
self-adjoint operators on ${\cal H}_M$.  Formally the one-parameter
group of Euclidean time translation on ${\cal H}_E$ becomes a
contractive Hermitian semigroup \cite{Hille:1957} on ${\cal H}_M$ and
the one-parameter groups of rotations in Euclidean space-time planes
in ${\cal H}_E$ become local symmetric semigroups
\cite{Klein:1981}\cite{Klein:1983}\cite{Frohlich:1983kp} of Lorentz
transformations with imaginary rapidity on ${\cal H}_M$.  Both of these
semigroups have self-adjoint generators on ${\cal H}_M$ for
quasi-Schwinger functions where the continuity properties of the
semigroups can be established.  In what follows I assume that the
model quasi-Schwinger functions satisfy these conditions.

Explicit expressions for the Poincar\'e generators are constructed by
computing the generators of the one-parameter subgroups of $U_E(O,a)$
on suitable domains and multiplying by the appropriate factors of $1$
for the generators that commute with $\Theta$ and $-i$ by the
generators that anti-commute with $\Theta$.  The parameters of the
Euclidean group are angles of rotation and displacements of spacetime
coordinates.  The resulting Poincar\'e generators have the following
representations:
\[
\langle \x \vert H \vert \mathbf{f} \rangle :=
-{d \over da} \langle \x-(a,0,0,0)  \vert \mathbf{f} \rangle\vert_{a=0}e =
\]
\beq
\{0 ,{\partial \over \partial
\x^0_{11}} f_1 (\x_{11}), 
\left ( {\partial \over \partial
\x^0_{21}} + {\partial \over \partial
\x^0_{22}} \right )  f_2 (\x_{21},\x_{22}), \cdots  \}
\label{d.9}
\eeq

\[
\langle \x \vert \mathbf{P}  \vert \mathbf{f} \rangle = 
i \pmb{\nabla}_{\mathbf{a}} \langle \x-(0,\mathbf{a})  \vert \mathbf{f} \rangle\vert_{\vert_{\mathbf{a}=0}}=
\]
\beq
\{0 ,  - i {\partial \over
\partial \pmb{\x}_{11}} f_1 (\x_{11}), 
 - i \left ( {\partial \over
\partial \pmb{\x}_{21}}  + {\partial \over
\partial \pmb{\x}_{22}} \right ) f_2 (\x_{21},\x_{22}), \cdots  \}
\label{d.10}
\eeq

\[
\langle \x \vert \mathbf{J}  \vert \mathbf{f} \rangle :=
\{0 , - i \pmb{\x}_{11} \times
{\partial \over \partial \pmb{\x}_{11}} f_1 (\x_{11}), 
%\label{d.11}
\]
\beq
  - i\left (  \vec{\x}_{21} \times
{\partial \over \partial \pmb{\x}_{21}}  + \pmb{\x}_{22} \times
{\partial \over \partial \pmb{\x}_{22}} 
\right  ) f_2 (\x_{21},\x_{22}), \cdots  \}
\label{d.12}
\eeq

\[
\langle \x \vert \mathbf{K}  \vert \mathbf{f} \rangle :=
\{0 , \left ( \pmb{\x}_{11} 
{\partial \over \partial \x_{11}^0}- \x_{11}^0  {\partial \over
\partial \pmb{\x}_{11}} \right )  f_1 (\x_{11}),
%\label{d.13}
\]
\beq
\left ( \pmb{\x}_{21} 
{\partial \over \partial \x_{21}^0}- \x_{21}^0  {\partial \over
\partial \pmb{\x}_{21}}+ \pmb{\x}_{22} 
{\partial \over \partial \x_{22}^0}- \x_{22}^0  {\partial \over
\partial \pmb{\x}_{22}} \right )  f_2 (\x_{21},\x_{22}), \cdots  \}.
\label{d.14}
\eeq
Here $H$ is the Hamiltonian, $\mathbf{P}$ is the linear momentum
operator, $\mathbf{J}$ is the angular momentum operator and
$\mathbf{K}$ is the rotationless boost generator.  These expressions
can be used to get the following representation for the square of the
mass operator
\beq
M^2 = H^2 - P^2 
\label{d.15}
\eeq
\beq
\langle \x \vert M^2 \vert \mathbf{f} \rangle :=
\{0 , -\nabla_{11}^2 f_1 (\x_{11}), 
-(\nabla_{21} + \nabla_{22})^2
f_2 (\x_{21},\x_{22}), \cdots  \}
\label{d.16}
\eeq
where $\nabla^2$ is the 4-dimensional Euclidean Laplacian.

The easiest dynamical quantities to calculate in this representation are matrix
elements of $e^{-\beta H}$, which are the result of translating all of
the initial Euclidean time variables to the right by the same positive
constant $\beta$:
\beq
\langle \x \vert e^{-\beta H} \vert \mathbf{f} \rangle :=
\{f_0 ,f_1 (\x_{21}- \beta\hat{\pmb{\tau}}), 
f_2 (\x_{21}-\beta\hat{\pmb{\tau}} ,\x_{22}-\beta\hat{\pmb{\tau}} ), \cdots  \}.
\label{d.17}
\eeq

The representation of the Poincar\'e Lie algebra (\ref{d.9}-\ref{d.14})
on the model Hilbert space ${\cal H}_M$ defines the Euclidean 
representation of a relativistic quantum theory.

In addition to giving a Euclidean representation of the 
Hilbert space scalar product, reflection positivity and
repeated applications of the Schwartz inequality \cite{glimm:1981} 
give the inequalities
\beq
\Vert e^{-\beta H } \vert \psi \rangle\Vert_M \leq
\Vert e^{-2^n \beta H } \vert \psi \rangle\Vert_M^{1/2^n}
\Vert \vert \psi \rangle\Vert_M^{1- 1/2^n} \leq
\Vert \vert \psi \rangle\Vert_E^{1/2^n}
\Vert \vert \psi \rangle\Vert_M^{1- 1/2^n} 
\label{d.18}
\eeq
for and $n$ and $\beta >0$.  Taking the limit as $n\to \infty$ 
\beq
\Vert e^{-\beta H } \vert \psi \rangle\Vert_M
\leq \Vert \vert \psi \rangle\Vert_M 
\eeq
shows that $e^{-\beta H}$ is a contractive semigroup on 
${\cal H}_M$,  which means that $H$ has a spectrum bounded from
below. 

For most purposes $e^{-\beta H}$ can be used to replace the
Hamiltonian, where $\beta$ is treated as an adjustable scale setting
parameter.  $e^{-\beta H}$ has the same eigenstates as $H$, the
eigenvalues are simply related, and $-e^{-\beta H}$ can be used in
place of the Hamiltonian to formulate a scattering theory. The
operator $e^{-\beta H}$ has the additional advantage that it is
bounded.

\section{Reflection positivity and particles}

The Minkowski designation of the scalar product $(,)_M$, where
everything is Euclidean, is surprising at first glance since
no analytic continuations have been performed.

The best way to understand this is to consider the example of the
two-point Schwinger function for a particle of mass $m$.  If this 
is the only Schwinger function in the model then the ``Minkowski''
inner product is \cite{glimm:1981}
\[
(f , f)_M = 
\int f^*(\x) S_{1:1}(\theta \x;\y) f(\y) d^4\x d^4\y 
\]
\beq
= {1 \over (2 \pi)^4} \int  
f^* (\x) { e^{i \p \cdot (\theta 
\x-\y)} 
\over \p^2 + m^2}  f(\y) d^4\x d^4\y d^4\p  
\label{e.1}
\eeq
\[
={1 \over (2 \pi)^4} 
\int d^4\x d^4\y d^4\p f (\x) { e^{-i \p_0 \cdot (\x_0+\y_0)
+ i \mathbf{\p} \cdot ( \mathbf{\x}-\mathbf{\y})}
\over  (\p^0 + i \omega_m (\mathbf{\p}\,))(\p^0 - i \omega_m (\mathbf{\p}\,))}
f(\y) 
\]
\beq
=\int {d^3 \p  \over 2\omega_{m} (\mathbf{\p} \,) }
\vert \psi (\mathbf{\p}) \vert^2 
\geq 0
\label{e.2}
\eeq
where $\omega_m (\mathbf{\p}) = \sqrt{m^2 + \mathbf{\p}^2}$ is the energy 
of a particle of mass $m$ and momentum $\mathbf{\p}$.  The wave 
function $\psi (\mathbf{p})$, expressed as a 
square integrable function of momentum, is related to the Euclidean 
positive-time support function $f(\x)$ by
\beq
\psi (\mathbf{p}) := 
{1 \over (2\pi)^{3/2} } \int_0^\infty d\x_0 \int d\pmb{\x} f(\x)
e^{-\omega_{m} (\mathbf{p})\y_0 -i \mathbf{p} \cdot \pmb{\x}}. 
\label{e.3}
\eeq
In this example the calculation of the norm involves only Euclidean 
quantities,  while the final result has the familiar Minkowski
form involving the Lorentz invariant measure for a free particle
of mass $m$.   This calculation also demonstrates 
that the Schwinger function for
a scalar field of mass $m$ is reflection positive.

