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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 exhibit a class of reflection positive Euclidean
  invariant four-point functions that can be used formulate a
  Poincar\'e invariant quantum theory.  I demonstrate the existence of
  scattering wave operators, which can be calculated without analytic
  continuation in this representation.


%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 the existence of scattering wave operators in
a representation of relativistic quantum mechanics where the dynamics
is introduced through a collection of reflection positive Euclidean
``Green functions''.

This work is motivated by the observation that the physical Hilbert
space inner product in the Osterwalder-Schrader reconstruction theorem
of Euclidean field theory \cite{Osterwalder:1973dx} can be expressed
as a sesquilinear form involving only Euclidean Green functions and
Euclidean test functions.  Furthermore, the properties of the
Euclidean Green functions that are needed to reconstruct a
relativistic quantum theory are subset of the properties needed to
construct a local field theory.  There are explicit representations of
all ten Poincar\'e generators as self-adjoint operators in this
Euclidean representation of the Hilbert space.  The important 
observation is that this representation of the relativistic 
quantum theory does {\it not} require analytic continuation.

The general formalism was presented in a previous paper
\cite{Kopp:2011vv}.  In that paper we introduced methods to compute
sharp-momentum transition operators using matrix elements of
$e^{-\beta H}$ in normalizable states.  These matrix elements can be
computed using only quadratures in the Euclidean representation.  The
computational methods that we introduced were tested using a solvable
model and convergence to the exact transition matrix elements was
demonstrated for energies between 50 MeV and 2 GeV.

The dynamical input is a set of multi-point Euclidean Green functions
that are defined as moments of Euclidean path integrals or
solutions of Schwinger-Dyson equations.  Actual computations use
finite collections of approximate or model Green functions.  The
Euclidean Green functions that define an acceptable quantum theory
satisfy a condition called reflection positivity, which is equivalent
to the requirement that vectors in the Euclidean representation of the
Hilbert space have non-negative norm.

Reflection positivity in continuum theories is a restrictive
condition.  This is expected since it ensures the existence of an 
analytic continuation of spacetime variables to Minkowski space. 
It is difficult to check because the sum of a reflection
positive Green function and a small Euclidean invariant perturbation
is not necessarily reflection positive \cite{vic}.  In addition, products of
reflection positive operators are not necessarily reflection positive.
These observations imply that the solution of the Bethe-Salpeter
equation with reflection positive input is not automatically
reflection positive.  A natural question that arises is are there real 
fundamental problems with finding reflection positive Green functions,
or do the issues discussed above just make it difficult to demonstrate? 
Having a class of non-trivial examples would make it clear that there are
no fundamental problems in constructing reflection positive 
Green functions. 

This paper addresses two questions which were not addressed in
\cite{Kopp:2011vv}.  The first question, discussed in the previous
paragraph, is ``Are there any non-trivial reflection-positive
multipoint Green functions?''  The second question, assuming that
first question is answered in the affirmative, is ``Do these
reflection positive Green functions lead to a non-trivial scattering
theory?''  This is the Euclidean version of determining what class of
potentials lead to a non-trivial scattering in non-relativistic
quantum mechanics.

This paper does not provide a general solution to this problem, but it
exhibits a representative class of reflection positive Euclidean
invariant four-point functions that are motivated by a one-dimensional
structure theorem. This paper also shows that scattering wave
operators exist for this class of four-point functions.  This paper
also includes a short discussion of issues related to applying these
methods to QCD as well as a discussion of how to extend the framework
presented in \cite{Kopp:2011vv} to treat states of any spin.

%We remark that this result seems to be a counter example to the Miani
%Testa theorem.  It is not - the point is that this ordinary quantum
%mechanics with a precisely defined Hamiltonian that just happens to be
%formulated in an abstract representation.  The formulation of the
%scattering theory uses strong limits, which is not common in field
%theory applications because they require having bound-state solutions.
%Applications of these methods to the lattice would require accurate
%bound-state wave functions to construct the scattering asymptotic
%condition with composite particles.

%Cluster properties, which are difficult to realize in direct
%interaction formulations of relativistic quantum mechanics, are
%relatively simple to realize in this Euclidean representation.  The
%play an important role in formulating scattering asymptotic conditions
%in the Euclidean framework.

The advantages 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 introduce notation and summarize needed results
from \cite{Kopp:2011vv}.  In section three I exhibit a class of reflection
positive four-point Green functions.  In section four I discuss the
formulation of the scattering problem when the dynamics is in the
kernel of the scalar product and demonstrate the existence of
scattering wave operators based on the Green functions introduced in
section three.  In section five I briefly review computational methods that
can be applied to the examples of sections three and four. Section six
provides a brief summary of the key results of this paper and a 
discussion of remaining open problems.  The appendix discusses 
the structure of two-point Euclidean Green functions for arbitrary spin.

\section{definitions and assumptions}

This section summarizes the essential background material from
\cite{Kopp:2011vv}. 

