Content-Type: multipart/mixed; boundary="-------------0307250648871" This is a multi-part message in MIME format. ---------------0307250648871 Content-Type: text/plain; name="03-349.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-349.keywords" Constructive field theory, thermal field theory, KMS states ---------------0307250648871 Content-Type: application/x-tex; name="KMSv1.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="KMSv1.tex" \documentclass[11pt, leqno]{article} \input amssym.def \input amssym.tex %\input montrere.tex \include{showlabels} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Symbols ( = symbols.tex ) % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % This is symbols.tex % the symbols not available in plain TeX are constructed % by overprinting some characters \def\sun{{\hbox{$\odot$}}} \def\la{\mathrel{\mathchoice 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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\theequation{\thesection.\arabic{equation}} \def\init{\setcounter{equation}{0}} %THEOREMLIKE ENVIRONNEMENTS \newtheorem{thmchap}{Theorem} \newtheorem{theoreme}{Theorem }[section] \newtheorem{proposition}[theoreme]{Proposition} \newtheorem{lemma}[theoreme]{Lemma} \newtheorem{definition}[theoreme]{Definition} \newtheorem{corollary}[theoreme]{Corollary} \newtheorem{remarque}{Remark}[section] \newtheorem{remark}{Remark}[section] %DECLARATION POUR LA FAMILLE DE FONTES DESTINEE AUX %ENSEMBLES USUELS %\font\tenblack=msym10 \font\sevenblack=msym7 %\font\fiveblack=msym5 %\font\twelveblack=msym10 at 12pt %\newfam\blackfam %\textfont\blackfam=\twelveblack %\scriptfont\blackfam=\sevenblack %\scriptscriptfont\blackfam=\fiveblack %\def\blackboard{\fam\blackfam} \def\rr{\bbbr} \def\cc{\bbbc} \def\nn{\bbbn} \def\zz{\bbbz} \def\one{\bbbone} \def\dd{{\bf d}} \def\DD{{\bf D}} \def\cdd{\check{\bf d}} \def\cDD{\check{\bf D}} \def\fin{{\rm fin}} \def\cGamma{{\check\Gamma}} \def\cad{\check{\rm ad}} \def\t{{\langle t\rangle}} \def\tchi{{\tilde\chi}} %MATHEMATICAL SYMBOLS %norme japonaise \newcommand{\jnorm}[1]{\langle #1 \rangle} \newcommand{\pscal}[2]{\langle #1,#2 \rangle} %norme droite \newcommand{\norme}[1]{\|#1\|} \newcommand{\module}[1]{|#1|} %ABBREVIATIONS DE LETTRES ET SYMBOLES %\newcommand{\rem}[1]{{\it Remark #1: }} \def\q{\hbox{\frak q}} \def\pp{{\rm pp}} \def\ess{{\rm ess}} \def\sch{Schr\"odinger } \def\schrod{Schr\"odinger } \def\edp{partial differential equations } \def\pdo{pseudodifferential operator } \def\pdos{pseudodifferential operators } \def\ie{{i.e.,} } \def\fld{\rightarrow} \def\flg{\leftarrow} \def\tU{\tilde{U}} \def\ham{Hamiltonian } \def\e{{\rm e}} \def\D{{\cal D}} \def\i{{\rm i}} \def\d{{\rm d}} \def\12{\frac{1}{2}} \def\cinf{C^{\infty}} \def\proof{{\bf Proof. }} \def\slim{\hbox{\rm s-}\lim} \def\coinf{C_{0}^{\infty}} \def\intc{\frac{i}{2\pi}\int\partial_{\overline z}\tilde{\chi}(z)} 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\def\tsig{{\tilde \sigma}} \def\tcH{\cH} \def\tch{\ch} \def\tv{v} \def\tH{H} \def\th{h} \def\slimt{\slim_{t\to +\infty}} \def\gfh{\G(g^{t})} \def\cGam{{\check\Gamma}} \newcommand{\sur}[2] {\mathop{#1}\limits_{\smallsmile\atop{#2}}} \def\tphi{\phi} \def\tto{t\to+\infty} \def\ta{a} \def\tOmega{ \Omega} \def\cf{{\frak f}} \def\cq{{\frak q}} \def\cg{{\frak g}} \def\cD{{\cal D}} \def\cN{{\cal N}} \def\cV{{\cal V}} \def\cM{{\cal M}} \def\subsubset{\Subset} \def\rx{{\rm x}} \def\c{{\rm c}} \def\e{{\rm e}} \def\tcK{{\cal K}} \def\At{\frac{A}{t}} \def\Att{\frac{A_{t}}{t}} \def\tto{t\to+\infty} \def\pfi2{P(\varphi)_{2}} \def\cf{{\frak f}} %ABBREVIATIONS DE COMMANDES \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\bet}{\begin{theoreme}} \newcommand{\eet}{\end{theoreme}} \newcommand{\bel}{\begin{lemma}} \newcommand{\eel}{\end{lemma}} \newcommand{\bep}{\begin{proposition}} \newcommand{\eep}{\end{proposition}} \newcommand{\bear}[1]{\begin{array}{#1}} \newcommand{\ear}{\end{array}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % SIZE OF THE PAGE % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %LARGEUR ET HAUTEUR DU TEXTE \setlength{\textwidth}{16cm} \setlength{\textheight}{22cm} \setlength{\oddsidemargin}{0cm} \setlength{\topmargin}{-1cm} %\addtolength{\footskip}{1cm} % 1 INCH = 2.54 CM !!!! %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % END OF PERSONAL MACROS % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} %\def\j{{\frak i}} \def\j{{\rm j}} \def\bc{{\rm c}} \def\bp{{\rm p}} \def\bt{{\rm t}} \def\q{{\rm q}} \def\chbar{\overline{\ch}} \def\cO{{\cal O}} \def\cA{{\cal A}} \def\cF{{\cal F}} \def\stp{ stochastic process} \def\stps{stochastic processes} \def\cUbar{\overline{\cal U}} \def\stpos{stochastically positive} \def\kms{ KMS system} \def\Xbar{\overline{X}} \def\cBbar{{\overline \cB}} \def\cFbar{{\overline \cF}} \def\cRbar{{\overline \cR}} % % %new macros: hermitian form, ie, eg % \def\hf{{(\, . \, ,\, . \,)}} \def\eg{{e.g.\ }} \def\tq{{\sl q}} \def\maath{\mathsurround=0pt } \def\eqalign#1{\null\, \vcenter{\openup1\jot \maath \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil \crcr#1\crcr}}\,} % Set the beginning of a LaTeX document \title{Thermal Quantum Fields with Spatially Cut-off Interactions in 1+1 Space-time Dimensions\protect\footnotetext{AMS 1991 {\it{Subject Classification}}. 81T08, 82B21, 82 B31, 46L55} \protect\footnotetext{{\it{Key words and phrases}}. Constructive field theory, thermal field theory, KMS states. }} % Enter your title between \author{Christian G\'{e}rard\footnote{ christian.gerard@math.u-psud.fr, Universit\'e Paris Sud XI, F-91405 Orsay, France} \ and Christian D.\ J\"akel\footnote{ christian.jaekel@math.polytechnique.fr, \'Ecole Polytechnique, F-91128 Palaiseau, France}} % Enter your name between curly braces \date{June 2003} % Enter your date or \today between curly braces \maketitle \abstract{We construct interacting quantum fields in 1+1 space-time dimensions, representing char\--ged or neutral scalar bosons at positive temperature and zero chemical potential. Our work is based on prior work by Klein and Landau and H\o egh-Krohn. Generalized path space methods are used to add a spatially cut-off interaction to the free system, which is described in the Araki-Woods representation. It is shown that the interacting KMS state is normal w.r.t.\ the Araki-Woods representation. The observable algebra and the modular conjugation of the interacting system are shown to be identical to the ones of the free system and the interacting Liouvillean is described in terms of the free Liouvillean and the interaction.} \tableofcontents \section{Introduction} \init\label{introd} \noindent Thermal quantum field theory is supposed to unify both quantum statistical mechanics and elementary particle physics. The formulation of the general framework should be wide enough to allow a QED description of ordinary matter. It should also provide the necessary tools for the~QCD description of several experiments currently envisaged with the new Large Hadron Collider (LHC) at CERN. While the general theory of thermal quantum fields has made substantial progress in recent years, the actual construction of interacting models, which fit into the axiomatic setting, has not yet started (with the exception of the very early contributions by H\o egh-Krohn \cite{H-K} and Fr\"ohlich \cite{Fr2}). Let us briefly recall the formal description of charged scalar fields in physics. Examples of scalar particle-antiparticle pairs are the mesons $\pi^+$, $\pi^-$, $K^+$, $K^-$, or $K^0$, $\overline{K^0}$. (In the last case the `charge' is strangeness). One starts with the classical Lagrangian density \[ {\cal L} = (\p_\nu \varphi) (\p^\nu \varphi^*) - m^2 \varphi \varphi^* - {\lambda \over 4} (\varphi \varphi^*)^2 . \] Here $\varphi (t,x)$ is a complex scalar field over space--time. The Lagrangian density ${\cal L}(t,x)$ is invariant under the global gauge transformations $\varphi \mapsto {\rm e}^{\i \alpha} \varphi $, $\alpha \in \rr$. By Noether's theorem this invariance leads to a conserved current \[ j_\nu = i ( \varphi^* \partial_\nu \varphi - \varphi \partial_\nu \varphi^*), \qquad \nu = 0, \dots, 3, \] and to a conserved charge \[ q = \int {\rm d}^{3} x \, j_0 (t, x). \] The next step, according to the physics literature, is to setup real or imaginary time perturbation theory. The state of art of perturbative thermal field theory is covered in three recent books by Kapusta [K], Le Bellac [L-B] and Umezawa [U]. The authors concentrate on theoretical efforts to understand various hot quantum systems (e.g., ultra-relativistic heavy-ion collisions or the phase transitions in the very early universe) and various physical implications (e.g., spontaneous symmetry breaking and restoration, deconfinement phase transition). Constructive thermal field theory allows one to circumvent (at least in lower space-time dimensions) the severe problems (see, e.g., Steinmann \cite{St}) of thermal perturbation theory, which can otherwise only be removed partially by applying certain ``resummation schemes". A class of models representing scalar neutral bosons with polynomial interactions in 1+1 space--time dimensions was constructed by H\o egh-Krohn \cite{H-K} more than twenty years ago. As he could show, thermal equilibrium states for these models exist at all positive temperatures. For neutral particles, the particle density (and the energy density) adjust themselves to the given temperature; contrary to the non-relativistic case, a chemical potential adjusting the particle density can not be introduced, since the mass is no longer a conserved quantity. Shortly afterwards, several related results on the construction and properties of self-interacting thermal fields in 1+1 space--time dimensions were announced by Fr\"ohlich \cite{Fr2}. Our goal in this and a subsequent paper \cite{GeJ} was twofold: first we wanted to fully understand the neutral scalar thermal field with polynomial interaction as constructed by H\o egh-Krohn \cite{H-K}, with the aim to study thermal scattering theory, using the framework introduced by Bros and Buchholz in \cite{BB1}, \cite{BB2}. Secondly we wanted to generalize this construction to charged fields. This would allow us to study the system at different temperatures and chemical potentials, i.e., different charge densities. A possibility to change the charge density would put this model closer to non-relativistic models, where the mass is a conserved quantity, giving rise to the existence of a chemical potential. The construction of