Another simple calculation shows that 
the representation (\ref{d.9}) of the  Hamiltonian
has the correct interpretation:
\beq
(f ,H f)_M = 
\int f^*(x) S_{1:1}(\theta x;y) {\partial \over \partial \y^0} f(y) d^4x d^4y 
\eeq
\beq
=\int {d^3 \p  \over 2\omega_{m} (\mathbf{p} \,) }
\omega_{m} (\mathbf{p} \,)  \vert \Psi(\mathbf{p},m) \vert^2 
\label{e.4}
\eeq
where integration by parts was used to show that the Hamiltonian
has the interpretation of the energy of a free particle of mass $m$.
The spectral condition is clearly satisfied.

Direct calculations also shows that for this Schwinger function, any
$f$ is an eigenstate of $M^2$ with eigenvalue $m^2$,
\beq
(g ,M^2 f)_M = 
\int g^*(x) S_{1:1}(\theta x;y) {\nabla^2_\y} f(y) d^4x d^4y 
\eeq
\beq
=m^2 (g ,M^2 f)_M .
\label{e.5}
\eeq

Reflection positive two-point Schwinger functions for 
particles with any $m>0$ and spin are given in the appendix.
%This same calculation can be repeated for a free spin $1/2$ 
%Schwinger function.  In this case the ``Minkowski'' scalar product is 
%\[
%(f , f)_M = 
%\int f^*(\x) \gamma^0 S_{1:1}(\theta \x;\y) f(\y) d^4\x d^4\y 
%\]
%\[
%= {1 \over (2 \pi)^4} \int  
%f^* (\x) { e^{i \p \cdot (\theta 
%\x-\y)} \gamma^0(m- p\cdot \gamma_E )
%\over \p^2 + m^2}  f(\y) d^4\x d^4\y d^4\p  
%\]
%\beq
%={1 \over (2 \pi)^3} \int d\mathbf{p} 
%g^{\dagger}(\mathbf{p}) \Lambda^+(\mathbf{p}) g(\mathbf{p}) \geq 0 
%\label{e.5}
%\eeq
%where 
%\beq
%\Lambda^+(\mathbf{p})= 
%{\omega_m (\mathbf{p}) + \gamma^0 \pmb{\gamma} \cdot 
%\mathbf{p} - m\gamma^0 \over 2 \omega_m(\mathbf{p})} 
%\label{e.6}
%\eeq
%is the positive energy Dirac projector. Again we get the familiar 
%spin-$1/2$ Minkowski result.  Here a factor of $\gamma^0$ becomes part of
%the time reflection operator.  
%This is because the ass
%\Vert e^{-2^n \beta H } \vert \psi \rangle\Vert_M^{1/2^n}
%\Vert \vert \psi \rangle\Vert_M^{1- 1/2^n} \leq

This one-particle example illustrates how Lorentz invariant inner
products emerge from calculations involving {\it purely Euclidean
  quantities}. This example also shows how reflection positivity gives
both the Minkowski scalar product as well as the spectral condition on
the Hamiltonian.

% Examples of reflection positive quasi-Schwinger
%functions that lead to any positive mass representation
%of the Poincar\'e group are given in the appendix. 

\section{Reflection positivity}

One difficulty with constructing reflection positive multipoint
functions is that there are non-trivial functions associated with zero
norm vectors.  This can be seen in the integral, (\ref{e.3}), which
can be expressed as a partial Fourier transform over the spatial
variables and Laplace transform over the Euclidean time variable
\beq
\psi (\mathbf{p}) := 
\int_0^\infty d\y_0 \int d^3\y \tilde{f}(\y^0,\mathbf{p})
e^{-\omega_{m} (\mathbf{p})\y_0} .
\label{f.1}
\eeq
If $\tilde{f}(\y^0,\mathbf{p})$ is orthogonal to 
$e^{-\omega_{m} (\mathbf{p})\y_0}$ as a function of $\y_0$ for 
each value of $\mathbf{p}$, then this vanishes, leading to 
a non-trivial positive Euclidean-time-support function, 
$\tilde{f}(\y^0,\mathbf{p})$,
corresponding to a $0$ norm vector.

If a free quasi-Schwinger functions is perturbed by adding
a small connected perturbation that is only required to be Euclidean
invariant, then a function representing a zero norm vector with
respect to the product of the free quasi-Schwinger functions might
have a non-zero contribution due to the perturbation.  This
contribution can always be made negative using the freedom to adjust
the sign of the perturbation.  The means that reflection positivity
is not stable with respect to small Euclidean invariant perturbations.
The practical consequence of this observation is that the solution of 
the Euclidean Bethe-Salpeter equation,
\beq
S_{2:2} = S_{1:1} S_{1:1} + S_{1:1} S_{1:1} K S_{2:2},
\label{f.2}
\eeq
with a model Euclidean invariant kernel, is not automatically
reflection positivity, even if the kernel is small.  I am not aware of
any general results about what kind of restrictions are needed on
Euclidean invariant Bethe-Salpeter kernels $K$ for $S_{2:2}$ to be
reflection positive.

%The direct approach is also difficult because the quasi-Schwinger 
%functions $S_{1:1}$ do not preserve the support condition.

An alternative strategy is to explore the direct construction of
connected reflection positive four-point functions.  The resulting
four-point quasi-Schwinger functions could be used directly or used to
deduce sufficient properties of Bethe-Salpeter kernels by expressing
them in terms of known reflection-positive quasi-Schwinger functions:
\beq
K = S^{-1}_{1:1} S^{-1}_{1:1} - S^{-1}_{2:2} . 
\label{f.3}
\eeq 
In this expression the inverses are interpreted as inverses of 
operators on the ``Euclidean'' Hilbert space. 

Since cluster properties imply that the four-point function can be 
expressed as a sum of products of two-point functions and a 
connected four-point function,
\beq
S_{2:2} = \sum S_{1:1} S_{1:1} + S^c_{2:2}, 
\label{f.4}
\eeq  
in order to show $S_{2:2}$ is reflection positive it is sufficient to
show that the connected four-point function, $S^c_{2:2}$, is
reflection positive.

In one dimension there is a result due to Widder
\cite{Widder:1941}\cite{Widder:1931}\cite{Widder:1934} from classical
analysis that gives the general structure of reflection positive
two-point functions.  Widder's theorem points out that any kernel
$k(s)$ satisfying the reflection positivity condition
\beq
\int f(\theta s) k(s-t) f(t) ds dt =
\int f(s) k(-s-t) f(t) ds dt
\geq 0
\label{f.5}
\eeq
can be expressed in the exponential form 
\beq
k(-\tau'-\tau) = \int e^{- \lambda (\tau'+\tau) } \rho (\lambda ) d\lambda
\label{f.6}
\eeq
for some positive density $\rho(\lambda)$.  Since in this example, 
$\tau',\tau>0$, we can write the kernel as
\beq
k(-\tau'-\tau) =  \int_0^{\infty} {\lambda \over \pi} 
\rho (\lambda ) d\lambda \int_{-\infty}^\infty ds  
{e^{-is ( \tau'+\tau) }\over s^2 + \lambda^2 }.  
\label{f.7}
\eeq
This has the form of a one-dimensional version of the
K\"all\'en-Lehmann representation of a two-point Schwinger function.

The Widder result suggests that connected four-point 
quasi-Schwinger functions with the structure 
\[
\int 
d^4\p_1 d^4\p_2 d^4\p_3 dm_a dm_c dm_b
e^{-i \p_1 \cdot (\y' -\x')}
e^{-i \p_2 \cdot (\x' -\x)}
e^{-i \p_3 \cdot (\x -\y)}
\times 
\]
\beq
{ v(m_a, \p_1,m_c, \p_2, m_b, \p_3) \over 
(\p_1^2 + m_a^2 )(\p_2^2 + m_c^2) (\p_3^2 + m_b^2) 
}.
\label{f.8}
\eeq
would be reflection positive for suitable Euclidean invariant kernels,
$v(m_a, \p_1,m_c, \p_2, m_b, \p_3)$.  This structure does not provide
a general representation for a Euclidean invariant reflection positive
four point functions, as one gets in Widder's theorem.  On the other
hand Widder's theorem suggests that reflection positivity and
Euclidean covariance strongly constrain the class of reflection
positive four-point functions.

The ``Minkowski'' scalar product over Euclidean spacetime variables
for quasi-Schwinger functions of the form (\ref{f.8}) is
\[
\int 
d^4 \x d^4\y d^4\x' d^4\y d^4 \p_1 d^4\p_2 d^4\p_3 dm_a dm_c dm_b 
f_e^*(-\y^{0\prime}, \mathbf{y}')
g_e^*(-\x^{0\prime}, \mathbf{x}')
e^{-i \p_1 \cdot (\y' -\x')}
\times 
\]
\beq
e^{-i \p_2 \cdot (\x' -\x)}
e^{-i \p_3 \cdot (\x -\y)}
{ v(m_a, \p_1,m_c, \p_2, m_b, \p_3) \over 
(\p_1^2 + m_a^2 )(\p_2^2 + m_2^2) (\p_3^2 + m_b^2) 
}
f_e(\y^{0},\mathbf{y})
g_e(\x^{0} \mathbf{x}).
\label{f.9}
\eeq
I assume that the functions satisfy the support condition, 
$f_e (\y^{0}, \mathbf{y}) $ and $g_e (\x^{0},\mathbf{x}) $ 
can be non-zero only for $ 0 < \x^{0} < \y^{0}$. 