A relativistic Euclidean quantum theory is defined by a collection of
Euclidean invariant distributions, $S_{m:n}(\x_m, \cdots,
\x_1; \y_1, \cdots , \y_n)$, where the $\x_i$ and $\y_j$ are
final and initial Euclidean space-time variables.  The collection can
be finite or infinite.  

A dense set of Hilbert space vectors are represented by test functions
\beq
\{ f_m (\x_1, \cdots ,\x_m ) \}
\label{n:a1}
\eeq
with support for positive relative Euclidean times
\beq
0 < \x_1^0 < \x_2^0 < \cdots < \x_m^0 .
\label{n:a2}
\eeq
The physical Hilbert space inner product is 
\beq
\langle f\vert g\rangle  = 
\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{n:a3}
%\label{d.2}
\eeq
where $\theta \x = \theta (\x^0,\mathbf{x}) = (-\x^0,\mathbf{x})$
is the Euclidean time reflection operator.  The 
Hilbert space generated by the functions (\ref{n:a1}) with 
inner product (\ref{n:a3}) is denoted by ${\cal H}$.

The collection of Green functions in (\ref{n:a3}) 
is reflection positive if 
\beq
\langle f \vert f \rangle \geq 0.
\label{n:a4}
\eeq
This inner product has zero norm-vectors, so the actual 
Hilbert space vectors are equivalence classes of functions of the 
form (\ref{n:a1}-\ref{n:a2}), under the equivalence relation that the norm 
of the difference between equivalent functions is zero.

In quantum field theory microscopic locality requires that the
collection of Green functions is infinite and there is no distinction
between the initial and final variables; in relativistic quantum
mechanical models both of these conditions can be relaxed.  Relaxing
these conditions leads to violations of microscopic locality, however
all of the other observable requirements (axioms) of a relativistic
quantum theory remain satisfied \cite{Osterwalder:1973dx}. Computable
models involve finite numbers of degrees of freedom, so the full set
of requirements of the field theory will not be realized, but it is
still desirable to retain the relativistic invariance of the quantum
theory.

In addition, the collection of Green functions should be Hermitian, 
\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{n:a5}
%\label{c.4}
\eeq
so $\langle f\vert g \rangle = \langle g \vert f
\rangle^*$ and satisfy cluster properties,
\beq
S_{k:n} (\x_k , \cdots , \x_1; \y_1 , \cdots ,  \y_n) \to
\prod S_{k_i:n_i} (\x_{ki}, \cdots , \x_{1i}; \y_{1i}, \cdots ,  \y_{ni}),
\label{n:a6}
%\label{c.5}
\eeq
as distributions, which means that they become products of 
fewer-point Green functions when different clusters of initial and 
final variables are asymptotically separated.

The Poincar\'e generators on ${\cal H}$ have simple expressions.
The Hamiltonian and square of the invariant mass operator are
\[
\langle \x \vert H \vert \mathbf{f} \rangle :=
-{d \over da} \langle \x-(a,0,0,0)  \vert \mathbf{f} \rangle_{\vert_{a=0}} =
\]
\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{n:a7}
%\label{d.9}
\eeq
and
\beq
M^2 = H^2 - P^2 
\label{n:a8}
%\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{n:a9}
%\label{d.16}
\eeq
where $\nabla^2$ is the 4-dimensional Euclidean Laplacian.
The generators of spatial translations and rotations follow
in the usual way from the translational and rotational 
invariance of the Green functions.  Matrix elements of the 
form $\langle f \vert e^{-\beta H} \vert g \rangle$ are important
in applications and can be computed
by quadrature by replacing 
\beq
g_n (\y_1, \cdots, \y_n) \to 
g_n (\y_1-(\beta ,\mathbf{0}) , \cdots, \y_n-(\beta ,\mathbf{0}))
\label{n:a10}
\eeq 
in (\ref{n:a3}).


\section{Reflection positivity}

One difficulty with constructing reflection positive multipoint
functions is that there are non-trivial functions associated with zero
norm vectors.  

If a free Euclidean Green function 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 Euclidean Green 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, $K$, is not automatically
reflection positive, 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 for $S_{2:2}$ to be
reflection positive.  In addition, reflection positivity
ensures the existence of an analytic continuation to 
Minkowski space in space-time variables, which suggests that 
it is a restrictive condition.    

%The direct approach is also difficult because the Euclidean Green function 
%functions $S_{1:1}$ do not preserve the support condition.

The absence of any general methods for constructing reflection
positive Green functions is a problem if one wants to make
phenomenological models of Euclidean Green functions to use in this
framework.  Fortunately, a general structure theorem exits for the
two-point function in the one-dimensional case.  In what follows this
one-dimensional result will be used to motivate the construction of a
class of reflection positive four-point functions.

%The relation of reflection positivity
%to the spectral condition suggesst a direct construction of a class of
%connected reflection positive four-point functions.
%The resulting four-point Euclidean Green function functions could be 
%used directly or used to
%deduce sufficient properties of Bethe-Salpeter kernels by expressing
%them in terms of known reflection-positive Euclidean Green function 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 four-point functions can be 
expressed as a sum of products of reflection-positive 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 Euclidean
Green function.