the full interacting thermal quantum field without cutoffs in \cite{GeJ} includes several of the original ideas of H\o egh-Krohn \cite{H-K}, but instead of starting from the interacting system in a box we start from the Araki-Woods representation for the free system in infinite volume. Using a general method developed by Klein and Landau \cite{KL1} to treat spatially cutoff perturbations of the free system in infinite volume, we can eliminate some cumbersome limiting procedures due to the introduction of boxes, when we remove the spatial cutoff. The present paper is devoted to the construction of neutral and charged thermal fields with {\em spatially cutoff} interactions in 1+1 space--time dimensions, using the method of Klein and Landau \cite{KL1}. Although the excellent paper \cite{KL1} is rather self contained, it did not include the discussion of examples. Twenty years ago it might have been evident for the experts in the field how to apply their method to thermal quantum fields, but we find it worthwhile to present this application in some detail. A difference between this paper and \cite{KL1} is the use of generalized path spaces as in \cite{K}, instead of stochastic processes. This compact formulation is convenient for our applications. In addition we prove several new results concerning the interacting KMS systems obtained by perturbations of path spaces. \vskip 1cm \goodbreak \subsection{Content of this paper} Our paper can be divided into several parts. The first part, presented in Section \ref{sec-1}, discusses the description of neutral and charged scalar fields in terms of operator algebras. Its application to Klein-Gordon fields is discussed in Section \ref{sec2}. As usual the starting point is a real symplectic space $(X, \sigma)$, which allows the construction of the Weyl algebra $\fW(X, \sigma)$. The next step is to introduce on $(X, \sigma)$ a K\"{a}hler structure, \ie a compatible Hermitian structure. For charged scalar fields, the symplectic space $(X, \sigma)$ possesses also a canonical `charge' complex structure $\j$ and a `charge' sesquilinear form $\cq$, such that $\sigma={\rm Im}\cq$. The maps $X\ni x\mapsto \e^{ \, \j \alpha}x$ for $\alpha\in \rr$ generate the {\em gauge transformations}. Given a regular CCR representation, complex quantum fields are defined. This leads to the notion of a {\em charged K\"{a}hler structure}, corresponding to the introduction of another complex structure $\i$ and of the {\em charge operator} $q$, relating the two complex structures. Finally the notion of {\em charge conjugation} is discussed in this abstract framework. For Klein-Gordon fields, a conjugation inducing charge-time reflections is used to distinguish an appropriate abelian sub-algebra of the Weyl algebra to which the interaction terms considered later on will be affiliated. Section \ref{stochas} recalls the characterization of a thermal equilibrium state by the KMS property. The GNS representation associated to a KMS state has a number of interesting properties which are briefly recalled. For instance, the GNS vector is cyclic and separating for the field algebra $\cF$ (in our case the weak closure of the Weyl algebra in the GNS representation), and therefore one can always go over to the weak closure of the relevant operator algebras, and we will do so in the sequel. Since a KMS state is invariant under time translations, a Liouvillean implementing the time evolution is always available. As has been shown by Araki, the KMS condition allows us to introduce Euclidean Green's functions. The notion of {\em stochastically positive KMS systems} due to Klein and Landau is presented. This notion rests on the introduction of a distinguished abelian subalgebra $\cU$ of the field algebra $\cF$. In physics, this algebra is the algebra generated by the time-zero fields. It is also shown that stochastically positive KMS systems are invariant under a {\em time reversal} transformation. In Section \ref{quasi} we recall the notion of a quasi-free KMS system associated to a positive selfadjoint operator acting on the one-particle space. The GNS representation for a quasi-free KMS system has been analyzed by Araki and Woods. We briefly recall this framework and its connection to the Fock representation in a modern notation. It is shown that the field algebra~$\cF$ is generated by the time-translates of the abelian algebra $\cU$. The observable algebra, consisting of elements of the field algebra which are invariant under gauge transformations, is introduced. In Subsection \ref{stocpos} it is shown that the KMS system for the (quasi-)free charged thermal field is indeed stochastically positive, if the chemical potential vanishes. However, if the chemical potential is non-zero, then the charge distinguishes a time direction, and consequently, the system is no longer invariant under time reversal. Thus it fails to be stochastically positive too, as we show in Subsection \ref{poskg}. Following Klein and Landau, a cyclicity property of the Araki-Woods representation, which will imply the so-called {\em Markov property} for the free system later on, is shown. The Markov property has the consequence that the physical Hilbert space can naturally be considered as an $L^{2}$-space. Section \ref{path} recalls the notion of a {\em generalized path space}, both for the 0-temperature case and the case of positive temperature. We follow here \cite{K}, \cite{KL1}. Although the $0$-temperature case is not needed in this paper, it will be useful later on in \cite{GeJ}. A generalized path space consists of a probability space~$(Q, \Sigma, \mu)$, a distinguished $\sigma$-algebra $\Sigma_0$, a one-parameter group $t \mapsto U(t)$ and a reflection $R$. We recall the definition of {\em OS-positivity} and the {\em Markov property} for both cases. Section \ref{secst1} is devoted to a discussion of the Osterwalder-Schrader {\em reconstruction theorem} in the framework of generalized path spaces. This reconstruction theorem associates to a $\beta$-periodic, OS-positive path space a stochastically positive $\beta$-KMS system. In Section \ref{secst3} we recall from \cite{KL1} how to deal with of perturbations, which are given in terms of {\em Feynman-Kac-Nelson kernels}. The main examples of FKN kernels are those obtained from a selfadjoint operator $V$ on the physical Hilbert space $\cH$, where $V$ is affiliated to $\cU$. We show that for a class of perturbations $V$ considered in \cite{KL1}, the perturbed Hilbert space can be canonically identified with the free Hilbert space in such a way that the interacting algebras $\cR$, $\cU$ and the modular conjugation $J$ coincide with the free ones. Moreover, we prove that the perturbed Liouvillean $L_{V}$ is equal to $\overline{\overline{L+V}-JVJ}$, if $L$ is the free Liouvillean. Here $\overline{H}$ denotes the closure of a linear operator $H$. Finally we show that the Markov property of a generalized path space is not destroyed by the perturbations associated to FKN kernels. In Section \ref{sec2} we apply the framework of Sections \ref{sec-1} and \ref{quasi} to charged and neutral Klein-Gordon fields at positive temperature. The case of the neutral Klein-Gordon field is well known and reviewed only for completeness. We give more details on the charged Klein-Gordon field which provides an example of a charge symmetric K\"{a}hler structure. We also compare our setup with the one used in physics textbooks. Using the results of Section~\ref{quasi}, we present the quasi-free KMS system describing a free charged or neutral Klein-Gordon field at positive temperature. Note that the conjugation used in the definition of the abelian algebra~$\cU$ corresponds to time reversal in the neutral case and to the composition of time-reversal and charge conjugation in the charged case. We show that the KMS system for the charged Klein-Gordon field is not stochastically positive, if the chemical potential is unequal to zero. The physical reason is that the dynamics of charged particles is only invariant under the combination of time reversal and charge conjugation. A non-zero chemical potential introduces a disymmetry between particles of positive and negative charge and hence breaks time reversal invariance, which itself is a property shared by all stochastically positive KMS systems. In Section \ref{kgint} we consider Klein-Gordon fields at positive temperature with spatially cutoff interactions in $1+1$ space-time dimensions. In the neutral case we will treat the $P(\phi)_{2}$ and the $\e^{\alpha\phi}\!\:_{2}$ interactions (the later being also known as the {\em H\o egh-Krohn model}\/). In the charged case we treat the (gauge invariant) $P(\overline{\varphi}\varphi)_{2}$ interaction. The UV divergences of the interactions are eliminated by Wick ordering, which is discussed in some details in Subsections \ref{int} and \ref{wickwick}. As it turns out, the leading order in the UV divergences is independent of the temperature. Thus it is a matter of convenience whether one uses thermal Wick ordering or Wick ordering w.r.t.