<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[aps,prc,reprint,groupedaddress,onecolumn]{revtex4}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amscd}
\begin{document}
\title{Euclidean representations of relativistic quantum theories}

\author{W.~N.~Polyzou}
\affiliation{Department of Physics and Astronomy, The University of
Iowa, Iowa City, IA 52242}

\thanks{Non-Perturbative QFT in Euclidean and Minkowski  Workshop\\
Mini-proceedings, 10$^{\mathrm{th}}$-12$^{\mathrm{th}}$ September, University of Coimbra, Portugal.
This research 
supported by the US Department of Energy, Office of Science, 
grant number DE-SC0016457
}
\begin{abstract}

This contribution discusses a Euclidean approach to relativistic quantum
mechanics.  The input is a collection of Euclidean covariant reflection
positive distributions.  The advantage of the Euclidean
approach are (1) cluster properties are easily realized (2) analytic
continuation is not needed and (3) the dynamics can be motivated by
appealing to local quantum field theory.

\end{abstract}

\maketitle

There are computational advantages to using Euclidean formulations of
relativistic quantum mechanics or local quantum field theory
\cite{schwinger}.  The challenges are treating problems like
scattering, where real-time evolution plays an important role.  In
addition, the analytic continuation back to real time requires a
detailed understanding of the analytic structure of the quantities
being continued.

The Osterwalder-Schrader reconstruction theorem \cite{os1} provides a
means for constructing a relativistic quantum field theory from a
collection of Euclidean covariant Green functions.  In their proof of
the reconstruction theorem Osterwalder and Schrader construct the
physical quantum mechanical Hilbert space {\it without any explicit
analytic continuation}.  In addition, the locality axiom is logically
independent of the rest of the axioms, suggesting the possibility of
using Euclidean methods to construct relativistic models of systems of
a finite number of degrees of freedom that satisfy cluster properties
and a spectral condition.  While these properties have been known
since the 1970's, they have not been exploited in applications.

The input to the reconstruction theorem is a collection of Euclidean
covariant distributions that satisfy reflection positivity.  These
distributions are formally functional derivatives of the functional
Fourier transform of a Euclidean path integral, which provides a
formal connection to the dynamics.  A corresponding connection to the
dynamics is not as direct with the Minkowski-space Wightman axioms.

Reflection positivity implies the spectral condition.  While reflection
positivity is not expected to hold gauge theories, it
is expected to hold if the theory is formulated in terms of gauge
invariant degrees of freedom, like Wilson loops.  This is the case in lattice
formulations of QCD, and is sufficient for computing physical
observables.

The structure of the {\it physical Hilbert space inner product} in terms of the
Euclidean covariant distributions has the form
\begin{equation}
\langle \psi \vert \phi \rangle =
\sum_{mn} \int \psi^*_m (\theta x) S_{m:n}(x,y) \phi_n (y) d^{4m}x d^{4n}y
\label{eq1}
\end{equation}
where the space-time-spin variables are Euclidean, $\theta$ changes the
sign of the Euclidean times and the wave functions have support
for positive relative Euclidean times.  Reflection positivity is
the condition that this inner product leads to a non-negative norm:
\begin{equation}
\Vert \vert \psi \rangle \Vert^2 := \langle \psi \vert \psi \rangle \geq 0 .
\label{eq2}
\end{equation}
It is a condition on the distributions, $S_{n:m}$.
The relativistic invariance of a quantum theory requires the existence of 
a dynamical unitary ray representation of the Poincar\'e group.
The covering group of both the complex Lorentz group and the complex orthogonal
group is $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$.  This relation
implies relations of the generators of Euclidean time translations and
rotations in space- Euclidean time planes with the generators of
Minkowski time evolution and Lorentz boosts \cite{gohin}.  By making the replacements
\begin{equation}
H_e \to H_m = i H_e \qquad
{J}^{i0}_{e}  \to 
{J}^{i0}_m  = -i
{J}_{e}^{i0}    
\label{eq3}
\end{equation}
the Euclidean generators can be transformed into operators satisfying
the Poincar\'e commutations relations.  The generators $H_e$ and $H_m$
or ${J}^{i0}_e$ and ${J}_{m}^{i0}$ cannot both be self-adjoint
operators on the same representation of the Hilbert space.  The time
reflection in the inner product (\ref{eq1}) breaks Euclidean
invariance and makes the Lorentz generators self-adjoint on the
Hilbert space (\ref{eq1}).  The inner product with the time reflection
cannot have a positive norm for general functions of Euclidean
variables since the norm of functions that are even or odd with
respect to time reflection cannot both be positive.  The projection on
states with positive relative times gives the required non-negative
inner product, provided the Euclidean distributions are reflection
positive.