The Widder result suggests that connected four-point 
Euclidean Green functions with the structure 
\[
S^c_{2:2} (\x_2,\x_2;\y_1,\y_2) = 
\]
\[
\int 
d^4\p_1 d^4\p_2 d^4\p_3 dm_a dm_c dm_b
e^{-i \p_1 \cdot (\x_2 -\x_1)}
e^{-i \p_2 \cdot (\x_1 -\y_1)}
e^{-i \p_3 \cdot (\y_1 -\y_2)}
\times 
\]
\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,
$v(m_a, \p_1,m_c, \p_2, m_b, \p_3)$.  The relevant observation is that 
this reduces to the one-dimensional case if there is enough symmetry 
between the initial and final variables.  
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 contribution to the Hilbert space norm 
from connected Green functions of the form (\ref{f.8}) is
\[
\Vert \vert fg \rangle \Vert^2_c = 
\]
\[
\int 
d^4 \x_1 d^4\x_2 d^4\y_1 d^4\y_2 d^4 \p_1 d^4\p_2 d^4\p_3 dm_a dm_c dm_b 
f_e^*(-\x_2^{0}, \mathbf{x}_2)
g_e^*(-\x_1^{0}, \mathbf{x}_1)
e^{-i \p_1 \cdot (\x_2 -\x_1)}
\times 
\]
\beq
e^{-i \p_2 \cdot (\x_1 -\y_1)}
e^{-i \p_3 \cdot (\y_1 -\y_2)}
{ 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_2^{0},\mathbf{y}_2)
g_e(\y_1^{0},\mathbf{y}_1)
\label{f.9}
\eeq
where the functions satisfy the support condition, 
$g_e (\y_1^{0},\mathbf{y}_1) $ and
$f_e (\y_2^{0}, \mathbf{y}_2) $  
can be non-zero only for $ 0 < \y_1^{0} < \y_2^{0}$. 

The most straightforward assumption is to choose $v(m_a, \p_1,m_2, \p_2,
m_b, \p_3)$ to be symmetric with respect to $\mathbf{p}_1 ,m_a \leftrightarrow 
\mathbf{p}_3,m_b$ and 
analytic in the upper-half $p_i^0$ planes.  In this case
the $\p^0_i$ integrals can be performed as in the one-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
\[
(2 \pi)^9 \int 
d\x^0 d\y^0 d\x^{0\prime} d\y^{0\prime}  d\mathbf{\p}_1 
d\mathbf{\p}_2 d\mathbf{\p}_3 dm_a dm_c dm_b 
\tilde{f}_e^*(\x_2^{0\prime },-\mathbf{p}_1 )
\tilde{g}_e^*(\x_1^{0\prime}, \mathbf{p}_1-\mathbf{p}_2) \times
\]
\[
e^{- \omega_{m_a} (\mathbf{p}_1) (\x_2^{0} - \x_1^{0}) }
e^{- \omega_{m_c} (\mathbf{p}_2) (\x_1^{0} + \y_1^{0} ) }
e^{- \omega_{m_b} (\mathbf{p}_3) (\y_2^{0} - \y_1^{0}) }
\times 
\]
\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_2^0, \mathbf{p}_3-\mathbf{p}_2)
\tilde{g}_e(\y_1^0, -\mathbf{p}_3).
\label{f.10}
\eeq
The  kernel, which was initially a Euclidean invariant function of Euclidean
scalar products becomes a Lorentz invariant function of Lorentz invariant inner 
products when the residues are evaluated at the poles
$\p^0\to i \omega_{m_i}(\mathbf{p}_i)$.
This connected Euclidean Green 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 construction demonstrates the existence of a large class
of reflection positive connected four-point Euclidean Green functions.
Exchange symmetry puts additional constraints on the Euclidean Green function
functions.  This construction provides a class of non-trivial 
reflection positive Green functions that can be used to investigate 
properties of non-trivial models.

The second question that can now be addressed is whether the relativistic
quantum models defined by these Green functions have non-trivial
scattering operators.
 
\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.

Instead, in order to formulate the scattering theory, 
cluster properties of the Green functions are utilized to formulate
the asymptotic conditions.
Cluster properties imply
that in each asymptotic region the Green functions break up into
products of subsystem Green functions.  Since the Green functions
define the structure of the Hilbert space, in each asymptotic region
the Hilbert space looks like the tensor product of Hilbert spaces
associated the dynamics of asymptotically separated subsystems.  
Scattering asymptotic
conditions can be formulated by finding eigenfunctions associated with
the point spectrum of each of the subsystem mass operators using
(\ref{n:a9}).  These can be further decomposed into simultaneous
eigenstates of mass, linear momentum, spin and spin projection using
the space translations and rotations (see (4.4) equations 
and (4.6) of \cite{Kopp:2011vv}).  These vectors are a basis for an irreducible
representation space of the Poincar\'e group.  Products of these
subsystem bound-state eigenfunctions define a mapping from a Hilbert
space of scattering asymptotes, which is the tensor product of
Poincar\'e irreducible representation spaces, to ${\cal H}$.