\ the vacuum state. The $L^{p}$-properties of the interactions needed to apply the abstract results of Section \ref{secst3} are shown in Subsections \ref{int.sub1}, \ref{int.sub11} and \ref{chargedp}. Finally, the main results of this paper, namely the construction and description of a KMS system representing a Klein-Gordon field at positive temperature with spatially cutoff interactions, is given in Subsection \ref{mainres}. In a forthcoming paper we will consider the translation invariant $P(\phi)_{2}$ model at positive temperature. Following again ideas of H\o egh-Krohn \cite{H-K}, Nelson symmetry will be used to establish the existence of the model in the thermodynamic limit. %Although one might argue that infrared problems are less severe in higher space-time dimensions it is %difficult to establish such a theorem; %a successful treatment of the %infrared divergences using Nelson symmetry %has only been achieved in 1+1 space-time dimensions. %It would be very interesting to extend our results to KMS states for non-zero chemical potential. \medskip \noindent {\bf Acknowledgments}. The second named author was supported under the FP5 TMR program of the European Union by the Marie Curie fellowship HPMF-CT-2000-00881. Both authors benefited from the IHP network HPRN-CT-2002-00277 of the European Union. \vfill \section{Real and complex quantum fields} \init \label{sec-1} In this section we present real and complex quantum fields in an abstract framework. Usually in the physics literature complex quantum fields are described in the case of Klein-Gordon fields. Although the results of this section are probably known, we have not found them in the literature. \subsection{Notation} \label{notat} Let $X$ be a real vector space. If $X$ is equipped with a complex structure $\i$, then we will denote by $(X, \i)$ the complex vector space $X$. If $(X, \i)$ is equipped with a hermitian form $\hf$, then we will denote by $(X, \i , \hf)$ the Hermitian space $X$. If it is clear from the context which complex or Hermitian structure is used, $(X, \i)$ or $(X, \i , \hf)$ will simply be denoted by $X$. As a rule the complex structure of a Hermitian space $X$ will be denoted by the letter~$\i$. Sometimes another `charge' complex structure appears; it will be denoted by the letter~$\j$. %----------------------------------------------------------------------------------------- %----------------------------------------------------------------------------------------- %----------------------------------------------------------------------------------------- \subsection{Real fields} \init \label{sec0} We start by recalling the formalism of real quantum fields. \vskip .3cm \noindent {\bf CCR Algebra} \vskip .2cm \noindent Let $(X, \sigma)$ be a real symplectic space. Let $\fW(X, \sigma)$ be the (uniquely determined) $C^*$-algebra generated by nonzero elements $W(x)$, $x \in X$, satisfying \[ \begin{array}{l}W (x_1)W (x_2)=\e^{-\i\sigma(x_1,x_2)/2} W (x_1+x_2),\\[3mm] W ^*(x)=W (-x),\ \ \ \ \ W (0)=\one . \\[3mm] \end{array} \] $\fW(X, \sigma)$ is called the {\em Weyl algebra} associated to $(X, \sigma)$. \vskip .3cm \noindent {\bf Regular representations} \vskip .2cm \noindent Let $\cH$ be a Hilbert space. We recall that a representation \[ \pi \colon \fW(X, \sigma)\ni W (x) \mapsto W_{\pi}(x)\in {\cal U}(\cH) \] is called a {\em regular CCR representation} if \[ \begin{array}{l} t\mapsto W_{\pi}(tx)\hbox{ is strongly continuous for any }x\in X. \end{array} \] One can then define {\em field operators} \[ \phi_{\pi}(x):= -\i\frac{\d}{\d t}W_\pi(tx)\Big|_{t=0},\: x\in X, \] which satisfy in the sense of quadratic forms on $\cD(\phi_{\pi}(x_{1}))\cap \cD(\phi_{\pi}(x_{2}))$ the commutation relations \beq [\phi_\pi(x_1),\phi_\pi(x_2)]=\i\sigma(x_1,x_2),\: x_{1}, \:x_{2}\in X. \label{e0.3} \eeq \vskip .3cm \goodbreak \noindent {\bf K\"{a}hler structures} \vskip .2cm \noindent Let $(X, \sigma)$ be a real symplectic space and $\i$ a complex structure on $X$. The space $(X, \i, \sigma)$ is called a {\em K\"{a}hler space} if \[ \sigma( \i x_{1}, x_{2})= -\sigma(x_{1}, \i x_{2}) \hbox{ and } \sigma(x, \i x)\hbox{ is positive definite}. \] If $(X, \i, \sigma)$ is a K\"{a}hler space, then $(X, \i, \hf )$ is a Hermitian space for \[ (x_{1}, x_{2}):= \sigma( x_{1}, \i x_{2})+ \i \sigma( x_{1}, x_{2}). \] The typical example of a K\"{a}hler space is a Hermitian space $(X, \i, \hf )$ with its natural complex structure and symplectic form $\sigma= {\rm Im} \hf$. \vskip .3cm \noindent {\bf Creation and annihilation operators} \vskip .2cm \noindent If $\pi$ is a regular CCR representation of the Weyl algebra $\fW(X, \sigma)$, and $(X, \sigma)$ is equipped with a K\"{a}hler structure, then the {\em creation} and {\em annihilation operators} are defined as follows: \[ %\begin{array}{l} a_\pi^*(x):=\frac{1}{\sqrt2} \bigl(\phi_\pi(x)-\i\phi_\pi(\i x) \bigr),\:%\\[3mm] a_\pi(x):=\frac{1}{\sqrt2}\bigl(\phi_\pi(x)+\i\phi_\pi(\i x)\bigr). %\end{array} \] Clearly, \[ \phi_\pi(x)=\frac{1}{\sqrt2} \bigl(a_\pi^*(x)+a_\pi(x) \bigr),\: x\in X . \] The operators $a_\pi^*(x)$ and $a_\pi(x)$ with domain $\cD(\phi_\pi(x))\cap\cD(\phi_\pi(\i x))$ are closed and satisfy canonical commutation relations in the sense of quadratic forms: \[ %\begin{array}{l} [a_\pi(x_1), a_\pi^{*}(x_{2})]= (x_1 , x_2)\one,\:%\\[3mm] [a_\pi(x_2),a_\pi(x_1)]=[a_\pi^{*}(x_2),a^*(x_1)]= 0. %\end{array} \] %----------------------------------------------------------------------------------------- %----------------------------------------------------------------------------------------- %----------------------------------------------------------------------------------------- \subsection{Complex fields} \label{sec1} Let $(X, \j)$ be a complex vector space. Let us assume that $X$ is equipped with a sesquilinear, symmetric non-degenerate form $\cq$. If $a\in L(X)$, we say that $a$ is isometric (resp.\ symmetric, skew-symmetric) if $[a, \j]=0$ and $\cq(ax_{1}, ax_{2})= \cq(x_{1}, x_{2})$ (resp.\ $ \cq(ax_{1}, x_{2})= \cq(x_{1}, ax_{2})$, $ \cq(ax_{1}, x_{2})= - \cq(x_{1}, ax_{2})$). Clearly $(X, {\rm Im} \cq)$ is a real symplectic space. The quadratic form~$ \cq$ is called the {\em charge quadratic form}. \vskip .3cm \noindent {\bf Gauge transformations} \vskip .2cm \noindent The maps $X\ni x\mapsto \e^{\, \j \alpha}x\in X$ for $\alpha\in \rr$ are called {\em gauge transformations}. They are symplectic on $(X, {\rm Im \cq})$ and isometric on $(X, \cq)$. We have \beq \cq(x_{1}, x_{2})= {\rm Im \, } \cq(x_{1}, \j x_{2})+ \i {\rm Im} \cq(x_{1}, x_{2}). \label{e0.1} \eeq \vskip .3cm \noindent {\bf Complex fields} \vskip .2cm \noindent Let now $\pi$ % \[ % \begin{array}{rl} %W_{\pi}: (X, {\rm Im} \cq)\to {\cal U}(\cH),\\[3mm] %x\mapsto W_{\pi}(x) %\end{array} %\] be a regular CCR representation of $\fW(X, {\rm Im} \cq )$ on a Hilbert space $\cH$ and let $\phi_{\pi}(x)$ be the associated field. \goodbreak Using the complex structure $\j$, we can define the {\em complex fields} \[ \begin{array}{l} \varphi_\pi^*(x):=\frac{1}{\sqrt2} \bigl(\phi_\pi(x)-\i\phi_\pi(\j x) \bigr),\\[3mm] \varphi_\pi(x):=\frac{1}{\sqrt2} \bigl(\phi_\pi(x)+\i\phi_\pi(\j x) \bigr), \end{array} \] with domains $\cD(\phi_{\pi}(x))\cap \cD(\phi_{\pi}(\j x))$. The maps $X\ni x\mapsto \varphi_{\pi}^{*}(x)\: (\hbox{resp.\ } x\mapsto \varphi_{\pi}(x))$ are $\j$-linear (resp.\ $\j$-antilinear). \begin{lemma} The operators $\varphi^{\sharp}_{\pi}(x)$ are closed. In the sense of quadratic forms on $\cD(\phi_{\pi}(x))\cap \cD(\phi_{\pi}(\j x))$ they satisfy the commutation relations \[ %\begin{array}{l} [\varphi_\pi(x_1), \varphi_\pi^{*}(x_{2})]= \cq(x_1, x_2)\one,\: %\\[3mm] [\varphi_\pi(x_1),\varphi_\pi(x_2)]=[\varphi_\pi^{*}(x_1),\varphi^*(x_2)]= 0. %\end{array} \] \end{lemma} \proof The commutation relations are easily deduced from (\ref{e0.3}). Let $u\in \cD(\phi_{\pi}(x))\cap \cD(\phi_{\pi}(\j x))$. To prove that $\varphi^{\sharp}_{\pi}(x)$ is closed, we write \[ 2\|\varphi_{\pi}(x)u\|^{2}= \|\phi_{\pi}(x)u\|^{2}+ \|\phi_{\pi}(\j x)u\|^{2}- \cq(x, \j x)\|u\|^{2}. \] This easily implies that $\varphi_{\pi}(x)$ is closed. The case of $\varphi^{*}_{\pi}(x)$ is treated similarly. \qed %----------------------------------------------------------------------------------------- %----------------------------------------------------------------------------------------- %----------------------------------------------------------------------------------------- \subsection{Charge operator} \label{sec1.2} \begin{definition} Let $(X, \j , \cq)$ be as in Subsection \ref{sec1} and $\i$ another complex structure on $X$. Then $(X, \j, \i, \cq)$ is called a {\em charged K\"{a}hler space} if $[\i,\j]=0$ and $(X, \i, {\rm Im} \cq)$ is a K\"{a}hler space. \label{0.1} \end{definition} Let $(X, \j, \i, \cq)$ be a charged K\"{a}hler space. Then $\i$ is antisymmetric for $ \cq$, \ie $\cq( x_{1}, \i x_{2})= - \cq( \i x_{1}, x_{2})$, and $\j$ is antisymmetric for $\hf$. We can introduce the {\em charge operator}\/: \[ \q := -\i \j. \] Note that $[\q, \i]= [\q,\j]=0$, $\q^{2}=1$ and that $\q$ is symmetric and isometric both for $ \cq$ and~$\hf$. Since $\i =\j \q$ we have $\e^{\, \j \alpha}= \e^{\i \alpha \q}$ and the gauge transformations $x \mapsto \e^{ \, \j \alpha}x$, $\alpha\in \rr$, form a unitary group on $(X, \i, \hf )$ with infinitesimal generator $\q$. The typical example of a charged K\"{a}hler space is a Hermitian space $(X, \i, \hf)$ with a distinguished symmetric operator $\q$ such that $\q^{2}=1$. Let us denote by $X^{\pm}:= {\rm Ker}(\q \mp \one)$ the spaces of positive (resp.\ negative) charge and by $x^{\pm}$ the orthogonal projection of $x\in X$ onto~$X^{\pm}$. If we set $ \cq(x_{1}, x_{2})= (x_{1}^{+}, x_{2}^{+})-(x_{2}^{-}, x_{1}^{-})$, then $(X, \i \q, \i, \cq)$ is a charged K\"{a}hler space. % %\vskip .2cm %\noindent %{\bf vergleiche} $(X, \i, \j, \cq)$ und $(X, \i \q, \i, \cq)$! %\vskip .2cm % Note that $X^{+}$ or $X^{-}$ may be equal to $\{0\}$. Using the fact that $\q$ is symmetric for $\hf$ and $ \cq$, we see that the spaces $X^{\pm}$ are orthogonal both for $\hf$ and $ \cq$. If we set $x^{\pm}= \12(x\pm \q x)$, then the map \[ \matrix {U\colon & X & \to & X^{+}\oplus X^{-} \cr & x & \mapsto & x^{+}\oplus x^{-} \cr} \] is unitary from $(X, \i, \hf )$ to $(X^{+}, \i , \hf )\oplus (X^{-}, \i , \hf )$ and isometric from $(X, \j , \cq)$ to~$(X^{+}, \i, \hf)\oplus (X^{-}, -\i, -\overline{\hf})$. If $\pi \colon \fW (X, {\rm Im} \cq)\to {\cal U}(\cH)$ is a regular CCR representation on a Hilbert space $\cH$, then we can introduce, just as in Subsection \ref{sec0}, {\em creation} and {\em annihilation operators} \[ %\begin{array}{l} a_\pi^*(x):=\frac{1}{\sqrt2}\bigl(\phi_\pi(x)-\i\phi_\pi(\i x)\bigr),\:%\\[3mm] a_\pi(x):=\frac{1}{\sqrt2}\bigl(\phi_\pi(x)+\i\phi_\pi(\i x) \bigr), %\end{array} \] with domains $\cD(\phi_{\pi}(x))\cap \cD(\phi_{\pi}(\i x))$. The maps $X\ni x\mapsto a_{\pi}^{*}(x)\: (\hbox{resp.\ }a_{\pi}(x))$ are $\i$-linear (resp.\ $\i$-antilinear). If $x= x^{+}+ x^{-}$, with $x^{\pm} \in X^{\pm}$, then \[ \varphi_{\pi}(x)= a_{\pi}(x^{+})+ a^{*}_{\pi}(x^{-}) \hbox{ \ and \ } \varphi_{\pi}^{*}(x)= a_{\pi}^{*}(x^{+})+ a_{\pi}(x^{-}). \] % Note that this is consistent with fact that the maps $X\ni x\mapsto \varphi_{\pi}^{*}(x)\: (\hbox{resp.\ } x\mapsto \varphi_{\pi}(x))$ are $\j$-linear (resp.