The most straightforward assumption is to choose $v(m_a, \p_1,m_2, \p_2,
m_b, \p_3)$ analytic in the upper-half $p_i^0$ planes.  In this case
the $\p^0_i$ integrals can be performed as in the 1 dimensional case,
where the convergence in the upper-half plane is ensured by the Euclidean
time-support constraints provided the kernel is polynomially bounded,
and the Minkowski boundary value is a tempered distribution.
The integral over the vector variables leads to Fourier transforms of
the vector variables and results in the expression
\[
(2 \pi)^9 \int 
d\x^0 d\y^0 d\x^{0\prime} d\y^{0\prime}  d\mathbf{\p}_1 
d\mathbf{\p}_2 d\mathbf{\p}_3 dm_a dm_c dm_b 
\tilde{f}_e^*(\y^{0\prime },-\mathbf{p}_1 )
\tilde{g}_e^*(\x^{0\prime}, \mathbf{p}_1-\mathbf{p}_2) \times
\]
\[
e^{- \omega_{m_a} (\mathbf{p}_1) (\y^{0\prime} - \x^{0\prime}) }
e^{- \omega_{m_c} (\mathbf{p}_2) (\x^{0\prime} + \x^{0} ) }
e^{- \omega_{m_b} (\mathbf{p}_3) (\y^{0} - \x^{0}) }
\times 
\]
\beq
{ v (m_1,(i \omega_{m_a}(\mathbf{p}_1), \mathbf{p}_1),m_c, 
(i \omega_{m_c}(\mathbf{p}_2), \mathbf{p}_2),
(i \omega_{m_b}(\mathbf{p}_3), \mathbf{p}_3),m_b)
\over 
2 \omega_{m_a} (\mathbf{p}_1)
2 \omega_{m_c} (\mathbf{p}_2)
2 \omega_{m_b} (\mathbf{p}_3) 
}
\tilde{f}_e(\y^0, \mathbf{p}_3-\mathbf{p}_2)
\tilde{g}_e(\x^0, -\mathbf{p}_3).
\label{f.10}
\eeq
The  kernel, which was initially a Euclidean invariant function of Euclidean
scalar product becomes a Lorentz invariant function of Lorentz invariant inner 
products when all of the $\p^0\to i \omega_{m_i}(\mathbf{p}_i)$.
This connected quasi-Schwinger function 
will be reflection positive if
\beq
v (m_a,(i \omega_{m_a}(\mathbf{p}_1), \mathbf{p}_1),m_c, 
(i \omega_{m_c}(\mathbf{p}_2), \mathbf{p}_2),
(i \omega_{m_b}(\mathbf{p}_3), \mathbf{p}_3),m_b)
\label{f.12}
\eeq
is a positive symmetric matrix in $\mathbf{p}_1, m_a$ and
$\mathbf{p}_3, m_b$ for all values of $\mathbf{p}_2$ and $m_c$.
This is not a difficult condition to realize.

This simple construction demonstrates the existence of a large class
of reflection positive connected four-point quasi-Schwinger functions.
Exchange symmetry puts additional constraints on the quasi-Schwinger
functions.  The elements of this construction provide a framework 
for investigating the reflection positivity of a larger class
of quasi-Schwinger functions.  In what follows 
this class of reflection-positive quasi-Schwinger
functions will be used to test the existence of scattering 
wave operators in this Euclidean formulation of relativistic quantum mechanics.
 
\section{Scattering theory}

In theories like quantum field theory, where the dynamics is encoded
in the kernel of the quantum mechanical scalar product, there is no
natural asymptotic dynamics on the physical Hilbert space to formulate
scattering asymptotic conditions.

One way to formulate scattering in this situation
\cite{baumgartl:1983}\cite{simon} is to use a method that is sometimes used
in non-relativistic scattering theory, which involves introducing a
secondary Hilbert space of scattering asymptotes \cite{Coester:1965zz}
\cite{Coester:1982vt}, where the asymptotic states are treated as free
particles on a second Hilbert space and wave functions defining the
internal structure of the asymptotic separated particles are treated
as mappings into the dynamical Hilbert space.  In the field theory
case \cite{Haag:1958vt}\cite{Ruelle:1962} this approach has the 
advantage that scattering observables can
be computed using strong limits, as in non-relativistic quantum
mechanics.  The price for this is that it is necessary to solve the
subsystem ``bound state'' problems to formulate the scattering
asymptotic condition.  The benefit is that it is
possible to do scattering calculations using purely Euclidean methods.

While the two-Hilbert space formulation of scattering can be used more
generally, I consider the special case of a model with a fixed number
of particles.  In this case the dynamics is given by a single 2n-point
quasi Schwinger function
\beq
S_{n:n} (\x_1, \cdots , \x_n:\y_1, \cdots ,\y_n).
\label{i.1}
\eeq
The cluster condition ({\bf E.3}) means that for a given partition of the
coordinates into asymptotically separated clusters of a partition $a$,
the quasi-Schwinger functions can be expressed as
a product of quasi-Schwinger functions for each cluster plus a
remainder that vanishes when the clusters are asymptotically
separated:
\beq
S_{n:n} = 
\prod_{i=1}^{n_a} S_{n_{a_i}:n_{a_i}} + S^a_{n:n} . 
\label{i.2}
\eeq
In this expression $n_a$ denotes the number of non-empty clusters in the
partition $a$, $n_{a_i}$ are the number of elements in the $i^{th}$
cluster of $a$, where $\sum_{i=1}^{n_a} n_{a_i} = n$.
$S^a_{n:n}$ is the remainder that vanishes as a
distribution when the coordinates in the different clusters are
asymptotically separated. 

There are scattering channels $\alpha$ associated with the partition
$a$ if the quasi-Schwinger functions, $S_{n_{a_i}:n_{a_i}}$, for each
cluster of $a$ has point spectrum eigenstates of the square of the
cluster mass operator (\ref{d.15}).

These asymptotic one-particle eigenstates are represented by functions of
the Euclidean variables, 
\beq
\langle \x_1, \cdots , \x_{n_{a_i}} \vert m_i \rangle.
\label{i.3}
\eeq
associated with the $i^{th}$ cluster of $a$ satisfying the mass
eigenvalue problem 
\beq
\langle \x_1, \cdots , \x_{n_{a_i}} \vert M^2 \vert m_i \rangle = 
m_{i}^2
\langle \x_1, \cdots , \x_{n_{a_i}} \vert m_i \rangle
\label{i.4}
\eeq
where $M^2$ is the square of the mass operator defined by (\ref{d.15}). 

These mass eigenstates can be decomposed into simultaneous eigenstates of 
this mass, spin, linear momentum, and $3$-component of the canonical 
spin using the translation and rotation subgroups of the Poincar\'e group: 
\beq
\langle \x_1, \cdots , \x_{n_{a_i}} \vert m_i,\mathbf{p}_i \rangle.
= 
\int {d \mathbf{\a} \over (2 \pi)^{3/2}}
e^{-i \mathbf{p} \cdot \mathbf{a}}
\langle \x_1+\pmb{\a}, \cdots , \x_{n_{a_i}}+\pmb{\a}) \vert m_i \rangle 
\label{i.5}
\eeq
and
\[
\langle \x_1 , \cdots , \x_{n_{a_i}}
\vert (m_i,j_i),\mathbf{p}_i,\mu_i \rangle=
\]
\beq 
\int_{SU(2)}  dR 
\sum_{\nu=-j_i}^{j_i}
\langle R^{-1}\x_1, \cdots , R^{-1}\x_{n_{a_i}} \vert m_i,
R^{-1}\mathbf{p}_i \rangle
D^{j_i*}_{\mu_i,\nu}(R) . 
\label{i.6}
\eeq
where the integral in (\ref{i.6}) is over the $SU(2)$ Haar measure.  Note that 
if the original mass eigenstates have compact positive Euclidean time support, 
this condition is unchanged by spatial translations or rotations.
The resulting irreducible states $\vert (m_i,j_i),\mathbf{p}_i,\mu_i \rangle$
can be renormalized as desired.