%In this framework the formultation 
%of the scattering asymptotic condition becomes a generalization 
%of what is done for two particles in a potential.  The same 
%methods that are used to establish the existence of limits
%that define scattering states can be applied in this case.
%
% 
%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, 
In order to illustrate the main concepts I consider the special 
case of two-particle scattering.  Also for simplicity I assume
that $1$ and $2$ correspond to different types of scalar particles.
In this case the dynamics is given
by a 4-point Euclidean Green function
\beq
S_{2:2} (\x_2,\x_1:\y_1,\y_2).
\label{i.1}
\eeq
The cluster condition means that the four-point function 
has the form  
\[
S_{2:2} (\x_2,\x_1:\y_1,\y_2) =
\]
\beq
S_{1:1} (\x_1:\y_1)S_{1:1} (\x_2:\y_2) +
S^c_{2:2} (\x_2,\x_1:\y_1,\y_2)
\label{i.2}
\eeq
where $S^c_{2:2} (\x_2,\x_1:\y_1,\y_2)$ is a connected four-point function
and $S_{1:1} (\x_i:\y_i)$ are two-point functions.  

I also assume that $S_{1:1} (\x_i:\y_i)$ is the two-point Euclidean Green
function for a free scalar particle of mass $m_i$.  In this case every vector 
in the Hilbert space associated with $S_{1:1} (\x_i:\y_i)$ 
is a mass eigenstate with eigenvalue $m_i$ so there is no need to 
solve the mass eigenvalue problem discussed above.
%
% mass 
%eigenvalue problem associated with the 
%
%for a given partition of the
%coordinates into asymptotically separated clusters of a partition $a$,
%the Green function (\ref{i.1})  can be expressed as
%a product of Green 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 Euclidean Green function 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{n:a8}).
These asymptotic one-particle eigenstates are represented by functions of
the Euclidean variables, 
\beq
\langle \x_i \vert m_i \rangle =  \psi_i(\pmb{\x_i}) h_i(\x_i^0) 
=
h_i(\x_i^0) \int {d \mathbf{p} \over (2 \pi)^{3/2}}
e^{i \pmb{\p} \cdot \pmb{\x}_i} \tilde{\psi}_i(\mathbf{p}) 
\eeq
where  $h_i(\x_i^0)$ is a smooth function that has compact support for 
positive Euclidean time. The freedom to choose the functions $h_i(\x_i^0)$ 
is related to the fact the vectors are represented by 
equivalence classes of functions.  Specifically, in the asymptotic
region, the different $h_i$'s are all associated with the same one-body 
Poincar\'e irreducible basis states. 
 
I define a mapping, $\Phi$ from the tensor 
product of the space of square integrable functions
of the $\pmb{\p}_i$ to the 
Hilbert space ${\cal H}$ by
\beq
\langle \x_1 ,\x_2 \vert \Phi \vert \psi_1 \otimes \psi_2 \rangle
=   
h_1(\x_1^0) h_2(\x_2^0) \int {d\pmb{\p}_1 \over (2 \pi)^{3/2}}
{d\pmb{\p}_2 \over (2 \pi)^{3/2}}
e^{i \pmb{\p}_1 \cdot \pmb{\x}_1} \tilde{\psi}_1(\pmb{\p}_1) 
e^{i \pmb{\p}_2 \cdot \pmb{\x}_2} \tilde{\psi}_2(\pmb{\p}_2) 
\eeq
where the $h_i$ can be chosen so their supports do not intersect.  Since 
the particles are spin zero particles, this defines a mapping from
the tensor product of two irreducible representation spaces
of the Poincar\'e group (mass $m_1$, spin 0 and mass $m_2$, spin 0),
${\cal H}_0$,
to the Hilbert space, ${\cal H}$.  The functions $h_1(\x_1^0) h_2(\x_2^0)$
are considered as part of the mapping.  

Scattering wave operators are mappings from the asymptotic 
Hilbert space ${\cal H}_{0}:={\cal H}_{m_1} \otimes {\cal H}_{m_2}$ 
to ${\cal H}$ 
defined by the strong limits 
\beq
\Omega_{\pm} := \lim_{t \to \pm \infty} e^{i Ht} \Phi e^{-i H_{0} t} 
\label{i.18}
\eeq
where $H_{0}= \sqrt{m_1^2 + \mathbf{p}_1^2} +
\sqrt{m_2^2 + \mathbf{p}_2^2}$ is the Hamiltonian associated with 
the asymptotically free particles.
The existence of these limits depends on properties of the
connected four-point Euclidean Green function.  This will be shown below.