A general unitary representation of the covering group of the
Poincar\'e group of a relativistic quantum theory can be transformed
into a direct integral of irreducible representations by a
unitary transformation.  This is the
relativistic analog of diagonalizing a non-relativistic
Hamiltonian. For a physically sensible theory the irreducible
representations are representations with positive energy and positive
or zero mass.  The fact that finite dimensional
representations of $SU(2)$ are entire functions of angles can be
used to show the equivalence of mass $m$, spin $s$ irreducible unitary
representations of the Poincar\'e group, covariant two-point Wightman
functions for particles with the same mass and spin and reflection
positive Euclidean distributions that give the same inner products as
the Wightman functions.  The equivalent Wightman and Euclidean kernels are
related by
\begin{equation}
\delta (p^2 +m^2 )\theta (p^0)D^{s}_{\mu \nu} (\sigma \cdot p)
\leftrightarrow 
{2 \over \pi}{D^{s}_{\mu \nu} (\sigma_e \cdot p_e)
\over p_e^2 + m^2},
\qquad
\sigma_{\mu} = (I,\pmb{\sigma}),
\qquad
\sigma_{e\mu} = (iI,\pmb{\sigma}).
\label{eq4}
\end{equation}
For systems of non-interacting particles the Euclidean distributions are
irreducible representation or tensor products of irreducible representations
so they are reflection positive.  The main challenge is to show that
reflection positivity is preserved after including interactions.

Widder's theorem \cite{widder} gives a representation for the
kernel of the positive quadratic form
\begin{equation}
\int f^* (\tau) k(\tau + \tau') f(\tau' ) \geq 0
\label{eq5}
\end{equation}
of the form
\begin{equation}
k(\tau + \tau') = \int_c^\infty  d\rho(\alpha) e^{-\alpha (\tau + \tau')}
\label{eq6}
\end{equation}
where $d\rho(\alpha)$ is a positive measure.  This can be transformed to
look like a Lehmann representation in 1 dimension.  This provides some
insight into the structure of the reflection positive Euclidean covariant
distributions, but we do not know of any general structure theorems for
reflection positive Euclidean covariant distributions. 

The Euclidean representation of the Hilbert space has some interesting
properties.  One is that four-dimensional delta functions are square
integrable.  This is a consequence of the structure of the kernel that
includes the Euclidean time-reflection.  In an irreducible
representation the square of the mass operator is the Euclidean
Laplacian.  If analog of the Lehmann weight is polynomially bounded,
polynomials in this Laplacian are complete \cite{gordon}.  This can be
used to construct operators that project out single particle states
that satisfy the relative time support condition.  These states are
needed to construct the asymptotic states in the Haag-Ruelle
formulation of scattering theory, which is the field theoretic
generalization of non-relativistic time-dependent scattering.  It
differs from the LSZ formulation in that the interpolating field can only
create one-particle states out of the vacuum.  The advantage is
that the weak limits in LSZ scattering are replaced by strong limits.

The Haag-Ruelle $S$ matrix has the standard form
\begin{equation}
S_{fi} = \Omega_{f+}^{\dagger} \Omega_{i-} := \lim_{t \to \infty} \langle \phi_f e^{i H_f t} \Pi_f^{\dagger}
e^{-2iHt} \Pi_i e^{i H_f t} \vert \phi_i \rangle  
\label{eq7}
\end{equation}
where $\Pi_f$ and $\Pi_i$ are channel injection operators.
The invariance principle \cite{cg} can be used to replace $H, H_f$ and $H_i$
by $-\exp (-\lambda H)$.  This means that $e^{-2iHt}$ can be replaced
by $e^{2ine^{-\lambda H }}$  which, because of the compact spectrum of
$-\exp (-\lambda H)$, can be uniformly approximated by a polynomial in
$\exp (-\lambda H)$ \cite{phil}.  Matrix elements of this polynomial can be computed
using elementary Euclidean time translations by integer multiples of
$\lambda$, which is a free parameter that sets an energy scale.
For gauge theories,  the interpolating field creates gauge invariant
one-particle states out of the vacuum, which means that the Hamiltonian
in (\ref{eq7}) 
will not couple to non-singlet states. 

The conclusion is that by taking advantage of the Euclidean
representation of the dynamics the difficulties associated with
analytic continuation can be avoided.  In addition, because the
construction gives the Hilbert-space inner product and
self-adjoint Poincar\'e generators that satisfy cluster properties, standard
methods can be used to compute any quantum mechanical observable.

%Blablabla blabla blablabla, blablabla \cite{webpage}
%and bla \cite{ref1} and bla \cite{ref2}.

\begin{thebibliography}{99}
\setlength{\itemsep}{0cm}

\bibitem{schwinger}
J. Schwinger, ``On the Euclidean structure of relativistic field theory'',
Proc. Nat. Acad. Sci., 44,956(1958).


\bibitem{os1}
K. Osterwalder and R. Schrader,,
``Axioms for Euclidean Green's functions'',
Commun. Math. Phys. 31,83(1973).
  
%\bibitem{os2}
%K. Osterwalder and R. Schrader,,
%``Axioms for Euclidean Green's Functions. 2'',
%Commun. Math. Phys., 42,281(1975).