The $SU(2)$ representation property follows by direct computation 
\[
\langle \x_1 , \cdots , \x_{n_{a_i}} \vert U(R) 
\vert (m_i,j_i),\mathbf{p}_i,\mu_i \rangle=
\]
\[
\int_{SU(2)}  dR' 
\sum_{\nu=-j_i}^{j_i}
\langle R^{\prime -1}R^{-1}\x_1, \cdots ,  R^{\prime -1} R^{-1} \x_{n_{a_i}} \vert m_i,R^{\prime -1}\mathbf{p}_i \rangle
D^{j_i*}_{\mu_i,\nu}(R')  = 
\]
\[
\int_{SU(2)}  dR'' 
\sum_{\nu,\mu_i'=-j_i}^{j_i}
\langle R^{\prime\prime  -1}\x_1, \cdots , R^{\prime\prime -1}\x_{n_{a_i}} 
\vert m_i,R^{\prime\prime  -1}R \mathbf{p}_i \rangle
D^{j_i*}_{\mu_i',\nu}( R'')D^{j_i}_{\mu_i',\mu_i}( R)  = 
\]
\beq
\sum_{\mu_i'=-j_i}^{j_i}
\langle \x_1 , \cdots , \x_{n_{a_i}}  
\vert (m_i,j_i),R \mathbf{p}_i,\mu_i' \rangle
D^{j_i}_{\mu',\mu_i}( R).
\label{i.6a}
\eeq

By construction, the states 
\beq
\langle \x_1 , \cdots , \x_{n_{a_i}}
\vert (m_i,j_i),\mathbf{p}_i,\mu_i \rangle 
\label{i.7}
\eeq
transform irreducibly with respect to the Poincar\'e group:    
\[
\langle \x_1 , \cdots , \x_{n_{a_i}}
\vert
U_{\alpha_i}(\Lambda ,a) 
\vert (m_i,j_i),\mathbf{p}_i,\mu_i \rangle =
\]
\beq
\sum_{\nu_i} 
\langle \x_1 , \cdots , \x_{n_{a_i}}
\vert (m_i,j_i),\pmb{\Lambda}{p}_i,\nu_i \rangle 
D^{j_i }_{\nu_i\mu_i} (R_w (\Lambda, p_i))
\sqrt{{\omega_{m_i}(\pmb{\Lambda} p_i)
\over \omega_{m_i}(\mathbf{p}_i)}} 
e^{i \Lambda p_i \cdot a}
\label{i.8}
\eeq
where the factor $\sqrt{{\omega_{m_i}(\pmb{\Lambda} p_i)
\over \omega_{m_i}(\mathbf{p}_i)}}$ arises because 
I have chosen to normalize these states with a
$\delta (\mathbf{p}- \mathbf{p}')$ normalization.  This demonstrates 
how to perform finite Poincar\'e transformations on Poincar\'e irreducible
eigenstates associated with masses in the point spectrum of the 
mass operator. 

These one-particle states can be used to construct a mapping from the
mass $m_i$ spin $j_i$ irreducible subspace of the Poincar\'e group to
the Hilbert space associated with cluster quasi-Schwinger function
$S_{n_{a_i}:n_{a_i}}$ 
for the variables in the $i^{th}$ cluster of $a$.  I define the mapping
$\Phi_{\alpha_i}$ from the irreducible representation 
space, ${\cal H}_{m_ij_i}$, of square
integrable functions of $\psi_{i} (\mathbf{p}_i, \mu_i)$ to functions
of the Euclidean space-time variables $(\x_1, \cdots ,\x_{n_{a_i}})$ by
\beq
\langle \x_1, \cdots \x_{n_{a_i}} \vert \Phi_{\alpha_i} \vert
\psi_i \rangle :=
\int d\mathbf{p}_i 
\sum_{\mu_i=-j_i}^{j_i}
\langle \x_1 , \cdots , \x_{n_{a_i}}
\vert (m_i,j_i),\mathbf{p}_i,\mu_i \rangle
\psi_i (\mathbf{p}_i, \mu_i) .
\label{i.9}
\eeq
Taking tensor products defines the channel 
$\alpha$ injection operator $\Phi_\alpha$: 
\beq
\Phi_{\alpha} := \otimes_{i=1}^{n_a} \Phi_{\alpha_i}
\label{i.11}
\eeq
from the tensor product of the
$n_a$ irreducible representation spaces of the Poincar\'e group
associated with the scattering channel $\alpha$:
\beq
{\cal H}_{\alpha} = \otimes_{i=1}^{n_a} {\cal H}_{m_{\alpha_i},j_{\alpha_i}}
\label{i.10}
\eeq
to the Hilbert space ${\cal H}_M$.  
In (\ref{i.10}) $\alpha$ denotes the scattering
channel labeled by the subsystem bound states $\alpha_1 \cdots 
\alpha_{n_i}$ corresponding to each cluster of the partition $a$.
In general a given partition $a$ can be associated with any number of
channels.  The multi-particle interpretation of these states is only
realized in asymptotic regions where the residual part, $S^a_{n:n}$,
of $S_{n:n}$ in (\ref{i.2}) does not contribute to the inner product.

In order to be in the Hilbert space ${\cal H}_M$ the function needs to 
have positive relative time support.  This is true for the 
individual cluster mass eigenstates.  The supports of each of 
these one-particle solutions can be shifted to the right using Euclidean time 
translations: 
\[
\langle \x_1 , \cdots , \x_{n_{a_i}} \vert  
e^{-\beta H} \vert (m_i,j_i),\mathbf{p}_i,\mu_i \rangle =
\langle \x_1 , \cdots , \x_{n_{a_i}} 
\vert (m_i,j_i),\mathbf{p}_i,\mu_i \rangle 
e^{-\beta \sqrt{\mathbf{p}_i^2 + m_i^2}}
\]
\beq
\langle \x_1-\beta\hat{\pmb{\tau}} , \cdots , \x_{n_{a_i}} -
\beta \hat{\pmb{\tau}}
\vert (m_i,j_i),\mathbf{p}_i,\mu_i \rangle .
\label{i.12}
\eeq
Since the Hamiltonian commutes with the linear momentum, all components of the 
spin and the mass, the resulting irreducible eigenstates are unchanged. 
Comparing these expressions leads to the identification 
\[
\langle \x_1 , \cdots , \x_{n_{a_i}} 
\vert (m_i,j_i),\mathbf{p}_i,\mu_i \rangle
=
\]
\beq
\langle \x_1-\beta \hat{\pmb{\tau}} , \cdots , \x_{n_{a_i}} - 
\beta \hat{\pmb{\tau}}
\vert (m_i,j_i),\mathbf{p}_i,\mu_i \rangle 
e^{\sqrt{\mathbf{p_i}^2 + m_i^2}\beta} .
\label{i.12b} 
\eeq
In this case the equality means that both functions belong to 
the same
equivalence class.  The support of the second function representing
this class is shifted to the right in the Euclidean time variables by
a factor $\beta$ relative to the function on the left.  Using
different Euclidean time translations on each cluster, the support of
each cluster bound state can be translated so they have positive
relative time support with respect to some ordering of the Euclidean
times.

This shows that Euclidean time translations behave something like
gauge transformations when they act on irreducible eigenstates in the
sense that they transform functions representing irreducible basis
vectors to different functions in the same equivalence class.

The asymptotic Hilbert space ${\cal H}_{\cal A}$ is defined to be the
direct sum of all of the channel subspaces (including one-cluster
channels if applicable)
\beq
{\cal H}_{\cal A} = \oplus_{\alpha} {\cal H}_{\alpha}. 
\label{i.14}
\eeq
The injection operator 
\beq
\Phi: {\cal H}_{\cal A} \to {\cal H}_M
\label{i.15}
\eeq
is a mapping from the asymptotic Hilbert space to the 
``Minkowski'' Hilbert space defined by the sum of the channel 
injection operators 
\beq
\Phi := \sum_{\alpha} \Phi_{\alpha}.
\label{i.16}
\eeq
There is a natural unitary representation of the Poincar\'e group
on ${\cal H}_{\cal A}$ defined by 
\beq
U_{\cal A} (\Lambda ,a) =
\oplus_{\alpha \in {\cal A}} \left (\otimes_{i=1}^{n_a} 
U_{\alpha_i} (\Lambda ,a) \right )
\label{i.17}
\eeq 
where the $U_{\alpha_i} (\Lambda ,a)$ are given by (\ref{i.8}).

Scattering wave operators are mappings from the asymptotic 
Hilbert space ${\cal H}_{\cal{ A}}$ to the ``Minkowski'' Hilbert space 
defined by the strong limits 
\beq
\Omega_{\pm} := \lim_{t \to \pm \infty} e^{i Ht} \Phi e^{-i H_{\cal_{A}} t} 
\label{i.18}
\eeq
where $H_{\cal A}$ is the Hamiltonian associated with (\ref{i.17}).
The existence of these limits depends on properties of the
quasi-Schwinger functions.