\ $\j$-antilinear). %----------------------------------------------------------------------------------------- %----------------------------------------------------------------------------------------- %----------------------------------------------------------------------------------------- \subsection{Charge conjugation} \label{sec1.3} Let $(X, \j,\i, \cq)$ be a charged K\"{a}hler space. Assume that there exists some ${\rm c}\in L(X)$ such that \beq {\rm c}^{2}=\one, \: {\rm c}\i =\i \bc , \: \bc \q =- \q \bc , \: (x_{1}, \bc x_{2})= (\bc x_{1},x_{2}),\: x_{1}, x_{2}\in X. \label{e0.4} \eeq I.e., $\bc$ is a symmetric involution for $\hf$, which anticommutes with the charge operator~$\q$. An operator $\bc$ satisfying (\ref{e0.4}) is called a {\em charge conjugation}. Charge conjugations exist in charge-symmetric quantum field theories. A charged K\"{a}hler space $(X, \j, \i, \cq, \bc)$ equipped with a charge conjugation $\bc$ will be called a {\em charge-symmetric K\"{a}hler space}. It follows from (\ref{e0.4}) that $ \cq(x_{1}, \bc x_{2})=- \cq(\bc x_{1}, x_{2})$, \ie $\bc$ is antisymmetric for $ \cq$. Since $\bc \q =- \q \bc$, we see that $\bc$ is a unitary map from $(X^{-}, \i, \hf )$ to $(X^{+}, \i, \hf)$. %------------------------------------------------------------------- %------------------------------------------------------------------- %------------------------------------------------------------------- \section{Stochastically positive KMS systems} \label{stochas} \init In this section we recall the notion of a {\em stochastically positive KMS system} due to Klein and Landau \cite{KL1}. We prove that stochastically positive KMS systems are invariant under time-reversal. \subsection{KMS systems} Let $\cR$ be a $C^{*}$-algebra and $\{\tau_{t}\}_{t\in \rr}$ a group of $^{*}$-automorphisms of $\cR$. Let $\omega$ be a $(\tau, \beta)$-{\em KMS state} on $\cR$, \ie a state such that for each $A,B\in \cR$ there exists a function $F_{A,B}(z)$ holomorphic in the strip $\{z\in \cc\mid 0<{\rm Im}z<\beta\}$ and continuous on its closure such that \[ F_{A, B}(t)= \omega(A\tau_{t}(B)),\:F_{A, B}(t+\i \beta)= \omega(\tau_{t}(B)A),\: t\in \rr. \] A triple $(\cR, \tau, \omega)$ such that $\omega$ is a $(\tau, \beta)$-KMS state is called a $\beta$-{\em KMS system}. Let us now recall some standard facts about KMS systems. By the GNS construction, one associates to $(\cR, \tau, \omega)$ a Hilbert space $\cH_{\omega}$, a representation $\pi_{\omega}$ of $\cR$ on $\cH_{\omega}$, a unit vector~$\Omega_{\omega}$, cyclic for $\pi_{\omega}$, and a strongly continuous unitary group $\{\e^{-\i tL}\}_{t\in \rr}$ such that \[ \omega(A)= (\Omega_{\omega}, \pi_{\omega} (A)\Omega_{\omega}),\:\pi_{\omega}(\tau_{t}(A))= \e^{\i t L}\pi_{\omega}(A)\e^{-\i tL},\:L\Omega_{\omega}=0. \] The KMS condition implies that $\Omega_{\omega}$ is separating for the von Neumann algebra $\pi_\omega (\cR)''$, \ie $A \Omega _{\omega} = 0 \Rightarrow A=0$ for $A \in \pi_\omega(\cR)''$. Consequently, the image of $\cR$ under $\pi_{\omega}$ is isomorphic to~$\cR$; it will therefore not be distinguished from $\cR$. Moreover, we will identify an element $A$ of~$\cR$ with its image $\pi_{\omega}(A)$. The selfadjoint operator $L$ is called the {\em Liouvillean} associated to the KMS system $(\cR, \tau, \omega)$. It is the unique selfadjoint operator whose associated unitary group generates the dynamics~$\tau$ and such that $L\Omega_{\omega}=0$ (see \eg \cite[Prop. 2.14]{DJP}). \begin{proposition} \label{propliou} Let $\cR_{1}\subset \cR$ be the set of $A\in \cR$ such that $\tau \colon t\mapsto \tau_{t}(A)$ is $C^{1}$ for the strong topology on $\cB(\cH_{\omega})$. Then $\cR_{1}\Omega_{\omega}\subset \cD(L)$ %, $\cR_{1}\Omega_{\omega}$ is a core for $L$. %\vskip .3cm %\halign{ \indent # \hfil & \vtop { \parindent =0pt \hsize=34.8em % \strut # \strut} \cr %{\rm (i)} & $\cR_{1}\Omega_{\omega}\in \cD(L)$, $\cR_{1}\Omega_{\omega}$ %is a core for $L$, %\cr %{\rm (ii)} & if $A\in \cR$, $A\Omega_{\omega}\in \cD(L)$ iff $A\in %\cR_{1}$. %\cr} \end{proposition} \proof Note first that $A\in \cR_{1}$ iff $A$ is of class $C^{1}(L)$ (see \cite[Def.\ 6.2.2]{ABG}). % Let us first prove {\rm (i)}. Clearly $\cR_{1}$ is dense in $\cR$ for the strong operator topology. In fact, if $A\in \cR$, then the strong integral $A_{\epsilon}=\epsilon^{-1}\int_{0}^{\epsilon}\tau_{t}(A)\d t$ belongs to $\cR_{1}$ and converges strongly to $A$ when $\epsilon\to 0$. Since $\Omega_{\omega}$ is cyclic for $\cR$, this implies that $\cR_{1}\Omega_{\omega}$ is dense in $\cH_{\omega}$. Moreover, since $L\Omega_{\omega}=0$, we have $\e^{\i tL}\cR_{1}\Omega_{\omega}=\cR_{1}\Omega_{\omega}$ and $\cR_{1}\Omega_{\omega}\subset \cD(L)$. Thus Nelson's theorem implies that $\cR_{1}\Omega_{\omega}$ is a core for $L$. % Let us now prove {\rm (ii)}. We have already seen the $\Leftarrow$ % part, so it remains to prove the $\Rightarrow$ implication. If $A\in % \cR$ and $A\Omega_{\omega}\in \cD(L)$, the map $t\mapsto \e^{\i % tL}A\e^{-\i tL}\Omega_{\omega}$ is $C^{1}$. For $C\in\cR'$, we see % then % that the map $t\mapsto \e^{\i tL}A\e^{-\i tL}C\Omega_{\omega}$ is % $C^{1}$. % Let us denote by $\cR_{1}'$ the subspace of $C^{1}$ vectors in $\cR'$ for % $\tau$. Then arguing as in {\rm (i)}, we see that % $\cR_{1}'\Omega_{\omega}$ is a core for $L$. Using then \cite[??]{ABG}, % this implies that the quadratic form $[L, A]$, defined on % $\cR_{1}'\Omega_{\omega}$ is bounded for the topology of % $\cH_{\omega}$. Since $\cR_{1}'\Omega_{\omega}$ is a core for $L$, % this implies that the quadratic form $[L, A]$ defined on $\cD(L)$ is % bounded for the topology of $\cH_{\omega}$, and hence that $A\in % % \cR_{1}$ (see \cite{thm.??}{ABG}). \qed \vskip .3cm \noindent {\bf Euclidean Green's functions} \vskip .2cm \noindent Let \beq \label{inbeta} I_{\beta}^{n+}:=\{(z_{1},\dots, z_{n})\in \cc^n \:| \:{\rm Im}z_{j}<{\rm Im}z_{j+1}, \: {\rm Im}z_{n}- {\rm Im}z_{1}<\beta\}. \eeq It follows from a result of Araki \cite{Ar1, Ar2} that, for $A_{1}, \dots, A_{n}\in \cR$, the Green's function \[ G(t_{1}, \dots, t_{n};A_{1},\dots, A_{n}): =\omega \bigl(\prod_{1}^{n}\tau_{t_{i}}(A_{i}) \bigr) \] extends to an holomorphic function in $I_{\beta}^{n+}$, continuous on $\overline{I_{\beta}^{n+}}$. In particular, one can uniquely define the {\em Euclidean Green's functions} \[ \EG(s_{1}, \dots, s_{n};A_{1}, \dots, A_{n}):=G(\i s_{1}, \dots, \i s_{n}; A_{1}, \dots, A_{n}) \] for all $(s_{1}, \dots, s_{n})$ such that $s_{1}\leq \cdots\leq s_{n}$ and $s_{n}-s_{1}\leq \beta$. The correct way to view such an n-tuple $(s_{1}, \dots, s_{n})$ is as an n-tuple of points on the circle of length $\beta$, {\em ordered counter-clockwise}. %------------------------------------------------------------------- %------------------------------------------------------------------- %------------------------------------------------------------------- \subsection{Stochastically positive KMS systems} In \cite{KL1} Klein and Landau introduced a class of KMS systems which they called {\em stochastically positive KMS systems}. To a stochastically positive KMS system one can associate a (unique up to equivalence) {\em generalized path space} $(Q, \Sigma, \Sigma_0, U(t), R, \mu)$ (see Section 5) %\ref{secst5}) which has some special properties, the most important being the $\beta$-periodicity in $t$ and the {\em Osterwalder-Schrader {\rm (OS)}-positivity}. Conversely Klein and Landau have shown in \cite{KL1} that to a generalized path space satisfying the properties in Definition 5.1 %\ref{defstp}, one can associate a (unique up to unitary equivalence) stochastically positive KMS system. This is an example of a {\em reconstruction theorem}; similar results are well-known in Euclidean QFT. A reconstruction theorem allowing to go from Euclidean Green's functions to a KMS system has recently been proved in a general context by Birke and Fr\"ohlich in \cite{BF}. The advantage of the Klein and Landau formalism is that it is relatively easy to perturb the stochastic process associated to a KMS system, using functional integral methods. \begin{definition} \label{defstocpos} Let $(\cR, \tau, \omega)$ be a KMS system and $\fU\subset \cR$ an abelian $^{*}$-subalgebra. The KMS system $(\cR, \fU, \tau , \omega)$ is called {\em stochastically positive} if \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & the $C^{*}$-algebra generated by $\bigcup_{t\in \rr}\tau_{t}(\fU)$ is equal to $\cR$; \cr {\rm (ii)} & the Euclidean Green's functions $\EG(s_{1}, \dots, s_{n}; A_{1}, \dots, A_{n})$ are positive for all $A_{1}, \dots, A_{n}\in \fU^{+}=\{A\in \fU\mid A\geq 0\}$ and for all $(s_{1}, \dots, s_{n})$ such that $s_{1}\leq \cdots\leq s_{n}$ and $s_{n}-s_{1}\leq \beta$. \cr} \end{definition} It is often more convenient to consider instead of the $C^{*}$-algebras $\cR$ and $\fU$ their weak closures in the GNS representation, which we denote by $\cRbar$ and $\fUbar$. We denote by $\overline{\tau}$ the group $\{\overline{\tau}_{t}\}_{t \in \rr} $ of $^{*}$-automorphisms of $\cRbar$ defined by $\overline{\tau}_{t}(A) := \e^{\i tL}A\e^{-i tL}$. The state $\omega$ extends to $\cRbar$ by setting $\overline{\omega}(A):= (\Omega_{\omega}, \pi_{\omega}(A)\Omega_{\omega})$. The following fact has been shown in \cite[Prop. 3.4]{KL1}. \begin{proposition} \label{vonneumann} Let $(\cR, \fU, \tau, \omega)$ be a stochastically positive KMS system. Then $(\cRbar, \fUbar, \overline{\tau}, \overline{\omega})$ is also a stochastically positive KMS system (in the $W^*$-sense). I.e., \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & the $W^{*}$-algebra generated by $\bigcup_{t\in \rr}\tau_{t}(\fUbar)$ is equal to $\cRbar$; \cr {\rm (ii)} & the Euclidean Green's functions $\EG(s_{1}, \dots, s_{n}; A_{1}, \dots, A_{n})$ are positive for all $A_{1}, \dots, A_{n}\in \fUbar^+$ and for all n-tuples $(s_{1}, \dots, s_{n})$ such that $s_{1}\leq \cdots\leq s_{n}$ and $s_{n}-s_{1}\leq \beta$. \cr} \end{proposition} Now we show that stochastically positive KMS systems are invariant under {\em time reversal}, a fact that is well known for $0$-temperature field theories (see for example \cite{Si}). \begin{proposition} \label{timereversal} Let $(\cR, \fU, \tau, \omega)$ be a {\em stochastically positive} KMS system. Then there exists an anti-unitary involution $T$ on $\cH_{\omega}$ such that \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & $T\cRbar T^{-1}= \cRbar $, $TAT^{-1}= A^{*}$ for $A\in \fUbar$; \cr {\rm (ii)} & $T\Omega_{\omega}= \Omega_{\omega}$, $T \: \overline{\tau}_{t}(A)= \overline{\tau}_{-t}(A)T $ for $A\in \overline{\cR}, \: t\in \rr$. \cr} \end{proposition} From the properties of $T$ we see that $T$ implements the {\em time reversal transformation}. \medskip \noindent \proof Let $A_{1},A_{2}\in \fU$. The map $z\mapsto \omega(A_{1}\tau_{t}(A_{2}))_{|t = iz}$ is holomorphic in $\{0<{\rm Re}z<\beta\}$. By stochastic positivity it is real on $\{{\rm Im}z=0\}$ if $A_{i}=A_{i}^{*}$. The Schwarz's reflection principle implies \[ \omega \bigl(A_{1}\tau_{t}(A_{2})\bigr)_{|t = iz}= \overline{\omega(A_{1}\tau_{t}(A_{2}))_{|t = i \bar z}} \hbox{ \ for \ }A_{i}\in \fU, \:A_{i}= A_{i}^{*}. \] For $z=-\i t$ this yields \beq \label{timerev} \omega\bigl(A_{1}\tau_{t}(A_{2})\bigr)= \overline{\omega(A_{1}\tau_{-t}(A_{2}))}=\omega\bigl(\tau_{-t}(A_{2})A_{1}\bigr) \hbox{ \ for \ }A_{i}\in \fU, \:A_{i}= A_{i}^{*}. \eeq By $\cc$-linearity this identity extends to all $A_{i}\in \fU$. We can now define the antilinear operator \beq T \colon \sum_{j=1}^{n}\e^{\i t_{j}L}A_{j}\Omega_\omega \mapsto \sum_{j=1}^{n}\e^{-\i t_{j}L}A^{*}_{j}\Omega_\omega. \label{reversal} \eeq For $u=\sum_{j=1}^{n}\e^{\i t_{j}L}A_{j}\Omega_\omega$ identity (\ref{timerev}) implies \[ \begin{array}{rl} \|u\|^{2}=&\bigl(\sum_{j=1}^{n}\e^{\i t_{j}L}A_{j}\Omega_\omega, \sum_{k=1}^{n}\e^{\i t_{k}L}A_{k}\Omega_\omega \bigr)\\ =& \sum_{j,k}\bigl(\Omega_\omega, A_{j}^{*}\e^{\i (t_{k}-t_{j})L}A_{k}\Omega_\omega\bigr)= \sum_{j,k}\omega\bigl(A_{j}^{*}\tau_{t_{k}-t_{j}}(A_{k})\bigr)\\ =&\sum_{j,k}\omega\bigl(\tau_{t_{j}-t_{k}}(A_{k})A_{j}^{*}\bigr)= \sum_{j, k}\bigl(\Omega_\omega, A_{k}\e^{\i (t_{k}-t_{j})L}A_{j}^{*}\Omega_\omega\bigr)\\ =&\sum_{j,k}\bigl(\e^{-\i t_{k}L}A_{k}^{*}\Omega_\omega, \e^{-\i t_{j}L}A_{j}^{*}\Omega_\omega \bigr)=\|Tu\|^{2}. \end{array} \] Thus $T$ is a well defined antilinear operator. Moreover, using property {\rm (i)} of Definition \ref{defstocpos} and the fact that $\Omega_{\omega}$ is cyclic for~$\cR$, we conclude that $T$ has a dense domain and a dense range. Hence $T$ extends uniquely to an anti-unitary operator. Clearly $T$ is an involution. The other properties of $T$ follow directly from (\ref{reversal}). \qed \section{Quasi-free KMS states} \init \label{quasi} In this section we recall some well-known facts about quasi-free KMS states and describe a class of quasi-free KMS states which generate stochastically positive KMS systems (see \cite{KL2}, \cite{OGie}). \init \label{sec4} \subsection{Quasi-free KMS states} \label{sec4.1} Let $X_{0}$ be a pre-Hilbert space, $X$ the completion of $X_{0}$. Then $(X_{0}, \sigma)$ is a real symplectic space for $\sigma={\rm Im}\hf$, and we denote by $\fW(X_{0})$ the Weyl algebra $\fW(X_{0}, \sigma)$. Let ${\rm a}\geq 0$ be a selfadjoint operator on $X$ such that $X_{0}\subset \cD({\rm a}^{-\12})$ and $\e^{-\i t{\rm a}}$ preserves $X_{0}$. Given ${\rm a}\geq 0$ the canonical choice for $X_{0}$ is $\cD({\rm a}^{-\12})$. For $\beta>0$ one defines a state $\omega_{\beta}$ on $\fW(X_{0})$ by the functional \beq \label{statedef} \omega_{\beta}(W(x)):= \e^{-\frac{1}{4}(x, (1+ 2\rho)x)}, \: x\in X_{0}, \eeq where $\rho:= ( \e^{\beta {\rm a}}-1)^{-1}$. Since $1+2\rho= \frac{1+ \e^{-\beta{\rm a}}}{1- \e^{-\beta{\rm a}}}$ and ${\rm a}\geq 0$ the form domain of $1+ 2\rho$ is equal to $D({\rm a}^{-\12})\supset X_{0}$. The state $\omega_{\beta}$ is a $(\tau^\circ, \beta)$-KMS state for the dynamics $\tau^\circ \colon t \mapsto \tau^\circ_{t}$ defined by \[ \matrix{ \tau^\circ_{t}\colon & \fW(X_{0}) & \to & \fW(X_{0}) \cr & W(x)& \mapsto & W(\e^{\i t{\rm a}}x). \cr} \] The state $\omega_{\beta}$ is {\em quasi-free} (see [BR]) and the KMS system $(\fW(X_{0}), \tau^\circ, \omega_{\beta})$ defined above is called the quasi-free KMS system {\em associated to }${\rm a}$. The standard example is the following one: let ${\rm h}\geq 0$ be a selfadjoint operator representing the {\em one particle energy}. Assume that there exists a selfadjoint operator $\tq$ on $X$ representing the {\em one particle charge} such that $\tq^{2}=\one$, $[{\rm h}, \tq]=0$. Then we can associate a group of {\em gauge transformations} $\{Ê\alpha_{t} \}_{ t \in [0, 2\pi[}$, \[ \matrix{ \alpha_{t}\colon & \fW(X_{0}) & \to & \fW(X_{0}) \cr & W(x) & \mapsto & W(\e^{\i t\tq}x), \cr} \] to the charge operator $\tq$. Let $\mu\in \rr$ such that ${\rm h}- \mu \tq\geq \lambda>0$. Thus the range for the value of the chemical potential~$\mu$, which we consider, excludes Bose-Einstein condensation. It follows that~${\rm a}:={\rm h}-\mu \tq>0$ and hence~$X_{0}=\cD({\rm a}^{-\12})=X$. Therefore the unique quasi-free KMS state on~$\fW(X)$ at inverse temperature~$\beta$ and chemical potential~$\mu$ is the state~$\omega_{\beta}$ defined by (\ref{statedef}). \subsection{Araki-Woods representation} \label{sec4.3} Let us consider a quasi-free KMS system associated to a selfadjoint operator~${\rm a}$ as in Subsection~\ref{sec4.1}. Let $\Xbar$ be the conjugate Hilbert space to $X$. Elements of $\Xbar$ will be denoted by~$\overline{x}$. Equivalently, we denote by $X\ni x\mapsto \overline{x}\in \Xbar$ the identity operator, which is antilinear. If ${\rm a}$ is a linear operator on $X$, we denote by $\overline{{\rm a}}$ the linear operator on $\Xbar$ defined by $\overline{{\rm a}}\:\overline{x}:= \overline{{\rm a}x}$. If $\ch$ is a Hilbert space, then \[ \G(\ch)=\bigoplus_{n=0}^{+\infty}\otimes^{n}_{\rm s}\ch \] denotes the bosonic Fock space over $\ch$. We set: \[ \begin{array}{l} \cH_{\omega}:= \G(X\oplus \Xbar),\\[3mm] \Omega_{\omega}:= \Omega, \\[3mm] W_{\omega, {\rm l}}(x):= W_{\rm F}\bigl( (1+ \rho)^{\frac{1}{2}}x\oplus \overline{\rho}^{\frac{1}{2}}\overline{x}\bigr),\: x\in X_{0},\\[3mm] W_{\omega, {\rm r}}(x):= W_{\rm F}\bigl( \rho^{\frac{1}{2}}x\oplus (1+ \overline{\rho})^{\frac{1}{2}}\overline{x}\bigr),\: x\in X_{0},\\[3mm] \end{array} \] where $W_{F}(.)$ denotes the Fock space Weyl operator on $\G(X\oplus\Xbar)$ and $\Omega\in \G(X\oplus \Xbar)$ denotes the Fock vacuum. The following facts are well known: \noindent \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & The map $W (x)\mapsto W_{\omega,{\rm l/r}}(x)\in U(\cH_{\omega})$ defines a regular CCR representations; \cr {\rm (ii)} & $[W_{\omega, {\rm l}}(x), W_{\omega, {\rm r}}(y)]=0$ for $x,y\in X_{0}$; \cr {\rm (iii)} & $(\Omega_{\omega},W_{\omega, {\rm l}}(x)\Omega_{\omega})= \omega(W(x))$ for $x\in X_{0}$; \cr {\rm (iv)} & Let $L:= \dG({\rm a}\oplus-\overline{{\rm a}})$ act on $\cH_{\omega}$. Then \[ \e^{-\i tL}\Omega_{\omega}= \Omega_{\omega}, \: \e^{\i tL}W_{\omega, {\rm l}}(x)\e^{-\i tL}= W_{\omega,{\rm l}}(\e^{\i ta}x), \: x\in X_{0}; \] \vskip -.4cm \cr {\rm (v)} & The vector $\Omega$ is cyclic for the representations $W_{\omega, {\rm l}/{\rm r}}(.)$. \cr} \vskip .2cm \noindent In particular, the Araki-Woods representation is the GNS representation for the KMS system $\bigl(\fW(X, \sigma), \tau^\circ , \omega\bigr)$ and $L$ is the associated Liouvillean. We will only consider the left Araki-Woods representation, thus will forget the subscript ${\rm l}$ and write $W_{\omega}(x):= W_{\omega, {\rm l}}(x)$, $x\in X$. The creation-annihilation operators associated to $W_{\omega}(.)$ are \[ \begin{array}{l} a^{*}_{\omega}(x)= a^{*}_{F}\bigl( (1+\rho)^{\12}x\oplus 0) + a_{F}(0\oplus \overline{\rho}^{\12}\overline{x}\bigr),\\ a_{\omega}(x)= a_{F}\bigl( (1+\rho)^{\12}x\oplus 0) + a_{F}^{*}(0\oplus \overline{\rho}^{\12}\overline{x}\bigr). \end{array} \] \subsection{Field algebras} \label{sec4.2} We recall that a {\em conjugation} on a Hilbert space $X$ is an anti-unitary involution on $X$. Let us assume that $X$ is equipped with a conjugation $\kappa$. To $\kappa$ we associate the real vector space $X_{\kappa}: =\{x\in X\:| \:\kappa x=x\}$. Let $\omega$ be the quasi-free state associated to a selfadjoint operator~${\rm a}$, as defined in Subsection \ref{sec4.1}, and let $\cH_{\omega}$ be the Araki-Woods space introduced in Subsection~\ref{sec4.3}. We will denote by $\cW\subset \cB(\cH_{\omega})$ the {\em field algebra}, \ie the von Neumann algebra generated by the $\{W_{\omega}(x) \mid x\in X\}$ and by $\cW_\kappa \subset \cB(\cH_{\omega})$ the von Neumann algebra generated by $\{W_{\omega}(x)\:| \: x\in X_{\kappa}\}$. Since the symplectic form $\sigma$ vanishes on $X_{\kappa}$, the algebra $\cW_\kappa $ is abelian. \begin{lemma}\label{4.1} Assume that ${\rm a}= {\rm h}-\mu \tq$, where ${\rm h}$ and $ \tq $ are selfadjoint operators such that $[{\rm h}, \tq ]=0$, $\tq^{2}=1$, ${\rm h}\geq m> 0$ and $|\mu|< m$. Let $\kappa$ be a conjugation on $X$ such that $[{\rm h} ,\kappa]=0$. Then $\cW$ is the von Neumann algebra generated by $\{\e^{\i tL}A\e^{-\i tL} \mid t\in \rr, \: A\in \cW_\kappa \}$. \end{lemma} \proof Clearly $\{\e^{\i tL}A\e^{-\i tL} \mid t\in \rr, \: A\in \cW_\kappa \}\subset \cW$, so it suffices to prove the converse inclusion. Using the CCR, the facts that $(1+ \rho)^{\12}$ and $\rho^{\12}$ are bounded, and the fact that the map \[ X\oplus \overline{X}\ni x_{1}\oplus \overline{x_{2}}\mapsto W_{F}( x_{1}\oplus \overline{x_{2}})\in \cB(\cH_{\omega}) \] is continuous for the strong topology, it suffices to verify that \beq E={\rm Vect}_{\rr}\{\e^{\i t( {\rm h}-\mu \tq)}x,\: t\in \rr, \; x\in X_{\kappa}\}\hbox{ is dense in }X. \label{e4.1} \eeq Clearly $\overline{E}$ contains $X_{\kappa}$, and by differentiating with respect to $t$, we see that $\overline{E}$ contains also $\{\i({\rm h}- \mu \tq)x \mid x\in X_{\kappa}\cap \cD({\rm h})\}$. We now claim that for each $x\in X$ there exists $x_{1}\in X_{\kappa}$ and $x_{2}\in X_{\kappa}\cap \cD({\rm h})$ such that \[ x= x_{1}+ \i ({\rm h}-\mu \tq)x_{2}. \] This will imply (\ref{e4.1}). In fact, the $\rr$-linear map $r=\12 \mu \tq{\rm h}^{-1}(1-\kappa)$ has norm less than~$|\mu|m^{-1}<1$, so for $x\in X$ we can find $y\in X$ such that $y- ry=x$. If $x_{1}= \12(y+ \kappa y)$ and~$x_{2}= \12 (\i {\rm h})^{-1}( y-\kappa y)$, then both are elements of $ X_{\kappa}$, since $[{\rm h}, \kappa]=0$. Now \[ x_{1}+ \i ({\rm h}- \mu \tq)x_{2}= y -\frac{\i}{2}\mu {\tq} {\rm h}^{-1}( y- \kappa y)= y -ry=x \square . \] \subsection{Observable algebras} \label{sec4.3b} The gauge transformations $\alpha_{t}$ on $\fW(X_{0}, \sigma)$ can be unitarily implemented in the Araki-Woods representation: \[ \alpha_{t} \bigl(W_{\omega}(x)\bigr)= \e^{\i tQ}W_{\omega}(x)\e^{-\i tQ}, \] where $Q:= \dG( \tq\oplus -\overline{\tq})$. We denote by $\cA$ the {\em observable algebra} \[ \cA:= \bigl\{ A\in \cW \mid \e^{\i tQ}A\e^{-\i tQ}= A, \: t\in [0, 2\pi[ \bigr\} \] and by $\cA_{\kappa}$ the {\em abelian observable algebra} $\cA_{\kappa}:= \cA \cap \cW_\kappa $. \begin{lemma}\label{4.2} Assume that ${\rm h}\geq m> 0$ and $|\mu|< m$. Let $\kappa$ be a conjugation on $X$ such that $[{\rm h}, \kappa]=0$. Then $\cA$ is the von Neumann algebra generated by $\{\e^{\i tL}A\e^{-\i tL} \mid t\in \rr, \: A\in \cA_{\kappa}\}$. \end{lemma} \proof Clearly $\e^{\i tL}A\e^{-\i tL}\in \cA$, if $A\in \cA_{\kappa}$, since $[L, Q]=0$. Conversely, let $A\in \cA$. By Lemma~\ref{4.1} there exists a net $\{ A_{i} \}_{i\in I}$ in the algebra generated by $ \{\e^{\i tL}A\e^{-\i tL}, \: t\in \rr, \: A\in \cA_{\kappa}\}$ such that $A= \slim A_{i}$. For $R\in \cB(\cH_\omega)$, let $R^{\rm av}:= (2\pi)^{-1}\int_{0}^{2\pi}\e^{\i t Q}R\e^{-\i tQ}\d t$ be the average of $R$ with respect to the gauge group. Then by dominated convergence $\slim A_{i}^{\rm av}= A^{\rm av}=A$. Since $[L, Q]=0$, we have $(\e^{\i tL}R\e^{-\i tL})^{\rm av}= \e^{\i tL}R^{\rm av}\e^{-\i tL}$, which implies the lemma \qed . \begin{lemma}\label{4.3} We have $\overline{\cA \Omega_{\omega}}=\{u\in \cH_\omega \mid Qu=0\}$. \end{lemma} \proof Since $Q\Omega_{\omega}=0$ we have $\overline{\cA\Omega_{\omega}}\subset {\rm Ker}Q$. Let now $u\in {\rm Ker}Q$. If $\{ A_{i} \in \cW\}_{i \in I}$ is a net such that $\lim A_{i}\Omega_{\omega}= u$, then \[ u=\frac{1}{2\pi}\int_{0}^{2\pi}\e^{\i tQ}u \, \d t= \lim\frac{1}{2\pi}\int_{0}^{2\pi}\e^{\i tQ}A_{i}\e^{-\i tQ}\Omega_{\omega} \, \d t= \lim_{n \to \infty} A_{i}^{\rm av}\Omega_{\omega}, \] which proves the lemma since $A_{i}^{\rm av}\in \cA$ \qed . \subsection{Stochastic positivity} \label{stocpos} In this subsection we give a criterion for the stochastic positivity of a quasi-free KMS system. The following lemma is due to Klein and Landau \cite{KL2}. \begin{lemma} Let ${\rm a}\geq 0$ be a selfadjoint operator on a Hilbert space $X$. Let $\rr\ni s\to r(s)\in \cB(X)$ be the $\beta$-periodic operator-valued function defined by \[ r(s)= \frac{\e^{- s{\rm a}}+ \e^{(s-\beta){\rm a}}}{1-\e^{-\beta {\rm a}}}, \: 0\leq s< \beta. \] Then, for $x_{i}\in X$ and $s_{i}\in \rr$, one has \[ \sum_{i,j}\bigl( x_{i}, r(s_{j}-s_{i})x_{j} \bigr)\geq 0. \] \label{p1.0} \end{lemma} \proof Using the spectral decomposition of ${\rm a}$, we can assume that $x_{i}\in \cc$ and ${\rm a}\geq 0$ is a positive real number. Hence it is sufficient to verify that $r(s)$ is a distribution of positive type. But this follows from Bochner's theorem and the fact that the Fourier transform of~$r$ is $\sum_{n\in \zz}r_{n}\delta(. -2\pi/n)$, where $r_{n}= \frac{2{\rm a}}{ {\rm a}^{2}+ (2\pi n/\beta)^{2}}\geq 0$ \rm \qed . \begin{theoreme} \label{p1.1} Let $X$ be a Hilbert space equipped with a conjugation $\kappa$ and ${\rm a}\geq m>0$ a selfadjoint operator on $X$ such that $[{\rm a}, \kappa]=0$. Let $X_{\kappa}\subset X$ be the real vector space associated to $\kappa$. Let $\bigl(\cW, \tau^\circ, \omega\bigr)$ be the quasi-free KMS system associated to ${\rm a}$ and let $\cW_\kappa\subset \cW$ be the abelian von Neumann algebra generated by $\{W_\omega(x) \mid x\in X_{\kappa}\}$. Then the KMS system $\bigl(\cW, \cW_\kappa, \tau^\circ,\omega \bigr)$ is stochastically positive. \end{theoreme} \proof We start by computing the Euclidean Green's functions. Using the CCR we get, for $x_{j}\in X$ and $1\leq j\leq n$, \[ \prod_{1}^{n}W(x_{j})= \prod_{1\leq i\leq j\leq n}\e^{-\frac{\i}{2}\sigma(x_{i}, x_{j})}W \bigl(\sum_{1}^{n}x_{j} \bigr). \] We denote by \[ G\bigl(t_{1}, \dots, t_{n}; W(x_{1}), \dots, W(x_{n}) \bigr)=\omega \bigl(\prod_{j=1}^{n}W(\e^{\i t_{j}{\rm a}}x_{j}) \bigr) \] the Green's functions for the Weyl operators $W(x_{j})$, $1\leq j\leq n$. Now \[ \begin{array}{rl} & G\bigl(t_{1}, \dots, t_{n}; W(x_{1}), \dots, W(x_{n})\bigr)\\[2mm] = &\lPi_{1\leq i< j\leq n}\e^{-\i {\rm Im}( x_{i}, \e^{\i (t_{j}-t_{i}){\rm a}}x_{j})}\e^{-\frac{1}{4}(\sum_{1}^{n}\e^{\i t_{j}{\rm a}}x_{j}, (1+ 2 \rho)\sum_{1}^{n}\e^{\i t_{j}{\rm a}}x_{j})}\\[2mm] =&\prod_{1}^{n}\e^{-\frac{1}{4}( x_{i}, (1+ 2 \rho) x_{i})} \lPi_{1\leq i< j\leq n}\e^{-\frac{1}{2}R(t_{j}- t_{i})( x_{i}, x_{j})}, \end{array} \] where \[ R( t)( x, y)= \bigl( x, (1-\e^{-\beta {\rm a}})^{-1}\e^{\i t{\rm a}}y \bigr)+ \bigl(y, \e^{-\beta {\rm a}}(1-\e^{-\beta {\rm a}})^{-1}\e^{\i t{\rm a}}x \bigr). \] For $x,y\in X$ the function $t\mapsto R(t)( x,y)$ has an holomorphic extension to $0<{\rm Im }z<\beta$ such that the function $(t_{1}, \dots, t_{n})\mapsto G\bigl(t_{1}, \dots, t_{n}; W(x_{1}), \dots, W(x_{n})\bigr)$ is holomorphic in the set~$I_{\beta}^{n+}$ defined in (\ref{inbeta}) and continuous on $\overline{I_{\beta}^{n+}}$ with holomorphic extension \[ (\zeta_{1}, \dots, \zeta_{n})\mapsto \prod_{1}^{n}\e^{-\frac{1}{4}( x_{i}, (1+2 \rho)x_{i})} \prod_{1\leq i< j\leq n}\e^{-\frac{1}{2}R(\zeta_{j}- \zeta_{i})( x_{i}, x_{j})}. \] Hence the euclidean Green's functions \[ \EG \bigl( s_{1}, \dots, s_{n}; W(x_{1}), \dots, W(x_{n})\bigr)= \prod_{1}^{n}\e^{- \frac{1}{2}C(0)( x_{i}, x_{i})}\prod_{1\leq i0$ a selfadjoint operator on $X$ such that $[{\rm a}, \kappa]=0$. Let $X_{\kappa}\subset X$ be the real vector space associated to $\kappa$. Let $(\cW(X), \tau^\circ, \omega )$ be the quasi-free KMS system associated to ${\rm a}$ and let $\cW_\kappa\subset \cW$ be the abelian von Neumann algebra generated by $\{W_\omega (x)\mid x\in X_{\kappa}\}$. Let $(\cH_{\omega}, L, \Omega_{\omega})$ be the Araki-Woods objects defined in Subsection \ref{sec4.3}. Then the space $\{A\e^{-\frac{\beta}{2}L}B\Omega, \: A, B\in \cW_\kappa\}$ is dense in $\cH_{\omega}$. \end{lemma} % \proof The function \[ \begin{array}{rl} \e^{\i tL} W_{\omega, l} (y) \Omega_\omega &= W_{\omega, l} (\e^{\i t{\rm a}}y) \Omega_\omega \\ &= W_{F} \bigl((1+\rho)^{\12} \e^{\i t{\rm a}} y \oplus (\overline{\rho})^{\12}\e^{-\i t\overline{\rm a}}\overline{y} \bigr)\\ &= \e^{\i a_F^* \bigl((1+\rho)^{\12} \e^{\i t{\rm a}} y \oplus (\overline{\rho})^{\12}\e^{-it\overline{\rm a}}\overline{y} \bigr)} \e^{- {1 \over 2} ( y, (1+ 2 \rho) y) } \Omega_\omega \end{array} \] % is analytic in $\{0<{\rm Im z}<\frac{\beta}{2}\}$ and continuous on $\{0\leq {\rm Im z}\leq \frac{\beta}{2}\}$, and \[ \begin{array}{rl} \e^{-\beta L/2} W_{\omega, l} (y) \Omega_\omega &= \e^{i a_F^* \bigl((1+\rho)^{\12} \e^{- \beta{\rm a}/2} y \oplus (\overline{\rho})^{\12}\e^{\beta \overline{\rm a}/2}\overline{y} \bigr)} \e^{- {1 \over 2} ( y, (1+ 2 \rho) y) } \Omega_\omega \\ &=W_{\omega, r} (y) \Omega_\omega. \end{array} \] % Hence, for $A=W_{\omega, {\rm l}}(x)$ and $B= W_{\omega, {\rm r}}(y)$, one has \beq \label{uti} A\e^{-\frac{\beta}{2}L}B\Omega= W_{\omega, {\rm l}}(x)W_{\omega, {\rm r}} (y) \Omega= W_{F} \bigl( (1+\rho)^{\12}x\oplus \overline{\rho}\overline{x} \bigr)W_{F} \bigl( \rho^{\12}y\oplus (1+\overline{\rho})^{\12} \: \overline{y} \bigr)\Omega. \eeq Let ${\cal M}$ be the von Neumann algebra generated by $\{W_{\omega, {\rm l}}(x), W_{\omega, {\rm r}}(y)\mid x, y\in X_{\kappa}\}$. By (\ref{uti}) the von Neumann algebra generated by $\bigl\{W_{F} \bigl((1+\rho)^{\12}x+ \rho^{\12}y\oplus \overline{\rho}^{\12}\overline{x}+ (1+\overline{\rho})^{\12}\overline{y} \bigr) \mid x,y\in X_{\kappa}\bigr\}$ is equal to ${\cal M}$. Since $[a, \kappa]=0$, the operator \[ \left(\begin{array}{cc}(1+\rho)^{\12}&\rho^{\12}\\ \rho^{\12} &(1+\rho)^{\12} \end{array}\right):X\oplus X\to X\oplus X \] sends $X_{\kappa}\oplus X_{\kappa}$ into itself. It is invertible with inverse \[ \left(\begin{array}{cc}(1+\rho)^{\12}&-\rho^{\12}\\ -\rho^{\12} &(1+\rho)^{\12} \end{array}\right). \] Thus ${\cal M}$ is equal to the von Neumann algebra generated by $\{W_{F}( x\oplus \overline{y}), \: x,y\in X_{\kappa}\}$. It is well known that if $\ch$ is a Hilbert space and ${\rm c}$ is a conjugation on $\ch$, then the vacuum vector~$\Omega$ is cyclic in the Fock space $\G(\ch)$ for the algebra generated by $\{W_{F}(h)\:| \: {\rm c}h=h\}$ (see e.g.\ \cite[Sect.\ 5.2]{DG} and references therein). We apply this result to $\ch= X\oplus \overline{X}$, ${\rm c}= \kappa\oplus\overline{\kappa}$ and obtain the lemma. \qed \section{Generalized path spaces} \init\label{path} In \cite{KL1} a canonical isomorphism is constructed between a stochastically positive $\beta$-KMS system $(\cW, \cW_\kappa, \tau^\circ , \omega)$ and a $\beta$-periodic {\em stochastic process} $(Q, \Sigma, \mu,X_{t})$ indexed by the circle~$S_{\beta}$ of length ${\beta}$, with values in the compact Hausdorff space $K= \hbox{\rm Sp} \: (\cW_\kappa)$, the spectrum of~$\cW_\kappa$. We recall that a {\em stochastic process} $(Q, \Sigma, \mu,X_{t})$ indexed by an interval $I\subset \rr$ with values in a topological space $K$ consists of \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & a probability space $(Q, \Sigma, \mu)$; \cr {\rm (ii)} & a family $\{X_{t}\}_{t\in I}$ of measurable functions $X_{t}\colon Q\to K$. \cr} \vskip .2cm \noindent Typically it is required that the map $I\in t\mapsto X_{t}$ is continuous in measure. \vskip .3cm The stochastic process $(Q, \Sigma, \mu,X_{t})$ associated to a stochastically positive $\beta$-KMS system in \cite{KL1} satisfies four important properties: {\em stationarity}, {\em symmetry}, $\beta$-{\em periodicity} and {\em Osterwalder-Schrader positivity} (see \cite[Sect.\ 4]{KL1}). It turns out that the only really important feature of such a stochastic process is the underlying {\em generalized path space}, which consists of the sub $\sigma$-algebra $\Sigma_{0}$ generated by the functions $F(X_{0})$ for $F\in C(K)$, the automorphism group $U(t)$ of $L^{\infty}(Q, \Sigma, \mu)$ generated by the time translations $U(t)\colon F(X_{t_{1}}, \dots, X_{t_{t}})\mapsto F(X_{t_{1}+t}, \dots, X_{t_{n}+t})$ for $F\in C(K^{n})$ and the automorphism $R$ of $L^{\infty}(Q, \Sigma, \mu)$ generated by $R\colon F(X_{t_{1}}, \dots, X_{t_{t}})\mapsto F(X_{-t_{1}}, \dots, X_{-t_{n}})$. In particular the detailed knowledge of the random variables $X_{t}$ and of the topological space $K$ is not necessary. (Note that time translations on the path space will correspond to imaginary time translations on the physical Hilbert space). The analog of the constructions of \cite{KL1} for $\beta=\infty$ done by Klein in \cite{K} is formulated in terms of generalized path spaces. Using this essentially equivalent formulation turns out to be more convenient in applications. We now proceed to a more precise description of this structure, taken from \cite{KL1} and~\cite{K}. If $\Xi_{i}$, for $i$ in an index set $I$, is a family of subsets of a set $Q$, we denote by $\bigvee_{i\in I} \Xi_{i}$ the $\sigma$-algebra generated by $\bigcup_{i\in J} U_{i}$, where $U_{i}\in \Xi_{i}$ and $J$ are countable subsets of $I$. \begin{definition} A {\em generalized path space} $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ consists of \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & a probability space $(Q, \Sigma, \mu)$; \cr {\rm (ii)} & a distinguished sub $\sigma$-algebra $\Sigma_{0}$; \cr {\rm (iii)} & a one-parameter group $\rr\ni t\mapsto U(t)$ of measure preserving $^{*}$-automorphisms of $L^{\infty}(Q, \Sigma, \mu)$, which