Conventional methods 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
\[
\lim_{t \to  \pm \infty} e^{i Ht} \Phi e^{-i H_{0} t} =
\Phi + 
\int_0^{\pm \infty}{d \over dt}\left (  
e^{i Ht} \Phi e^{-i H_{0} t} dt \right )   =
\]
\beq 
\Phi + i \int_0^{\pm \infty}  
e^{i Ht} (H \Phi - \Phi H_{0}) e^{-i H_{0} t} dt  . 
\label{i.19}
\eeq
A sufficient condition for the existence of the wave operator is the
strong 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_{0}) e^{\mp i H_{0} 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 Euclidean Green function 
functions.  In this case the integrand has the form
\[
\Vert (H \Phi - H_{0} \Phi ) e^{\mp i H_{0} t} 
\vert \psi \rangle \Vert = 
\]
\beq
( \psi_{}, e^{\pm i H_{0} t} 
(\Phi^{\dagger}H - \Phi^{\dagger}H_{0})
(\theta S_{1:1}S_{1:1}  + \theta S^c_{2:2}) 
(H \Phi- \Phi H_{0})
\psi_{} )^{1/2} 
\label{i.21}
\eeq
where $(,)$ represents the Euclidean inner product and  
I have used (\ref{i.2}) to express the four-point 
Green function as the sum of a product of two-point 
functions with a connected four-point function
that vanishes asymptotically.    
 
The second goal of this paper will be realized by
showing that 
condition (\ref{i.20}) holds for the four-point Green functions 
(\ref{f.8}), which is sufficient for the existence of the strong
limits (\ref{i.18}). To do this it is necessary to show 
\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. 

I will argue that 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 
Euclidean Green function is the sum of a 
product of two-point Euclidean Green functions and a connected term.  
I consider a
connected term that has the structure discussed in (\ref{f.8})
\[
S(\x_1,\x_2;\x_3,\x_4)= 
\]
\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 Euclidean Green 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 the 
asymptotic states $\Phi \vert e^{\mp i H_{0} t}
\vert \psi
\rangle $ have the 
form 
\beq
\langle \x_1, \x_2 \vert \Phi \vert e^{\mp i H_{0} t} \psi \rangle = 
\prod_{i=1}^2 h_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 I choose the functions $h_i(\x^0_i)$ to be sharply peaked with 
compact support and integrate to 1.

With this choice of wave packets
\[
\langle \x_1,\x_2 \vert 
(H \Phi - \Phi H_0)e^{\mp iH_0t}\vert  \psi \rangle =
\]
\[
(2 \pi)^3 \left ({\partial \over \partial \x^0_1} +
{\partial \over \partial \x^0_2} - \omega_{m_1} (\mathbf{p}_1) -
\omega_{m_2}(\mathbf{p}_2) \right )   
h_1(\x^0_1)h_2(\x^0_2) \times 
\]
\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_0 t} \psi ).
\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 energy factors (
see equations (6.4) and (6.5) of \cite{Kopp:2011vv})
that exactly cancel the asymptotic energy factors in (\ref{i.26}), 
making the $S_{1:1}S_{1:1}$ terms in (\ref{i.27}) vanish.

This means that only the connected part of the four-point
Euclidean Green function 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 
\[
(2 \pi )^{6}
\int d^4{\q}_1 d^4{\q}_2 d^4 {\q}_3 dm_a dm_c dm_b
d\x^0_1 d\x^0_2 d\x^0_3 d\x^0_4 
\left (\omega_{m_c} (\mathbf{q}_2)  - \omega_{m_1} (\mathbf{q}_1) -
\omega_{m_2}(\mathbf{q}_2-\mathbf{q}_1) \right ) \times
\]
\[   
h_1(\x^0_1)h_2(\x^0_2)
\psi^*_1(\mathbf{q}_1)
\psi_2^*(\mathbf{q}_2-\mathbf{q}_1)
e^{ \pm i \omega_{m_1} (\mathbf{q}_1)t  \pm i 
\omega_{m_2} (\mathbf{q}_2-\mathbf{q}_1)t} 
\times
\]
\[
{e^{- \omega_{m_a} (\mathbf{q}_1) (\x^0_1 -\x^0_2)  -\omega_{m_c}(\mathbf{q}_2) 
\cdot (\x^0_2+\x^0_3) 
- \omega_{m_b} (\mathbf{q}_3 ) (\x^0_3-\x^0_4 ) } \over
(\q_1^2 + m_a^2 ) (\q_2^2 + m_c^2)(\q_3^2 + m_b^2)} \times
\]
\[
e^{ \pm i \omega_{m_1} (\mathbf{q}_3)t  \mp i 
\omega_{m_2} (\mathbf{q}_2-\mathbf{q}_3)t} 
v(\q_1,m_a,\q_2,m_c,\q_3,m_b) \times
\]
\beq
\left (\omega_{m_c} (\mathbf{q}_2) - \omega_{m_1} (\mathbf{q}_3) -
\omega_{m_2}(\mathbf{q}_2-\mathbf{q}_3) \right )   
h_1(\x^0_3)h_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 
\[
- \omega_{m_b} (\mathbf{q}_3) (\x^0_3 -\x^0_2)
\pm i \omega_{m_1} (\mathbf{q}_3)t =
\]
\[
- (\omega_{m_b} (\mathbf{q}_3)
- \omega_{m_1} (\mathbf{q}_3)) (\x^0_3 -\x^0_2)
\]
\beq
- \omega_{m_3} (\mathbf{q}_3)( \mp i t +  (\x^0_1 -\x^0_2)) .
\label{i.27d}
\eeq
We assume that the $m_a, m_b \geq m_1$, which is the easiest 
case to consider.  These assumptions ensure that the integrals
are all convergent.