Conventional methods \cite{Cook:1957} can be used to derive
sufficient conditions for this limit to exist.  The first step is to 
write the limit as the integral of a derivative
\[
\lim_{t \to  \pm \infty} e^{i Ht} \Phi e^{-i H_{\cal A} t} =
\Phi + 
\int_0^{\pm \infty}{d \over dt}\left (  
e^{i Ht} \Phi e^{-i H_{\cal A} t} dt \right )   =
\]
\beq 
\Phi + i \int_0^{\pm \infty}  
e^{i Ht} (H \Phi - \Phi H_{\cal A}) e^{-i H_{\cal A} t} dt  . 
\label{i.19}
\eeq
A sufficient condition for the existence of the wave operator is the
convergence of the integral (\ref{i.19}).  A sufficient 
condition for the convergence of this integral is the Cook condition
\cite{Cook:1957}, which
exploits the unitarity of the time evolution operator 
\beq
\int_a^\infty \Vert (H \Phi - \Phi H_{\cal A}) e^{\mp i H_{\cal A} t} 
\vert \psi \rangle \Vert dt < \infty
\label{i.20}
\eeq
and provides a bound on the norm of the integral in (\ref{i.19}). 

The quantities appearing in (\ref{i.20}) depend on the quasi-Schwinger 
functions.  It is sufficient to consider this integral one channel at a time.  
Then the integrand becomes
\[
\Vert (H \Phi_{\alpha} - H_{\alpha} \Phi_{\alpha} ) e^{\mp i H_{\alpha} t} 
\vert \psi \rangle \Vert = 
\]
\beq
( \psi_{\alpha}, e^{\pm i H_{\alpha} t} 
(\Phi^{\dagger}H - \Phi^{\dagger}H_{\alpha})
(\Theta \prod S_{n_{a_i}:n_{a_i}}  + \Theta S_{n:n}^a) 
(H \Phi_{\alpha} - \Phi_{\alpha} H_{\alpha})
\psi_{\alpha} )^{1/2} 
\label{i.21}
\eeq
where I have used (\ref{i.2}) to express the 
quasi-Schwinger function  
as the sum of a term
that involves a product of the subsystem quasi-Schwinger functions 
associated with a partition $a$ into asymptotically separated subsystems
plus a remainder term $S_{n:n}^a$ that vanishes asymptotically.    
 
I consider the example of the four-point quasi-Schwinger function
\beq
\int_0^\infty  \left  (\psi,e^{\mp i H_{0} t}
( \Phi^{\dagger}H - H_0\Phi^{\dagger} 
) e^{\pm iH_0t} \Theta S_{2:2} 
(H \Phi - \Phi H_0) e^{\mp iH_0t} \psi \right )^{1/2} 
dt  < 
\infty .
\label{i.22}
\eeq
In (\ref{i.22}) $H \Phi - \Phi H_0$ replaces the non-relativistic
potential.  $U_0(t)$ is the time evolution operator for the 
asymptotically free eigenstates.  $H_0$ is the sum of the energies
of the particles in each asymptotically cluster 
\beq
H_0 = \sum \sqrt{m_{a_i}^2 + \mathbf{p}_{a_i}}.
\label{i.23}
\eeq
I will argue that for sufficiently well-behaved four-point
quasi-Schwinger functions the integrand in (\ref{i.22}) 
falls off like $t^{-3/2}$, which is
sufficient for the Cook condition (\ref{i.20}) to be satisfied.

To see this first note that cluster properties  imply that the 
quasi-Schwinger function is the sum of a 
product of two-point Schwinger functions and a connected term.  I consider a
connected term that has the structure discussed in (\ref{f.8})
\beq
\int d^4{\q}_1 d^4{\q}_2 d^4 {\q}_3 dm_a dm_c dm_b 
{e^{i \q_1 \cdot (\x_1 -\x_2) + i \q_2 \cdot (\x_2-\x_3) 
+ i \q_3 \cdot (\x_3-\x_4 ) } \over
(\q_1^2 + m_a^2 ) (\q_2^2 + m_c^2)(\q_3^2 + m_b^2)}
v(\q_1,m_a,\q_2,m_c,\q_3,m_b) 
\label{i.24}
\eeq
where $m_c > m_a,m_b$ and the spectrum of $m_c$ should be continuous
for scattering.  I have already established that 
this connected quasi Schwinger function is reflection positive for suitable 
$v(\q_1,m_a,\q_2,m_c,\q_3,m_b)$. 

First I consider the structure of the asymptotic states.  If the
two-point functions have only a single mass eigenstate, as in
(\ref{e.1}), then the one-body problem becomes trivial.  If the
two-point function has several discrete mass eigenvalues then
\beq
\Pi_j = \prod_{i\not=j} {\nabla^2 - m_i^2 \over m_j^2 - m_i^2}.
\label{i.24b}
\eeq
projects the initial or final state on the asymptotic $j$-th mass
subspace.  If the two-point Lehmann weight also has continuum
eigenstates then it is necessary to project on the appropriate one-particle
subspace.  In what follows I consider that simplest case where mass
operator associated with the two-point function has a single discrete
eigenvalue.  This means that the two point functions have the from
(\ref{e.1}).

In this case it is enough to consider 
asymptotic states $\Phi \vert e^{\mp i H_{0} t}
\vert \psi
\rangle $ of the 
form 
\beq
\langle \x_1, \x_2 \vert \Phi \vert e^{\mp i H_{0} t} \psi \rangle = 
\prod_{i=1}^2 f_i(\x^0_i) \int {d\mathbf{p}_i \over 
(2 \pi)^{3/2}} \psi_i(\mathbf{p}_i) e^{i \mathbf{p}_i \cdot \mathbf{x}_i \mp
i \omega_{m_i} (\mathbf{p}_i)t} 
\label{i.25}
\eeq
where the $f_i(\x^0_i)$ are sharply peaked with compact support 
and integrate to 1.

With this choice of wave packets
\[
\langle \x_1,\x_2 \vert 
(H \Phi - \Phi H_0)e^{\mp iH_0t}\vert  \psi \rangle =
\]
\[
(2 \pi)^3 \left ({\partial \over \partial \x^0_1} +
{\partial \over \partial \x^0_2} - \omega_{m_1} (\mathbf{p}_1) -
\omega_{m_2}(\mathbf{p}_2) \right )   
f_1(\x^0_1)f_2(\x^0_2) \times 
\]
\beq
\int {d\mathbf{p}_1 d\mathbf{p}_2 \over 
(2 \pi)^3} \psi_1(\mathbf{p}_1)\psi_2(\mathbf{p}_2) 
e^{i \mathbf{p}_1 \cdot \mathbf{x}_1 \mp i \omega_{m_1} (\mathbf{p}_1)t 
+i \mathbf{p}_2 \cdot \mathbf{x}_2 \mp i \omega_{m_2} (\mathbf{p}_2)t} .
\label{i.26}
\eeq
The integrand in the Cook condition uses (\ref{i.24}) and (\ref{i.26})  
in 
\beq
(\psi , e^{\mp iH_0t} (\Phi^{\dagger}H - H_0\Phi^{\dagger}) \Theta 
(S_{1:1} S_{1:1}+
S_{2:2}^c)
( H\Phi - \Phi H_0)e^{\pm iH_0t} \psi )_E.
\label{i.27}
\eeq
In this expression the partial derivatives in (\ref{i.26}) with
respect to the Euclidean times can be integrated by parts. 
When $(-)$ the Euclidean time derivatives act 
on the product of the two-point functions 
$S_{1:1} S_{1:1}$ they  generate a energy factors (\ref{e.4})
that exactly cancel the asymptotic energy factors in (\ref{i.26}), 
making (\ref{i.27}) vanish.

This means that only the connected part of the four-point
quasi-Schwinger functions contributes to the integral (\ref{i.27}).
Thus, the connected part of the four-point function plays an analogous
role to the interaction in the non-relativistic case.

What remains has the form 
\[
(2 \pi )^{6}
\int d^4{\q}_1 d^4{\q}_2 d^4 {\q}_3 dm_a dm_c dm_b
d\x^0_1 d\x^0_2 d\x^0_3 d\x^0_4 
\left (\omega_{m_c} (\mathbf{q}_2)  - \omega_{m_1} (\mathbf{q}_1) -
\omega_{m_2}(\mathbf{q}_2-\mathbf{q}_1) \right ) \times
\]
\[   
f_1(\x^0_1)f_2(\x^0_2)
\psi^*_1(\mathbf{q}_1)
\psi_2^*(\mathbf{q}_2-\mathbf{q}_1)
e^{ \pm i \omega_{m_1} (\mathbf{q}_1)t  \pm i 
\omega_{m_2} (\mathbf{q}_2-\mathbf{q}_1)t} 
\times
\]
\[
{e^{- \omega_{m_a} (\mathbf{q}_1) (\x^0_1 -\x^0_2)  -\omega_{m_c}(\mathbf{q}_2) 
\cdot (\x^0_2+\x^0_3) 
- \omega_{m_b} (\mathbf{q}_3 ) (\x^0_3-\x^0_4 ) } \over
(\q_1^2 + m_a^2 ) (\q_2^2 + m_c^2)(\q_3^2 + m_b^2)} \times
\]
\[
e^{ \pm i \omega_{m_1} (\mathbf{q}_1)t  \mp i 
\omega_{m_2} (\mathbf{q}_2-\mathbf{q}_1)t} 
v(\q_1,m_a,\q_2,m_c,\q_3,m_b) \times
\]
\beq
\left (\omega_{m_c} (\mathbf{q}_2) - \omega_{m_3} (\mathbf{q}_3) -
\omega_{m_4}(\mathbf{q}_2-\mathbf{q}_3) \right )   
f_1(\x^0_3)f_2(\x^0_4)  
\psi_1(\mathbf{q}_3)\psi_2(\mathbf{q}_2-\mathbf{q}_3) 
\label{i.27b}
\eeq
The large-time behavior of this integral is relevant for the Cook condition.  
To estimate the large time behavior write 
\[
- \omega_{m_a} (\mathbf{q}_1) (\x^0_1 -\x^0_2)
\pm i \omega_{m_1} (\mathbf{q}_1)t =
\]
\[
- (\omega_{m_a} (\mathbf{q}_1)
- \omega_{m_1} (\mathbf{q}_1)) (\x^0_1 -\x^0_2)
\]
\beq
- \omega_{m_1} (\mathbf{q}_1) (\mp i t +  (\x^0_1 -\x^0_2))
\label{i.27c}
\eeq
and 
\[
- \omega_{m_b} (\mathbf{q}_3) (\x^0_3 -\x^0_2)
\pm i \omega_{m_3} (\mathbf{q}_3)t =
\]
\[
- (\omega_{m_b} (\mathbf{q}_3)
- \omega_{m_3} (\mathbf{q}_3)) (\x^0_3 -\x^0_2)
\]
\beq
- \omega_{m_3} (\mathbf{q}_3)( \mp i t +  (\x^0_1 -\x^0_2)) .
\label{i.27d}
\eeq
We assume that the $m_b \geq m_3$ and $m_a \geq m_1$, which is the easiest 
case to consider.