is strongly continuous in measure; \cr {\rm (iv)} & a measure preserving $^{*}$-automorphism $R$ of $L^{\infty}(Q, \Sigma, \mu)$ such that $RU(t)= U(-t)R$, $R^{2}=\one$, $RE_{0}= E_{0}R$, where $E_{0}$ is the conditional expectation w.r.t. the $\sigma$-algebra $\Sigma_{0}$. \cr} Moreover one requires that %\vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (v)} & $\Sigma=\bigvee_{t\in \rr}U(t)\Sigma_{0}$. \cr} \end{definition} It follows from (iii) and (iv) that $U(t)$ extends to a strongly continuous group of isometries of $L^{p}(Q, \Sigma, \mu)$, and $R$ extends to an isometry of $L^{p}(Q, \Sigma, \mu)$, for $1\leq p<\infty$. We say that the path space $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ is $\beta$-{\em periodic} for $\beta>0$ if $U(\beta)=\one$. On a $\beta$-periodic path space we can consider the one-parameter group $U(t)$ as indexed by the circle $S_{\beta}=[-\beta/2, \beta/2]$. For $I\subset \rr$ we denote by $E_{I}$ the conditional expectation with respect to the $\sigma$-algebra $\Sigma_{I}:=\bigvee_{t\in I}U(t)\Sigma_{0}$. \begin{definition} {\rm ($0$-temperature case):} A path space $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ is {\em OS-positive} if \hfill\linebreak $E_{[0, +\infty[}RE_{[0, +\infty[}\geq 0$ as an operator on $L^{2}(Q, \Sigma, \mu)$. {\rm (Positive temperature case:)} A $\beta$-periodic path space $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ is {\em OS-positive} if $E_{[0,\beta/2]}RE_{[0, \beta/2]}\geq 0$ as an operator on $L^{2}(Q, \Sigma, \mu)$. \end{definition} In order to For simplify the notation we set $E_{0}= E_{\{0\}}$, $\Sigma_{+}=\Sigma_{[0, +\infty[},\:E_{+}=E_{[0, +\infty[}$, $\Sigma_{-}=\Sigma_{]-\infty, 0]}$ and $E_{-}= E_{]-\infty, 0]}$. If the path space $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ is $\beta$-periodic, we set $\Sigma_{+}=\Sigma_{[0, \beta/2]}, \:E_{+}= E_{[0,\beta/2 ]}$, $\Sigma_{-}=\Sigma_{[-\beta/2, 0]}$ and $E_{-}=E_{[-\beta/2, 0]}$. \begin{definition} A path space $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ is a {\em Markov path space} if it has the \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & {\em reflection property:} $RE_{0}= E_{0}$ (resp.\ $RE_{\{Ê0, \beta/2\} }= E_{\{Ê0, \beta/2\} }$); \cr {\rm (ii)} & {\em Markov property:} $E_{+} E_{-}= E_{+} E_{0} E_{-}$ (resp.\ $E_+ E_-= E_+ E_{\{Ê0, \beta/2\} }E_-$). \cr} \vskip .2cm \noindent It follows that $E_{+}RE_{+}= E_{-}E_{+}= E_{+}E_{-}= E_{0}$ (resp.\ $E_{+}RE_{+}= E_{-}E_{+}= E_{+}E_{-}= E_{\{Ê0, \beta/2\} }$) . \end{definition} A Markov path space is OS-positive because $E_{0} $ (resp.\ $E_{\{Ê0, \beta/2\} }$) is positive as an orthonormal projection. An OS-positive path space satisfies the reflection property (see \cite[Prop.\ 1.6]{K}). Let $(\cF, \cU, \tau, \omega)$ be a stochastically positive $\beta$-KMS system. Let $K:={\rm Sp} (\cU)$ be the spectrum of the abelian $C^{*}$-algebra $\cU$, which equipped with the weak topology is a compact Hausdorff space. Let $Q:=K^{[-\beta/2, \beta/2]}$ be equipped with the product topology and let $\Sigma$ be the Baire $\sigma$-algebra on $Q$. \begin{theoreme}{\rm \cite{KL1}.} Let $(\cF, \cU, \tau, \omega)$ be a stochastically positive $\beta$-KMS system. Then there exists a Baire probability measure $\mu$ on $Q$, a sub $\sigma$-algebra $\Sigma_{0}\subset \Sigma$, a measure preserving group $U(t)$ of $^{*}$-automorphisms of $L^{\infty}(Q, \Sigma, \mu)$ and a measure preserving automorphism~$R$ of $L^{\infty}(Q, \Sigma, \mu)$ such that $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ is an OS-positive $\beta$-periodic generalized path space. \end{theoreme} A more precise relationship between the $\beta$-KMS system and the generalized path space will be given in Theorem \ref{stp1.4}. \section{Reconstruction theorems} \label{secst1} \init In this section we recall reconstruction theorems of Klein \cite{K} and Klein and Landau \cite{KL1} which associate a stochastically positive $\beta$-KMS system to an OS-positive generalized path space $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$. To simplify notation, we allow the parameter $\beta$ to take values in $]0, +\infty]$. The case~$\beta=+\infty$ corresponds to the $0$-temperature case. If $\beta<\infty$, then the OS-positive path spaces will be assumed to be $\beta$-periodic. \subsection{Physical Hilbert space} Set $\cH_{OS}:= L^{2}(Q, \Sigma_{+}, \mu)$ and \[ (F, G):= \int_{Q}R(\overline{F})G\d \mu,\: F, G\in \cH_{OS}. \] By OS-positivity \[ 0\leq (F, F)\leq \|F\|^{2}_{\cH_{OS}}. \] If we set ${\cal N}:={\rm Ker}E_{+}RE_{+}$, then $(\cdot, \cdot)$ is a positive definite sesquilinear form on $\cH_{OS}/{\cal N}$. The {\em physical Hilbert space}, denoted by $\cH_{\rm phys}$ (or simply by $\cH$) is \[ \cH :=\hbox{ completion of }\cH_{OS}/{\cal N} \hbox{ for }(\cdot, \cdot). \] If we denote by ${\cal V}\colon \cH_{OS}\to \cH_{OS}/{\cal N}$ the canonical projection, then ${\cal V}$ extends uniquely to a contraction with dense range: $\cH_{OS}\to \cH$. In fact \[ ({\cal V}F, {\cal V}F)= (F, F) \leq \|F\|^{2}_{\cH_{OS}}. \] In the physical Hilbert space $\cH$ we find a {\em distinguished vector}\/: \[ \Omega:= \cV (1). \] \subsection{Selfadjoint operator} {\bf The $0$-temperature case} \begin{proposition}{\rm \cite[Thm.\ 1.7]{K}}. % Let $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ be an OS-positive generalized path space. For $t\geq 0$ the time evolution $U(t)$ maps $ {\cal N}\to {\cal N}$. Hence the linear operator \[ P(t)\colon \cH_{\rm OS}/{\cal N}\ni{\cal V}(F)\mapsto {\cal V}(U(t)F)\in \cH_{\rm OS}/{\cal N} \] is well defined for $t\geq 0$. The family $\{P(t)\}_{t\geq 0}$ uniquely extends to a strongly continuous selfadjoint semigroup of contractions $\{\e^{-tH}\}_{t\geq 0}$ on $\cH$, where $H$ is a positive selfadjoint operator such that $H\Omega=0$. \end{proposition} \noindent {\bf The positive temperature case} \vskip .3cm \noindent We first recall the definition of a local symmetric semigroup (\cite{KL3}, \cite{Fr}): \begin{definition} \label{st1.2ter} Let $\cH$ be a Hilbert space and $T>0$. A {\em local symmetric semigroup} \ \ $(P(t), {\cal D}_{t}, T)$ is a family $\{P(t), \cD_{t}\}_{t\in [0, T]}$ of linear operators $P(t)$ and vector subspaces $\cD_{t}$ of~$\cH$ such that \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & $D_{0}=\cH$, $\cD_{t}\supset\cD_{s}$ if $0\leq t\leq s\leq T$ and $\cD= \cup_{00$, \[ J_{\beta}^{n+}:= \{(t_{1}, \dots, t_{n})\in \rr^{n}\mid t_{i}\geq 0, \: t_{1}+ \cdots+ t_{n}\leq \beta/2\}.\] \begin{theoreme}{\rm \cite{KL1}}. \label{stp1.4} Let $L$ be the selfadjoint operator associated to the local symmetric semigroup $(P(t), \cD_{t}, \beta/2)$. It follows that \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & $\Omega\in \cD(L)$ and $L\Omega=0$; \cr {\rm (ii)} & if $n\in \nn$, $(t_{1}, \dots, t_{n})\in J_{\beta}^{n+}$ and $A_{1}, \dots, A_{n}\in {\cal U}$, then $A_{n} \bigl( \prod_{n-1}^{1}\e^{-t_{j}L}A_{j} \bigr) \Omega\in \cD(\e^{-t_{n}L})$. The vector span of these vectors is dense in $\cH$; \cr {\rm (iii)} & if $f_{1}, \dots, f_{n}\in L^\infty ( Q, \Sigma_0, \mu)$ and $0\leq s_{1}\leq \cdots\leq s_{n}\leq \beta/2$, then \[ \cV \bigl(\prod_{1}^{n}U(s_j) f_{j} \bigr)= \e^{-s_{1}L}\tilde{f}_{1} \bigl(\prod_{2}^{n}\e^{-(s_{j}- s_{j-1})L}\tilde{f}_{j} \bigr) \Omega, \] where $\tilde{f}_{j}$ is defined in Proposition \ref{st1.2}. \cr {\rm (iv)} & if $n\in \nn$, $(t_{1}, \dots, t_{n})\in J_{\beta}^{n+}$ and $A_{1}, \dots, A_{n}$, $B_{1}, \dots, B_{n}\in \cU^{+}$, then \[ \Bigl( A_{n} \bigl( \prod_{n-1}^{1}\e^{-t_{j}L}A_{j} \bigr) \Omega \, , \, %\prod_{n}^{1}A_{j}\e^{-t_{j}L}\Omega, %\prod_{n}^{1}B_{j}\e^{-t_{j}L}\Omega B_{n} \bigl( \prod_{n-1}^{1}\e^{-t_{j}L}B_{j} \bigr) \Omega \Bigr)\geq 0; \] \cr {\rm (v)} & $\|\e^{-\beta/2 L}A\Omega\|=\|A^{*}\Omega\|$ for all $A\in \cU$. \cr} \end{theoreme} \begin{theoreme}{\rm \cite{KL1}}. Let $\omega_\Omega$ be the state on $\cF$ defined by $\omega_\Omega (B)= (\Omega, B\Omega)$. Then $(\cF, \cU, \tau, \omega_\Omega)$ is a stochastically positive $\beta$-KMS system. \label{st1.5} \end{theoreme} Finally let $J$ be the modular conjugation associated to the KMS system $(\cF, \tau , \omega_\Omega)$. \begin{proposition}{\rm \cite{KL1}}. The modular conjugation $J$ is the unique extension of \beq J\cV(F)= \cV(R_{\beta/4}\overline{F}), \label{est1.2} \eeq where \[ R_{\beta/4}:=U(\beta/4)RU(-\beta/4)= RU(-\beta/2)= U(\beta/2)R \] is the reflection at $t=\beta/4$ in $\cH_{OS}$. \end{proposition} \subsection{Markov property for $\beta$-periodic path spaces} \label{secst2} We recall a characterization of the Markov property for a $\beta$-periodic path space in terms of the associated stochastically positive $\beta$-KMS system due to Klein and Landau \cite{KL1}. \begin{theoreme} \label{st2.2} A $\beta$-periodic OS-positive path space $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ satisfies the Mar\--kov property iff the vectors $A\e^{-\frac{\beta}{2}L}B\Omega$ for $A, B\in \cU$ are dense in $\cH$. In this case \[ \cH =L^{2}( Q, \Sigma_{\{0, \beta/2\}}, \mu) .\] \end{theoreme} \proof The first statement of the theorem is shown in \cite[Thm.