\bibitem{widder}
D. V. Widder, {\it The Laplace Transform},
p. 272-275, Dover, 1941.

\bibitem{gohin}
Gohin Shaikh Samad and W. N. Polyzou, 
``Relativistic invariance in Euclidean formulations of quantum mechanics'',
nucl-th 1906.10083.

\bibitem{gordon}
Gordon Aiello and W. N. Polyzou,
``Scattering asymptotic conditions in Euclidean
relativistic quantum theory'',
Phys. Rev. D93,056003(2016).

\bibitem{cg}
C. Chandler and A. Gibson,
``Invariance principle for scattering with long-range (and other) potentials'',
Indiana Journal of Mathematics.", 25,443(1976).


\bibitem{phil}
P. Kopp and W. N. Polyzou,
"{A Euclidean formulation of relativistic quantum mechanics}",
Phys. Rev. D85,016004(2012).


\end{thebibliography}
\end{document}
\newpage


\subsection{Second Talk}
\addtocontents{toc}{\hspace{2cm}{\sl Y.~YYY}\par}

\vspace{5mm}

{\sl Y.~YYY}

\vspace{5mm}

\noindent
Centre of \\

\vspace{5mm}

Blableble bleble blabla \cite{ref1,ref2} and blablebliblo \cite{ref3}.

\begin{thebibliography}{99}
\setlength{\itemsep}{0cm}

\bibitem{ref1}
  referencia I
    
\bibitem{ref2}
 referencia II  

\bibitem{ref3}
 Qualquer Coisa
   
 \end{thebibliography}


\newpage



\newpage



\section{List of participants}

\begin{flushleft}
\begin{itemize}
\item Tobias Frederico, ITA, Brazil ({\tt tobias@ita.br})
\item Pieter Maris, Iowa State University, USA ({\tt pmaris@iastate.edu})
\item Orlando Oliveira, University of Coimbra, Portugal ({\tt orlando@uc.pt})
\item Wayne Polyzou, University of Iowa, USA ({\tt polyzou@uiowa.edu})
\end{itemize}
\end{flushleft}

\end{document}

  
  

@article{Aiello:2015jgc,
      author         = "Aiello, Gordon and Polyzou, Wayne",
      title          = "{Scattering asymptotic conditions in Euclidean
                        relativistic quantum theory}",
      journal        = "Phys. Rev.",
      volume         = "D93",
      year           = "2016",
      number         = "5",
      pages          = "056003",
      doi            = "10.1103/PhysRevD.93.056003",
      eprint         = "1512.03651",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-th",
      SLACcitation   = "%%CITATION = ARXIV:1512.03651;%%"
}


@Article{Osterwalder:1973dx,
     author    = "Osterwalder, Konrad and Schrader, Robert",
     title     = "{Axioms for Euclidean Green's functions}",
     journal   = "Commun. Math. Phys.",
     volume    = "31",
     year      = "1973",
     pages     = "83-112",
     doi       = "10.1007/BF01645738",
     SLACcitation  = "%%CITATION = CMPHA,31,83;%%"
}


@Article{Osterwalder:1974tc,
     author    = "Osterwalder, Konrad and Schrader, Robert",
     title     = "{Axioms for Euclidean Green's Functions. 2}",
     journal   = "Commun. Math. Phys.",
     volume    = "42",
     year      = "1975",
     pages     = "281",
     doi       = "10.1007/BF01608978",
     SLACcitation  = "%%CITATION = CMPHA,42,281;%%"
}


@Book{Widder:1941,
  title = {The Laplace Transform},
  author = {D. V. Widder },
  journal = {},
  volume = {},
  number = {},
  pages = {272-275},
  numpages = {},
  year = {1941},
  month = {},
  publisher = {Dover}
}

@article{Schwinger:1958qau,
      author         = "Schwinger, J.",
      title          = "{ON THE EUCLIDEAN STRUCTURE OF RELATIVISTIC FIELD
                        THEORY}",
      journal        = "Proc. Nat. Acad. Sci.",
      volume         = "44",
      year           = "1958",
      number         = "9",
      pages          = "956-965",
      doi            = "10.1073/pnas.44.9.956",
      SLACcitation   = "%%CITATION = PNASA,44,956;%%"
}


@article{Kopp:2011vv,
      author         = "Kopp, Philip and Polyzou, Wayne",
      title          = "{A Euclidean formulation of relativistic quantum
                        mechanics}",
      journal        = "Phys. Rev.",
      volume         = "D85",
      year           = "2012",
      pages          = "016004",
      doi            = "10.1103/PhysRevD.85.016004",
      eprint         = "1106.4086",
      archivePrefix  = "arXiv",
      primaryClass   = "nucl-th",
      SLACcitation   = "%%CITATION = ARXIV:1106.4086;%%"
}

@Article{Chandler:1976,
     author    = "Chandler, C. and Gibson, A.",
     title     = "{}",
     journal   = "Indiana Journal of Mathematics.",
     volume    = "25",
     year      = "1976",
     pages     = "443",
}

</pre></body></html>