To put (\ref{i.27}) in a manageable form I make some simplifying assumptions.
First I assume that
the $f_i(\x^0)$ are sharply peaked to factor the integrand out of the
integral.  The resulting approximation leads to 
in a constant multiplied by the integrand 
evaluated at  points at $\x^0_i$ in the support of $f_i(\x^0)$.  
Similarly I change
to total and relative momentum variables and use the translational
invariance to eliminate the center of momentum degrees of freedom.
Specifically I assume that the total 3-momentum support of the wave
functions is near zero.  One total momentum integral is eliminated by
the momentum conserving delta function.  The second is approximated by
setting the total momentum equal to zero and multiplying by the volume of
the total momentum support.  What remains, up to a multiplicative 
constant, has the form
\[
\left (\psi ,U_0(\pm t)( H\Phi - \Phi H_0) U^{\dagger}_0(t) \psi ,\theta S^c_{2:2} 
(H \Phi - \Phi H_0)U_0(\mp t) \psi \right ) \to
\]
\[
\int \left ( 
\omega_{m_c} ( \mathbf{0}) -
\omega_{m_3}(\mathbf{k}') -
\omega_{m_4}(\mathbf{k}') \right )
\times 
\]
\[   
\psi_3^*(- \mathbf{k}' )\psi^*_4 
(\mathbf{k}' ) 
e^{i \omega_{m_4} ({1 \over 2}\mathbf{k}')t
+i \omega_{m_3} ({1 \over 2}\mathbf{k}')t}
\]
\[
e^{-\omega_{m_a} (\mathbf{k}') (\x_4^0 - \x_3^0)
-\omega_{m_c} (\mathbf{0} )(\x_3^0 + \x_1^0) 
-\omega_{m_b} (\mathbf{k}) ( \x_1^0 - \x_2^0) }
\]
\[
{d\mathbf{k} d\mathbf{k}'
\over \omega_{m_a} (\mathbf{k}' )
\omega_{m_c} (\mathbf{0})
\omega_{m_b} (\mathbf{k})} 
\]
\[
\left (
\omega_{m_c} (\mathbf{0}) - \omega_{m_1} ({1 \over 2}\mathbf{k}) -
\omega_{m_2}({1 \over 2}\mathbf{k}) \right )    
\times
\]
\beq
\psi_1(\mathbf{k})\psi_2(
-\mathbf{k}) 
e^{ \mp i \omega_{m_1} (\mathbf{k})t  
\mp i \omega_{m_2} (\mathbf{k})t}. 
\label{i.28}
\eeq 
where $\mathbf{k}$ is the momentum of one of the particles in the zero 
momentum frame.

The time dependence in this expression comes from the $\mathbf{k}$ and 
$\mathbf{k}'$ integrals.  If we use (\ref{i.27c}-\ref{i.27d}) in these
expression, assuming that $m_a\geq m_1$ and $m_b \geq m_2$ the integral 
$\ref{i.28}$ has the general form
\beq
\int_0^\infty  {k^2 dk \over \omega_{m_1} (k^2)}
h(k^2) 
e^{-\omega_{m_1} (k^2) ( \x_1^0 - \x_2^0 \pm 2it ) }
\label{i.29}
\eeq
where $h(k^2)$ is a well-behaved function of $\mathbf{k}^2$.
This can be put in a form where it is possible to see the asymptotic behavior 
by making the substitution $k = m_1 \sinh (\eta)$ and  
$\omega_{m_1} (k^2) = m \cosh (\eta)$ which leads to an 
integral of the form
\beq
m_2^3 \int_0^\infty \sinh^2 (\eta) 
e^{-m_1 \cosh (\eta) ( \x_1^0 - \x_2^0 -2it ) }
h(\sinh^2 (\eta)) d \eta .
\label{i.30}
\eeq
If I ignore the $h(\sinh^2 (\eta))$  this integral can be done exactly.
The time dependence is 
\beq
I \to {1 \over m_1 ( \x_1^0 - \x_2^0 \mp 2it )} 
K_1 (m_1( \x_1^0 - \x_2^0 \mp 2it ))
\label{i.31}
\eeq
where for large arguments 
\beq
K_n (z) \to \sqrt{\pi \over 2z} e^{-z} (1 + {n^2 \over 2z} + \cdots ).
\label{i.32}
\eeq
Using (\ref{i.32}) in (\ref{i.31}) shows that this integral falls off
like ${1 \over t^{3/2}}$ for large time.  Each extra power of $k$ in
$h(\sinh^2 (\eta))$ introduces another factor of $t^{-1/2}$.  The same
factor is generated by both the $k$ and $k'$ integrals.  This shows
that the Cook condition (\ref{i.20}) is satisfied for the reflection
quasi-Schwinger functions of the form (\ref{f.8}).  The large-time
behavior is asymptotically identical to what one finds in
non-relativistic scattering theory, however in this case it is
controlled in part by the relative Euclidean time support condition.

The relativistic invariance of the $S$ matrix can be established 
using similar methods.  The required condition in terms of the
wave operators are the intertwining conditions
\beq
U(\Lambda ,a)\Omega_{\pm\alpha} = \Omega_{\pm\alpha}U_{\alpha}(\Lambda ,a)
\label{i.33}
\eeq
for both asymptotic conditions.  For the space translations and rotations
this condition is a consequence of the translational and rotational 
invariance of the injection operators, $\Phi_{\alpha}$.  
For time translations this 
follows from the existence of the wave operators.     For the 
boosts a sufficient condition is
\beq
\lim_{t \to \pm \infty} 
\Vert (\mathbf{K}\Phi_\alpha - \Phi_\alpha \mathbf{K}_{\alpha}) 
e^{\mp i H_{\alpha} t} 
\vert \psi \rangle \Vert =0 
\label{i.34}
\eeq
As in the scattering case, the non-zero contributions to this expression 
before taking the limit come from the part $S_{n:n}^a$  (see \ref{i.21})
of the quasi-Schwinger function that vanishes as the clusters of $a$ are 
separated.   In the two-body example above, this depends on properties 
of the connected four-point function.  When (\ref{i.34})  holds finite 
Lorentz transformations on the scattering eigenstates 
can be realized by transforming the asymptotic states using (\ref{i.8})


\section{Computational issues}

This paper gives a representation of a model Hilbert space whose
vectors are equivalence classes of functions of {\it Euclidean spacetime
variables} with certain support conditions, and a representation of the
Poincar\'e Lie algebra on this space.  Given a Hilbert space
representation and a set of Poincar\'e generators in this
representation it is in principle possible to calculate any dynamical
quantity.

The new features in this Euclidean formulation are that the dynamics
enters in the kernel of the Hilbert space inner product and the
Hilbert space vectors are represented by equivalence classes of
functions.  Computational strategies must be designed with these
properties in mind.

What is easy to calculate in this formalism are matrix elements of
$e^{-\beta H}$ which only involve a translation of the all of the
Euclidean time variables in the wave functions by a common constant
and a quadrature.

The brute force way to treat the large degeneracy would be use start with
a dense set of vectors, use the Gram Schmidt method to generate an
orthonormal basis and compute matrix elements of $e^{-\beta H}$ in
that basis.  Working in an orthonormal basis eliminates the large
degeneracy, and having matrix elements of $^{-\beta H}$ in a complete
orthonormal basis is in principle sufficient to compute any dynamical
quantity.