To put (\ref{i.27}) in a manageable form I make some simplifying assumptions.
First I assume that
the $h_i(\x^0)$ are sharply peaked to factor the integrand out of the
integral.  The resulting approximation leads to 
a constant multiplied by the integrand 
evaluated at  points at $\x^0_i$ in the support of $h_i(\x^0)$.  
Similarly, I change
to total and relative momentum variables and use the translational
invariance to eliminate the center of momentum degrees of freedom.
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 other total momentum 
integral is approximated by setting the total momentum 
to zero and multiplying by the volume of the support of the 
total momentum.
What remains, up to a multiplicative constant, has the form
\[
\left (\psi ,U_0(\pm t)( H\Phi - \Phi H_0) U^{\dagger}_0(t) \psi ,\theta S^c_{2:2} 
(H \Phi - \Phi H_0)U_0(\mp t) \psi \right ) \to
\]
\[
\int \left ( 
\omega_{m_c} ( \mathbf{0}) -
\omega_{m_1}(\mathbf{k}') -
\omega_{m_2}(\mathbf{k}') \right )
\times 
\]
\[   
\psi_3^*(- \mathbf{k}' )\psi^*_4 
(\mathbf{k}' ) 
e^{\pm i \omega_{m_2} (\mathbf{k}')t
\pm i \omega_{m_1} (\mathbf{k}')t}
\]
\[
e^{-\omega_{m_a} (\mathbf{k}') (\x_4^0 - \x_3^0)
-\omega_{m_c} (\mathbf{0} )(\x_3^0 + \x_1^0) 
-\omega_{m_b} (\mathbf{k}) ( \x_1^0 - \x_2^0) }
\]
\[
{d\mathbf{k} d\mathbf{k}'
\over \omega_{m_a} (\mathbf{k}' )
\omega_{m_c} (\mathbf{0})
\omega_{m_b} (\mathbf{k})} 
\]
\[
\left (
\omega_{m_c} (\mathbf{0}) - \omega_{m_1} (\mathbf{k}) -
\omega_{m_2}(\mathbf{k}) \right )    
\times
\]
\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 I use (\ref{i.27c}-\ref{i.27d}) in this
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 
{d \mathbf{k} \over \omega_{m_1} (\mathbf{k})}
g(\mathbf{k}) 
e^{-\omega_{m_1} (\mathbf{k}^2) ( \x_1^0 - \x_2^0 \pm 2it ) }
\label{i.29}
\eeq
where $g(\mathbf{k})$ is a well-behaved function of $\mathbf{k}$.

It follows from the lemma on page 157 of \cite{Ruelle:1962} that, for the 
case that the wave functions $\psi_i(\mathbf{k})$ are smooth with compact 
support, integrals of this form fall off like $t^{-3/2}$ for large time.

This shows that the Cook condition (\ref{i.20}) is satisfied for
the reflection positive Euclidean Green functions of the form
(\ref{f.8}).  The asymptotic large-time behavior is identical to
the behavior found in non-relativistic scattering theory.

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} = \Omega_{\pm}U_{0}(\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, $\Phi$.  
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 - \Phi \mathbf{K}_{0}) 
e^{\mp i H_{0} t} 
\vert \psi \rangle \Vert =0 
\label{i.34}
\eeq
where expressions for the boost generators are given in
\cite{Kopp:2011vv}.
As in the scattering case, the non-zero contributions to this expression 
before taking the limit come from $S^c_{2:2}$  (see \ref{i.21}).
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.33}).

\section{Computational issues}

In this paper we have demonstrated the existence of a class of
reflection positive Euclidean Green functions and shown that this
class of Euclidean Green functions leads to non-trivial scattering
operators.  The scattering operators were constructed using
conventional time-dependent methods, where cluster
properties of the Green functions were used to formulate the scattering
asymptotic conditions.

The result is that having established the existence of wave
operators and knowing how to compute matrix elements of
$e^{-\beta H}$ in a basis of normalizable states (see \ref{n:a10}),  there 
is enough information to compute transition matrix elements.

The strategy adopted in \cite{Kopp:2011vv} to perform this
computation utilized three controlled approximations.  The first is to
use narrow wave packets to extract sharp momentum transition matrix
elements from $S$-matrix elements
\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
The convergence of these approximations is determined by the 
smoothness of the transition matrix elements.

The $S$ matrix elements needed as input 
were expressed, using the invariance 
principle \cite{kato:1966}\cite{Chandler:1976},  as matrix 
elements of $e^{i 2n e^{-\beta H}}$ in normalizable 
eigenstates 
\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
The convergence with $n$ depends on the width of the wave packets.
Ten significant figure accuracy was achieved 
in the test model of ref (\cite{Kopp:2011vv}).