The most interesting observables are scattering observables.  While
these could in principle be calculated using the method discussed in
the previous paragraph, it is not the most efficient way to proceed.
An alternative is to utilize the time-dependent representation of the $S$
matrix elements, which in the two-Hilbert space representation has
the form
\beq
\langle \psi_{\alpha f} \vert S \vert \psi_{\gamma i} \rangle =
\lim_{t \to \infty} \langle \psi_{\alpha f} \vert 
e^{iH_\alpha t} \Phi_{\alpha}^{\dagger}e^{-2iHt} \Phi_{\gamma} 
e^{iH_\gamma t} \vert \psi_{\gamma_i} \rangle .
\label{j.1}
\eeq
In normal formulations of scattering theory \cite{Brenig:1959} this is
the starting point.  The derivation of the usual time-independent
formula assumes that the wave packets are sufficiently narrow and the
kernel of the transition operator is sufficiently smooth that the
kernel of the transition operator can be factored out of the integral,
resulting in a representation of the cross section that is independent
of the choice of wave packet.

This can be turned around by using sufficiently narrow wave packets
centered about $\mathbf{p}_1, \cdots ,\mathbf{p}_{n_\alpha}$ and  
$\mathbf{p}_1, \cdots ,\mathbf{p}_{n_\gamma}$  and keeping them for the
entire calculation.  In both cases the result leads to (approximate) 
sharp-momentum values the transition matrix
\beq
\langle \mathbf{p}_1', \cdots ,\mathbf{p}'_{n_\alpha} \vert t(E_\gamma +i0) 
\vert 
\mathbf{p}_1, \cdots ,\mathbf{p}_{n_\gamma}  \rangle  \approx
{i \over 2 \pi} { \langle \psi_{\alpha f} \vert (S-I) 
\vert \psi_{\gamma i} \rangle \over \langle \psi_{\alpha f} 
\vert \delta (E_\alpha - E_\gamma ) \vert \psi_{\gamma i} \rangle}.
\label{j.2}
\eeq
All of the quantities in this expression can be computed
from (\ref{j.1}).

While (\ref{j.2}) may be difficult to compute directly, one expects that 
the Kato-Birman invariance principle \cite{kato:1966}\cite{Chandler:1976}    
can be used to make the replacement
$H \to - e^{-\beta H}$ in (\ref{j.1}) giving the following equivalent 
expression for the $S$-matrix elements   
\beq
\langle \psi_{\alpha f} \vert S \vert \psi_{\gamma i} \rangle =
\lim_{n \to \infty} \langle \psi_{\alpha f} \vert 
e^{-i n e^{-\beta H_\alpha}} \Phi_{\alpha}^{\dagger}e^{2ine^{-\beta H}} 
\Phi_{\gamma} 
e^{-in e^{-\beta H_\gamma}} \vert \psi_{\gamma_i} \rangle . 
\label{j.3}
\eeq
This is an identity rather than an approximation.  $\beta$ is a 
parameter that can be used to set an energy scale.  The result 
is independent of $\beta$, but the wrong choice of $\beta$ 
would make the limit difficult to evaluate.

While (\ref{j.3}) looks more complicated than (\ref{j.1}), 
what is relevant is that
because of the spectral condition (\ref{d.18}), $e^{-\beta H}$,
has a spectrum in the unit interval.  Since $e^{2inx}$ can be uniformly 
approximated by a polynomial on the compact interval $[0,1]$, it follows
that if 
\beq
\vert P(x) - e^{2inx} \vert < \epsilon  \qquad \forall \, x \in [0,1]
\label{j.4}
\eeq   
then 
\beq
\Vert P(e^{-\beta H} ) - e^{2ine^{-\beta H}} \Vert < \epsilon  
\label{j.5}
\eeq
where the same $\epsilon$ appears in both expressions.  This is important
because it means that it is sufficient to test the convergence of the 
approximation using (\ref{j.4}) rather than the more complex (\ref{j.5}). 

The result of this sequence of approximations is that sharp transition
matrix elements can formally be expressed in terms of matrix elements
of polynomials in $e^{-\beta H}$, which in the Euclidean framework can
be calculated using (\ref{d.17}).

The calculation outlined above involves four approximations that 
must be performed in a specific order.  
\begin{itemize} 

\item [1.] Solve for $\Phi_\alpha$.  This requires solving the
  one-body problem for subsystems.  This can be trivial or difficult
  depending on the spectral properties of the quasi-Schwinger
  functions.  The accuracy of the subsequent approximations will be
  sensitive to how well the one-body solution is isolated from the
  rest of the subsystem mass eigenstates.   

\item [2.] The next approximation is to choose sufficiently narrow
  wave packets.  These are needed to extract sharp momentum transition
  operators.  The error on the sharp momentum transition matrix
  elements decreases as the width of the wave packet decreases.

\item [3.] The large $n$ limit in (\ref{j.3}).  This replaces the
  large time limit in the expressions for the wave operators.  The
  value of $n$ needed for convergence will increase as the wave
  packets become narrower.  The parameter $\beta$ can be used to
  minimize the values of $n$ need for convergence.  The convergence is
  strong for well-behave connected quasi-Schwinger functions. 

\item [4.] The degree of the polynomial needed to approximate 
$e^{2inx}$ increases as $n$ increases. This can be tested directly using 
(\ref{j.4}).


\end{itemize} 

The virtue is that each of the above approximations converges.  
Since each approximation effects the subsequent approximations
it is clear that these approximations must be performed in the order
prescribed above.  The last three approximations were tested in the
context of a solvable model, where it was possible to independently
investigate the accuracy of each of the last three approximations
\cite{Kopp:2011vv}.  What was found, in the context of a simple model,
was that in order to get three-significant figure accuracy on
sharp-momentum two-body transition matrix elements, the wave packet
width needed to be about 1/10 of the beam momentum.  This was observed
for incident momenta between .05 to 2 GeV.

Ten significant figure accuracy could be realized in the approximation
of $S$-matrix elements with $n$.  These results were realized using
wave packets that were sufficiently narrow to get three significant figure
accuracy in the sharp-momentum transition operators.  Values of $n$
needed for ten significant figure accuracy for these wave packets 
were in the $100-400$ range
for incident momenta between .05 - 2.0 GeV, when $\beta$ was chosen
close to the inverse of the working energy scale.

Ten significant figure accuracy could be realized in the approximation
of $e^{2ine^{-\beta H}}$ by Chebyshev polynomials in $e^{-\beta H}$.
In the model calculations of \cite{Kopp:2011vv} the degree of the
polynomials was between 300-700.  The coefficients of the Chebyshev
polynomial expansion of $e^{2inx}$ were determined using
Gauss-Chebychev quadrature, which made it possible to handle 
polynomials of large degree. 

The demonstrated convergence of the last three of the four
approximations listed above in the model calculations of
\cite{Kopp:2011vv} strongly suggest that few-GeV scale scattering
calculations are possible.  The largest source of error is associated
with the width of the wave packets. The three significant figure accuracy
for the sharp momentum transition matrix elements 
achieved in these model calculations is sufficient for few-GeV
scattering calculations.  In the two-body case, where the two-point
functions are given by {\it free} Euclidean Green functions, the
one-body problem becomes trivial.  What remains are the three
approximations that were successfully tested in the model
calculations.  This suggests that the approximation methods discussed
above should be sufficient to solve a two-body model with a four-point
function of the form (\ref{f.4}),  where $S_{1:1}$ are free field Schwinger
function and the connected four-point function has the form
(\ref{f.8}).  The existence of the limits that define the wave
operators suggest the possibility that time-independent equations
based on (\ref{j.3}) might also be possible.

A number of open problems remain.  The first one is to develop a strategy for
constructing quasi-Schwinger functions that define realistic
models.  The difficulty with the Bethe-Salpeter approach, in addition
to the stability discussed in section V, is that because
$S_{1:1}$ does not map the positive-time subspace to the positive-time
subspace, the reflection positivity applied to the kernel involves the
full matrix of the kernel, rather than the restriction the positive
Euclidean time subspace.  The construction of realistic reflection 
positive quasi-Schwinger functions is most important remaining open
problem.  

A second open problem is how to obtain accurate solutions
to the one-body problem when the subsystem quasi-Schwinger functions
also contains continuum mass eigenstates.  The Lehmann weights for 
realistic two-point Schwinger functions have both discrete and 
continuous mass eigenstates.  The challenge is how to construct a
projection operator on the one-body subspace of ${\cal H}_M$. 
These are needed in order to realize the strong convergence of the 
scattering theory.  

The third open problem is how to perform quadratures with sufficient
accuracy to successfully implement the above strategy.  This problem 
is not special to this formalism. 

Finally the ultimate goal is to apply these methods to QCD, where the
connection with the spectral condition (\ref{d.18}) suggests that
reflection positivity may only be satisfied for initial and final
states that are color singlets.  The methods discussed above can
in principle be applied in this case, since the asymptotic 
one-body states would be color singlets, and the $S$ matrix should
map singlets to singlets.  

\section{Summary} 

In this paper I exhibited a class connected four-point quasi-Schwinger
functions satisfying reflection positivity.  Their structure was
motivated by a theorem of Widder which applies only to the
one-dimensional case.  Widder's result suggests that reflection
positivity is a fairly restrictive condition.  The general structure
of reflection positive four-point functions is still an open problem.
More importantly, the construction of quasi-Schwinger functions for
realistic models remains an open problem.  In this paper I used
time-dependent methods to demonstrate the existence of scattering wave
operators for models based on the reflection positive quasi Schwinger
functions of the form (\ref{f.8}).  The basic observation is that the
Cook condition that is normally used as a sufficient condition for the
existence of non-relativistic wave operators can be applied in this
formulation of Euclidean relativistic quantum mechanics.  The
$t^{-3/2}$ asymptotic behavior that ensures the existence of the wave
operator for short-ranged potentials in the non-relativistic case is
realized in the relativistic case for sufficiently well behaved
connected quasi-Schwinger functions. 

Time-dependent methods for computing transition matrix elements using
purely Euclidean methods were suggested.  Models calculations suggest
that these methods should in principle be applicable to few-GeV scale
scattering calculations.

This work was supported by the U.~S.
Department of Energy, Office of Nuclear Physics, under contract
No. DE-FG02-86ER40286.

\section{appendix - spin}

Quasi-Schwinger functions that lead to any positive-mass irreducible 
representation space of the Poincar\'e group are constructed in this 
appendix.

A basis for vectors in a positive-mass irreducible representation
space of the Poincar\'e group are simultaneous eigenstates of
the mass, spin, linear momentum, and $z$-component of some
kind of spin (canonical, Jacob-Wick helicity, light-front, $\cdots$).  
These states have the following transformation property
\[
U(\Lambda ,a) \vert (m,j) \mathbf{p} \mu \rangle =
\]
\beq
\sum_{\nu} e^{-i \Lambda p \cdot a}  \vert \pmb{\Lambda} p , \nu
\rangle
D^j_{\mu \nu} [ B^{-1}(\Lambda p/m) ,a) \Lambda B(p/m) ]
\sqrt{{\omega_m (\Lambda p) \over \omega_m(p)}}
\label{k.1}
\eeq
where $B^{-1}(\Lambda p/m) ,a) \Lambda B(p/m)$ is a Wigner rotation.
The choice of Lorentz boost, $B(p/m)^{\mu}{}_{\nu}$, in the 
Wigner rotation 
determines the type of spin \cite{Keister:1991sb}.  For any kind of spin 
the Wigner $D$ functions, which are also finite dimensional 
representations of $SL(2,\mathbb{C})$ and it can be factored into products.
Multiplication of both sides of (\ref{j.1}) by 
\beq
D^j_{\mu \nu} [ B^{-1}(p/m) ]
\eeq
leads to 
\[
U(\Lambda ,a) \vert (m,j) \mathbf{p} \mu \rangle
D^j_{\mu \sigma} [ B^{-1}(p/m)] 
\sqrt{\omega_m (p)}=
\]
\beq
e^{-i \Lambda p \cdot a}  \vert \pmb{\Lambda} p , \nu
\rangle
D^j_{\mu \sigma'} [ B^{-1}(\Lambda p/m)]  
\sqrt{\omega_m (\Lambda p)}
D^j_{\sigma' \sigma} [\Lambda  ].
\label{k.2}
\eeq
The vectors
\beq
\vert p ,j , \sigma \rangle := 
\vert (m,j) \mathbf{p} \mu \rangle
D^j_{\mu \sigma} [ B^{-1}(p/m)] 
\sqrt{\omega_m (p)}
\label{k.3}
\eeq
transform in a Lorentz covariant manner
\beq
U(\Lambda, 0) \vert p ,j , \sigma \rangle =
e^{-i \Lambda p \cdot a}
\vert \Lambda p ,j , \sigma \rangle
D^j_{\sigma' \sigma} [\Lambda  ].
\label{k.4}
\eeq
The transformation $U(\Lambda, 0)$
is unitary with respect to the inner product
\beq
\psi(p,j,\sigma) = 
\langle p ,j , \sigma \vert \psi \rangle ,
\label{k.5}
\eeq
\beq
\langle \psi \vert \phi \rangle = 
\int
\psi^*(p,j,\mu)  D^j_{\mu \sigma} [ B(p/m) B^{\dagger}(p/m)] m {d\mathbf{p}
\over  \omega_m (p,j,\sigma) } 
\phi (p)  
\label{k.6}
\eeq
where $p_0=\omega_m(\mathbf{p})$ is the energy.
The kernel simply removes the momentum-dependent $SL(2,C)$
Wigner functions from the covariant representation.  Because 
the $SL(2,\mathbb{C})$ matrices cancel in computing matrix elements
- the result is the same independent of whether the right or 
left-handed representations 
of $SL(2,\mathbb{C})$ are used.

Note that in $SL(2,\mathbb{C})$ a general boost has a polar decomposition
\beq
B(p) = P(p) R(p)  
\label{k.7}
\eeq
where $P(p)$ is the positive Hermitian operator, 
\beq
P(p) = e^{ \pmb{\rho}\cdot \pmb{\sigma} /2},
\label{k.8}
\eeq 
$\pmb{\rho}$ is the rapidity vector and $R(p)$ is an $SU(2)$
matrix  (generalized Melosh rotation) that determines the type of spin. 
It follows that 
\beq
B(p/m) B^{\dagger}(p/m) = P(p) R(p) R^{\dagger}(p) P(p )=
P^2 (p) = e^{\pmb{\rho} \cdot \pmb{\sigma}} = \sigma \cdot p .
\label{k.9}
\eeq
In this expression the Melosh rotations cancel, so the result is
independent of the choice of spins. 
Thus this scalar product can be expressed as
\beq
\int
\psi^*(p,j, \alpha)  D^j_{\alpha \beta} [ p \cdot \sigma ] m {d\mathbf{p}
\over 
\omega_m (p) } 
\psi (p,j,\beta) = 
\label{k.10}
\eeq
\beq 
\int
\psi^*(p,j,\alpha)  D^j_{\alpha \beta } [ p \cdot \sigma ] 2m d^4{p} 
\delta (p^2 + m^2)  
\psi (p,j,\beta). 
\label{k.11}
\eeq
This is essentially identical to the form found in \cite{Wightman:1980}
(see eq. 1.57).
The important observation is that $\sigma \cdot p$ is a positive 
Hermetian matrix for timelike $p$.  The same holds for 
$D^j_{\mu \sigma} [ p \cdot \sigma ]$,
$D^j_{\mu \sigma} [ p \cdot \sigma^* ]$, and
$D^j_{\mu \sigma} [ p \cdot \sigma^{-1}]$. 

The following quasi-Schwinger function is a Euclidean 
covariant rather than Euclidean invariant distribution
\beq
\int {2m D^j_{\alpha \beta } [ p_e \cdot \sigma_e ] \over
p_e^2 +m^2 } 
d^4 p_e e^{i p_e \cdot (\x-\y)}
\label{k.12}
\eeq
that leads exactly to the representation (\ref{k.6}) of 
a mass $m$ spin $j$ irreducible representation.

These considerations show that the following Euclidean scalar product is 
reflection positive  
\[
\int g^*_\alpha (\Theta \x) 
{2m D^j_{\alpha \beta } [ p_e \cdot \sigma_e ] \over
p_e^2 +m^2 } 
d^4 p_e e^{i p_e \cdot (\x-\y)} g_\beta (\y)d^4x d^4y d^4p_e =
%\label{i.13}
\]
\beq
\psi^*(p,j,\alpha)  D^j_{\alpha \beta } [ p \cdot \sigma ] 2m d^4{p} 
\delta (p^2 + m^2)  
\psi (p,j,\beta)
\label{k.13}
\eeq
with 
\beq
\psi (p,j,\beta) = \int g(\x^0,\mathbf{\x} )e^{-\omega_m (\mathbf{p}) \x^0}
e^{i \mathbf{p}\cdot \mathbf{\x}}.
\label{k.14}
\eeq
This shows how to construct reflection-positive quasi-Schwinger
functions for any irreducible representation of the Poincar\'e group.
While I did not choose to double the representation, doubled
representations can be realized by replacing
$
D^j_{\alpha \beta } [ p \cdot \sigma ] 
$
by a direct sum of a right and left handed representation ,
$
D^j_{\alpha' \beta' } [ p \sigma_2 \cdot \sigma* \sigma_2  ] 
$, 
which is also positive for positive energy timelike $p$.

The kernel of the resulting Lorentz covariant measure 
is modified by the presence of 
$D^j_{\mu \mu'} [p \cdot \sigma]$ where 
$p = ( \omega_m (\mathbf{p}),\mathbf{p})$: 
\[
{d \mathbf{p} \over 2 \omega_m (\mathbf{p})}
D^j_{\mu \mu'} [p\cdot \sigma] 
D^j_{\nu \nu'} [p\cdot \sigma_\pi]] 
\]
where 
$\pmb{\sigma}_\pi = \sigma_2 \pmb{\sigma}^* \sigma_2$.  The transformation 
properties of multipoint quasi-Schwinger function are the same as 
products of these two-point quasi Schwinger functions. 

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