The third approximation that we used was to uniformly approximate
$e^{2ine^{-\beta H}}$ by a polynomial in $e^{-\beta H}$.  This was
possible because the spectrum of $e^{-\beta H}$ is bounded.  
Because of the uniform convergence the error is identical to 
the error in approximating 
\beq
\vert e^{2inx} - P(x) \vert < \epsilon  \qquad x \in [0,1]
\eeq
by a polynomial.  Ten significant figure accuracy was again realized using 
Chebyshev polynomials with Gauss-Chebyshev quadratures.  This method
requires that the approximations be performed in the specified order.

In more realistic models an additional approximation is needed,
which is the solution of point spectrum mass eigenstates of the
subsystem Green functions.  These appear in the multi-channel
generalization of the mapping $\Phi$ and are needed to get the strong
convergence needed to satisfy the Cook condition (\ref{i.20}).
They can also appear in the two-point function if it has a 
non-trivial Lehmann weight. 

In general, given an explicit Hilbert space representation and knowing
that the scattering theory exists, there are many other techniques
that could be used to calculate scattering observables without
analytic continuation.  The method discussed above provides one 
method that has been tested, but it may not be the most efficient 
method available.

\section{QCD}

Ultimately one would like to use Euclidean methods to compute GeV scale
scattering observables in QCD.  Lattice, path integral, and Schwinger-Dyson 
formulations of QCD all yield Euclidean Green functions.

In QCD, because of confinement, the Euclidean Green functions of the
theory are not expected to be reflection positive.  However reflection
positivity should hold for color singlet initial and final states.  In
addition the scattering asymptotic states should also be reflection
positive.  The Euclidean methods discussed in this paper are still
be applicable if these two conditions hold.


\section{Summary} 

In this paper the existence of non-trivial reflection positive
Euclidean Green functions was demonstrated by exhibiting an explicit
class of reflection positive connected four-point Euclidean Green
functions.  The structure of this class of Green functions was
motivated by a theorem of Widder that exhibited the structure of a
general one-dimensional reflection-positive two-point function.  The
general structure of reflection positive four-point functions is still
an open problem.  More importantly, the structure of Euclidean Green
functions for realistic models remains an open problem.

In this paper time-dependent scattering methods were used to
demonstrate the existence of scattering wave operators for models
based on the reflection positive Euclidean Green 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 of the integrand in (\ref{i.20}) 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 Euclidean Green function functions.

The results of this paper imply that the Euclidean methods tested in
\cite{Kopp:2011vv}, when applied to models defined by the class of
reflection positive Green function in section three, should converge
to transition matrix elements for a range of energies up to the few
GeV scale.

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

\section{appendix - spin}

Two point Euclidean Green 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 consists of 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_{\nu \mu} [ B^{-1}(\Lambda p/m) \Lambda B(p/m) ]
\sqrt{{\omega_m (\Lambda p) \over \omega_m(p)}}
\label{k.1}
\eeq
where $B^{-1}(\Lambda p/m)\Lambda B(p/m)$ 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 $SL(2,\mathbb{C})$,  can be factored into products.
Multiplication of both sides of (\ref{k.1}) by 
\beq
D^j_{\mu \nu} [ B^{-1}(p/m) ]
\eeq
leads to 
\[
\sum_{\mu}  U(\Lambda ,a) \vert (m,j) \mathbf{p}, \mu \rangle
D^j_{\mu \sigma} [ B^{-1}(p/m)] 
\sqrt{\omega_m (p)}=
\]
\beq
\sum_{\nu \sigma'} e^{-i \Lambda p \cdot a}  \vert \pmb{\Lambda} p , \nu
\rangle
D^j_{\nu \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 := 
\sum_{\mu} \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 =
\sum_{\sigma'}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 = 
\sum_{\mu \sigma} \int
\psi^*(p,j,\mu)  D^j_{\mu \sigma} [ B(p/m) B^{\dagger}(p/m)]  {md\mathbf{p}
\over  \omega_m (p) } 
\phi (p,j,\sigma)  
\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
\sum_{\alpha \beta}\int
\psi^*(p,j, \alpha)  D^j_{\alpha \beta} [ p \cdot \sigma ] m {d\mathbf{p}
\over 
\omega_m (p) } 
\phi (p,j,\beta) = 
\label{k.10}
\eeq
\beq 
\sum_{\alpha \beta} \int
\psi^*(p,j,\alpha)  D^j_{\alpha \beta } [ p \cdot \sigma ] 2m d^4{p} 
\delta (p^2 + m^2)  
\phi (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 
$D^j_{\mu \sigma} [ p \cdot \sigma^t ]$,
$D^j_{\mu \sigma} [ p \cdot \sigma^* ]$, and
$D^j_{\mu \sigma} [ p \cdot \sigma^{-1}]$. 

The following Green 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  
\[
\sum_{\alpha \beta}
\int g^*_\alpha (\theta \x) 
{2m D^j_{\alpha \beta } [ p_e \cdot \sigma_e ] \over
p_e^2 +m^2 } 
d^4 p_e e^{i p_e \cdot (\x-\y)} g_\beta (\y)d^4x d^4y d^4p_e =
%\label{i.13}
\]
\beq
\sum_{\alpha \beta}
\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_\beta (\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 two-point
Euclidean Green 
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 \cdot \sigma_2  \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 Euclidean Green function are the same as 
%products of these two-point quasi Schwinger functions. 

\begin{thebibliography}{23}
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\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

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\bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Kato}},
  \emph{\bibinfo{title}{Perturbation theory for linear operators}}
  (\bibinfo{publisher}{Spinger-Verlag, Berlin}, \bibinfo{year}{1966}).

\bibitem[{\citenamefont{Chandler and Gibson}(1976)}]{Chandler:1976}
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  \bibinfo{author}{\bibfnamefont{A.~S.} \bibnamefont{Wightman}},
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\bibliography{master_bib_file.bib}


xxxxxxxxxxxxxxxx




\[
\langle \x \vert \mathbf{P}  \vert \mathbf{f} \rangle = 
i \pmb{\nabla}_{\mathbf{a}} \langle \x-(0,\mathbf{a})  \vert \mathbf{f} \rangle\vert_{\vert_{\mathbf{a}=0}}=
\]
\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 
 




In \cite{} we exhibited the Poincar\'e Lie algebra and invariant mass
operators on this space.  We showed how to perform finite Poincar\'e
transformation on point-spectrum eigenstates of the mass operator
without analytic continuation.  We gave methods to establish the 
existence and compute scattering observables without analytic continuation.
Finally we demonstrated, in the convergence of the proposed method
for computing GeV scale scattering observables in the context of a
solvable model, where it was possible to test the convergence 
of the computational algorithms.

What was not done in this paper was to exhibit model Green functions
satisfying the reflection positivity condition.  In addition, given
such a Green function, it is not automatic that it will satisfy the
sufficient conditions needed to establish the existence of scattering
wave operators.  Finally, we did not discuss techniques for finding
point spectrum eigenstates in this framework.  It turns out that each
one of these problems has some complications.  One difficulty is that
the solution of the Bethe Salpeter equation with a reflection positive
driving term and a sufficiently small Euclidean invariant kernel is
not automatically reflection positivity, independent of the size of
the kernel.  It turns out that this is not the case.  The reason for
this is that even when the driving term is positive, there are
functions in the test function space that have zero norm with respect
to the driving term but may have finite negative norm with respect to
the full Green function.  Perturbation theory can also be tricky
because products of reflection positive operators are not necessarily
reflection positive.

Another issue is that the Euclidean Green functions are formally
distributions rather than functions.  Because of this it is not
automatic that the conditions for the existence of wave operators will
be as simple as the corresponding non-relativistic conditions.

Finally, the construction of point mass eigenfunctions is not entirely
trivial.  These are needed to formulate scattering asymptotic
conditions when the initial and final states are composite.
They are important to get the strong convergence needed to do 
scatting in the Euclidean domain.  The
problem is to exhibit solutions that satisfy the support conditions.

While all of these concerns should in principle be resolvable, the
goal of this research is to eventually use these methods to do
GeV-scale scattering calculations in models that have the structure of
Euclidean QCD.  Our toy model calculations demonstrated the
convergence of propose computational methods in the context of a
quantum mechanical model; in this paper we exhibit a class of
reflection positive four point Green functions.

 and show that


suggest that such calculations
are feasible provided  


 For example if one tries to niavely transform 
 
quasilocal Haag Ruelle fields, which 


from the
Lehmann representation of a two-point Euclidean Green function that
also has a continuum weight is not entirely trivial because some of
the most obvious candidates do not satisfy the support condition.

 


function in not even trivial

  
are 




 significant number of 
0 norm vectors.  
  Each one of these
omissions is


 
without analytic continuation

 




point spectrum eigenstates
of the mass operator, the formulations of 



\section{Input}

The input to a Euclidean relativistic quantum theory is a collection of 
Euclidean invariant distributions, which I refer to as 
Euclidean Green function 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 Euclidean Green function 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 Euclidean Green function 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 Euclidean Green function 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 Euclidean Green function 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 Euclidean Green function functions
replace the $n$-body interactions in conventional potential models.
The problem is to construct a set of model Euclidean Green function 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 Euclidean Green function functions satisfying
{\bf E.1-E.3} is given.  The Euclidean Green function 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
Euclidean Green function 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 Euclidean Green function 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 Euclidean Green function 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 Euclidean Green function 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
Euclidean Green function functions where the continuity properties of the
semigroups can be established.  In what follows I assume that the
model Euclidean Green function 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 Euclidean Green function
%functions that lead to any positive mass representation
%of the Poincar\'e group are given in the appendix. 




\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{n:a8}). 

These ``subsystem'' 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 Euclidean Green function 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
as a mapping 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 ${\cal H}$.  
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 ${\cal H}$ the function needs to 
have positive relative time support.  This is true for the 
individual cluster mass eigenstates, since they are vectors in the 
Hilbert space.  The support of each of 
the 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 representing the same vector.

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}).