\ 11.2]{KL1}. The second statement is obvious: it follows from the Markov property that $E_{[0, \beta/2]}RE_{[0, \beta/2]}=E_{\{0, \beta/2\}}$ is a projection, hence $\cH_{OS}/{\cal N}$ is canonically identified with $E_{\{0, \beta/2\}}\cH_{OS}= L^{2}( Q, \Sigma_{\{0, \beta/2\}}, \mu)$. \begin{theoreme} Let $(\cW, \cW_\kappa, \tau^\circ, \omega_\beta )$ be the quasi-free KMS system associated to a selfadjoint operator ${\rm a}\ge0$ and a conjugation $\kappa$ with $[a, \kappa]=0$. Then the OS-positive generalized path space $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ associated to $(\cW(X), \cW_\kappa (X), \tau^\circ, \omega_\beta )$ satisfies the Markov property. \end{theoreme} % \proof Stochastic positivity of the quasi-free KMS system $(\cW, \cW_\kappa, \tau^\circ, \omega_\beta )$ was shown in Theorem 4.5. The Markov property follows from Lemma 4.6 and Theorem 6.10 \qed . \section{Perturbations of generalized path spaces} \init\label{secst3} In this section we recall some results concerning perturbations of OS-positive path spaces. \subsection{FKN kernels} \label{fkn} Let $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ be an OS-positive path space. \begin{definition} \label{fkn1} A {\em Feynman-Kac-Nelson} (FKN) {\em kernel} is a family $\{F_{[a, b]}\}$ of real measurable functions on $(Q, \Sigma, \mu)$ such that, for $0\leq b-a\leq \beta$, \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & $F_{[a,b]}>0$ $\mu$-a.e.; \cr {\rm (ii)} & $F_{[a,b]}\in L^{1}(Q, \Sigma, \mu)$ and $F_{[a,b]}$ is continuous in $L^{1}(Q, \Sigma, \mu)$ as a function of $b$; \cr {\rm (iii)} & $ F_{[a,b]}F_{[b,c]}= F_{[a,c]}$ for $a\leq b\leq c$, $c-a\leq \beta$; \cr {\rm (iv)} & $U(s)F_{[a,b]}= F_{[a+s, b+s]}$ for $s\in \rr$; \cr {\rm (v)} & $RF_{[a,b]}=F_{[-b, -a]}$. \cr} \end{definition} The main examples of FKN kernels are those associated to a selfadjoint operator $V$ affiliated to $\cU$. In \cite{KL1} and \cite{K} perturbations associated to more general FKN kernels are considered. However, the present case is sufficient for our applications. Let $V$ be a selfadjoint operator affiliated to $\cU$. Since by Proposition 6.5 the algebra %\ref{st1.4} $\cU$ is isomorphic to $L^{\infty}(Q, \Sigma_{0}, \mu)$, we can uniquely associate to $V$ a real function on $Q$, measurable with respect to $\Sigma_{0}$, which we will still denote by $V$. \begin{proposition} \label{st3.1b} Let $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ be a $\beta$-periodic OS-positive path space and let $V$ be a selfadjoint operator affiliated to $\cU$ such that $V\in L^{1}(Q, \Sigma_{0}, \mu)$, and $\e^{-TV}\in L^{1}(Q, \Sigma_{0}, \mu)$ for some $T>0$ if $\beta=\infty$ or $\e^{-\beta V}\in L^{1}(Q, \Sigma_{0}, \mu)$ if $\beta<\infty$. Then \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & the family of functions $F_{[a, b]}:=\e^{-\int_{a}^{b}U(t)V\d t}$ for $0\leq b-a\leq{\rm inf}(T, \beta)/2$ is a FKN kernel; \cr {\rm (ii)} & $F_{[0, s]}\in L^{2}(Q, \Sigma_{[0, s]},\mu)$ for $0\leq s\leq {\rm inf}(T, \beta)/2$ and the map $s \mapsto F_{[0, s]}$ is continuous in $L^{2}(Q, \Sigma_{[0, \beta/2]}, \mu)$. \cr} \end{proposition} \proof All properties required in Definition \ref{fkn1} except from property {\rm (ii)} follow directly from the definition of $U(t)$ and the properties of the path space $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$. Let us now verify {\rm (ii)}. Writing $V= V_{+}-V_{-}$, where $V_{\pm}$ is the positive/negative part of $V$, we have $F_{[a,b]}\leq \exp \bigl( \int_{a}^{b}U(t)V_{-}\d t \bigr)$, and hence $F_{[0, s]}^{2}\leq \exp \bigl(2\int_{0}^{\beta/2}U(t)V_{-}\d t \bigr)$. Since $\mu$ is a probability measure, we have $V_{-}, \:\e^{\beta V_{-}}\in L^{1}(Q, \Sigma_{0},\mu)$. We recall the following bound from \cite[Thm.~6.2 (i)]{KL0}: \beq \|\e^{-\int_{a}^{b} U(t)V\d t}\|_{L^{p}(Q, \Sigma, \mu)}\leq \|\e^{-(b-a)V}\|_{L^{p}(Q,\Sigma, \mu)},\: 1\leq p<\infty. \label{bound} \eeq This yields \[ \|F_{[0, s]}^{2}\|_{L^{1}(Q, \Sigma, \mu)}\leq \|\e^{2\int_{0}^{\beta/2}U(t)V_{-}\d t}\|_{L^{1}(Q, \Sigma, \mu)}\leq \|\e^{\beta V_{-}}\|_{L^{1}(Q, \Sigma,\mu)}<\infty. \] Hence $F_{[0,s]}\in L^{2}(Q, \Sigma_{[0, \beta/2]}, \mu)$ for $0\leq s\leq {\rm inf}(T, \beta)/2$. The continuity w.r.t.\ to $s$ follows from the dominated convergence theorem. This completes the proof of {\rm (ii)}. The proof of property {\rm (ii)} from Definition \ref{fkn1} for $0\leq a$ follows from {\rm (ii)} and the fact that $L^{2}(Q, \Sigma, \mu)\subset L^{1}(Q, \Sigma, \mu)$. The case $b\leq 0$ is reduced to the case $a\geq 0$ using property {\rm (v)}. Finally the case $a<00$. Let~$F_{[a, b]}$ be the associated FKN kernel. Let, for $0< t\epsilon\}}F_{[0, s']}^{2}|U(s')\psi- U(s)\psi|^{2}\d \mu\\ &+ \int_{\{|U(s')\psi- U(s)\psi|(q)\leq\epsilon\}}F_{[0, s']}^{2}|U(s')\psi- U(s)\psi|^{2}\d \mu \\ &+ \|F_{[0,s']}- F_{[0, s]}\|^{2}_{2}\|\psi\|^{2}_{\infty}. \end{array} \] The last term on the r.h.s.\ tends to $0$ if $s\to s'$ as a consequence of Proposition \ref{st3.1b}. The second term on the r.h.s.\ is less than $\epsilon^{2}\|F_{[0,s']}\|^{2}_{2}$. To estimate the first term, we write the function $f:= F_{[0,s']}^{2}$ as $f\one_{\{|f(q)|\leq M\}}+ f\one_{\{|f(q)|>M\}}$. It follows that \[ \begin{array}{rl} &\int_{\{|U(s')\psi- U(s)\psi|(q)>\epsilon\}} f |U(s')\psi - U(s)\psi|^{2}\d \mu \\[2mm] \leq &4M\|\psi\|^{2}_{\infty} \int\one_{\{|U(s')\psi- U(s)\psi|(q)>\epsilon\}}\d\mu + 4\|f\one_{\{|f(q)|>M\}}\|_{1}\|\psi\|^{2}_{\infty}. \end{array} \] Since $f\in L^{1}(Q, \Sigma_{+}, \mu)$, the second term tends to $0$ as $M\to \infty$. Since $U(t)$ is strongly continuous in measure, the first term tends to $0$ as $s\to s'$. Picking first $\epsilon\ll 1$, then $M\gg 1$ and finally $|s-s'|\ll 1$ we obtain {\rm (i)}. Let us now prove {\rm (ii)}. Let $0\leq s\leq t0$. Assume in addition that either $V\in L^{2+\epsilon}(Q, \Sigma_{0}, \mu)$ for $\epsilon>0$ or that $V\in L^{2}(Q, \Sigma_{0}, \mu)$ and $V\geq 0$. Let, for $\beta = \infty$, $H$ (resp.~$L$ for $\beta < \infty$) denote the selfadjoint generator of the unperturbed semi-group $t \mapsto P(t)$. Then $H+V$ (resp.\ $L+V$) is essentially selfadjoint and the operator $H_{V}$ (for both cases) constructed in Theorem \ref{pt1} is equal to $\overline{H+V}$ (resp.\ $\overline{L + V}$). \end{theoreme} \subsection{Perturbations in the positive temperature case} \label{otot} The following theorem is shown in \cite{KL1}: \begin{theoreme}{\rm \cite{KL1}}. \label{st3.2} Let $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ be a $\beta$-periodic OS-positive path space, $V$ a selfadjoint operator on $\cH$ affiliated to $\cU$, which satisfies the hypotheses of Proposition \ref{st3.1b}. Let $F=\{F_{[a,b]}\}$ be the associated $\beta$-periodic FKN kernel. Then the path space $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu_{V})$, where \[\d\mu_{V}:= {F_{[-\beta/2,\beta/2]}\d\mu \over \int_Q F_{[-\beta/2, \beta/2]} \d \mu \, ,} \] is a $\beta$-periodic OS-positive path space. \end{theoreme} By the reconstruction theorem recalled in Section \ref{secst2}, one can associate to the perturbed path space $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu_{V})$ a physical Hilbert space $\cH_{V}$, a distinguished vector $\Omega_{V}$, an abelian von Neumann algebra $\cU_{V}$, a selfadjoint operator $L_{V}$ and a von Neumann algebra $\cF_{V}$. If $\omega_{V}$ and $\tau_{V}$ are the state and $W^{*}$-dynamics associated to $\Omega_{V}$ and $ L_{V}$, then $(\cF_{V},\cU_{V}, \tau_{V}, \omega_{V})$ is a stochastically positive $\beta$-KMS system. Our next aim is to construct canonical identifications between the perturbed objects and perturbations of the original objects associated to the path space $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$. \medskip \noindent {\bf Identification of the physical Hilbert spaces} \medskip \noindent We first show that there is a canonical unitary operator between $\cH_{V}$ and $\cH$. \begin{proposition} \label{st3.3} Assume that $V, \:\e^{-\beta V}\in L^{1}(Q, \Sigma_{0}, \mu)$. Set \[ \matrix{ \hat{I}\colon & L^{\infty}( Q, \Sigma_{+}, \mu)/\cN_{V} & \to & \cH_{OS}/\cN \cr &&\cr & \cV_{V}(\psi) & \mapsto & {\cV(F_{[0, \beta/2]}\psi) \over \bigl(\int_Q F_{[-\beta/2, \beta/2]} \d \mu\bigr)^{\12}}. \cr} \] Then $\hat{I}$ is a well defined isometry from $\cH_{OS, V}/\cN_{V}$ into $\cH_{OS}/\cN$ with dense range and domain. Hence $\hat{I}$ uniquely extends to a unitary map $\hat{I} \colon \cH_{ V}\to \cH$. \end{proposition} \proof Note that $\mu_{V}$ is absolutely continuous w.r.t.\ $\mu$. Thus $L^{\infty}(Q, \Sigma, \mu_{V})= L^{\infty}(Q, \Sigma, \mu)$. If $\psi\in L^{\infty}(Q, \Sigma, \mu)\cap \cN_{V}$, then $\int_Q \, R\overline{\psi}\psi \d \mu_{V}= \int_Q \d \mu R\overline{F_{[0, \beta/2]}\psi}F_{[0, \beta/2]}\psi =0$. Hence~$F_{[0, \beta/2]}\psi\in \cN$. Consequently $\hat{I}$ is well defined. $\hat{I}$ is clearly isometric since \[ (\cV_{V}\psi, \cV_{V}\psi)_{V}= \frac{\int_Q R\overline{\psi}\psi \d \mu_{V} }{ \int_Q F_{[-\beta/2, \beta/2]} \d \mu } =\frac{ \int_Q R\overline{F_{[0, \beta/2]}\psi }F_{[0, \beta/2]}\psi \d \mu }{ \int_Q F_{[-\beta/2, \beta/2]} \d \mu } =(\hat{I}\cV_{V}\psi, \hat{I}\cV_{V}\psi). \] $\hat{I}$ is densely defined since $L^{\infty}(Q, \Sigma_{+}, \mu)$ is dense in $\cH_{OS, V}$. Since $\cV_{V}$ is a contraction, $L^{\infty}(Q, \Sigma_{+}, \mu)/\cN_{V}$ is dense in $\cH_{OS, V}/\cN_{V}$ and hence in $\cH_{ V}$. Finally, we note that ${\rm Ran}\hat{I}$ contains $\cV\bigl(F_{[0, \beta/2]} L^{\infty}(Q, \Sigma_{+}, \mu)\bigr)$. Since $F_{[0, \beta/2]}>0$ a.e., $F_{[0, \beta/2]} L^{\infty}(Q, \Sigma, \mu)$ is dense in~$\cH_{OS}$ and hence its image under $\cV$ is dense in $\cH$ \qed . \medskip \medskip \noindent {\bf Identification of the abelian algebra} \begin{proposition} \label{st3.4b} For $f\in L^{\infty}(Q,\Sigma_{0}, \mu)$ one has \[ \hat{I}\tilde{f}\psi= \tilde{f}\hat{I}\psi,\: \psi\in \cH_{V}, \] and, consequently, $\hat{I}\cU_{V}=\cU \hat{I}$. \end{proposition} \proof This follows immediately from the definitions of $\tilde{f}$ in Proposition \ref{st1.2} and $\hat{I}$ in %Theorem \ref{st3.4} Proposition 7.7 \qed . \medskip \noindent {\bf Identification of the $C^{*}$-dynamics} \medskip \noindent Applying Theorem \ref{pt1} we obtain a selfadjoint operator $H_{V}$ from the FKN kernel associated to $V$. It will be called the {\em pseudo-Liouvillean} generated by $V$. \begin{proposition} One has \label{st3.5} \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & $\hat{I}\Omega_{V}=\|\e^{-\beta H_{V}/2}\Omega\|^{-1}\e^{-\beta H_{V}/2}\Omega$; \cr {\rm (ii)} & for $0\leq s_{1}\leq\cdots\leq s_{n}\leq \beta/2$ and $A_{1}, \dots, A_{n}\in \cU$ \[ \begin{array}{rl} &\hat{I}\e^{-s_{1}H_{V}}A_{1} \bigl(\prod_{2}^{n}\e^{(s_{j-1}-s_{j})H_{V}}A_{j} \bigr)\Omega_{V}\\[2mm] =&{\e^{-s_{1}H_{V}}A_{1}\bigl(\prod_{2}^{n}\e^{(s_{j-1}-s_{j})H_{V}}A_{j} \bigr)\e^{(s_{n}-\beta/2)H_{V}}\Omega \over \|\e^{-\beta H_{V}/2}\Omega\| } \, \, ; \end{array} \] \cr {\rm (iii)} & for $t_{1}, \dots, t_{n}\in \rr$, $A_{1}, \dots, A_{n}\in \cU$ and $\psi\in \cH_{ V}$ \[ \hat{I} \bigl( \prod_{1}^{n}\e^{\i t_{j}L_{V}}A_{j}\e^{-\i t_{j}L_{V}}\bigr) \psi= \bigl(\prod_{1}^{n}\e^{\i t_{j}H_{V}}A_{j}\e^{-\i t_{j}H_{V}} \bigr)\hat{I}\psi \, ; \] \cr {\rm (iv)} & $\hat{I}J_{V}= J\hat{I}$. \cr} \end{proposition} Note that in {\rm (ii)} and {\rm (iii)} we identify $\cU$ with $L^{\infty}(Q, \Sigma_{0}, \mu)$. \medskip \noindent {\bf Identification of the observable algebras} \medskip \noindent We recall that the observable algebra and the dynamics associated to the perturbed path space $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu_{V})$ are the von Neumann algebra $\cF_{V}$ generated by $\{\e^{\i tL_{V}}A\e^{-\i tL_{V}} \mid A\in \cU_{V}, \: t\in \rr\}$ and the automorphism group $\tau_V \colon t \mapsto \tau_V (t)$, $t\in \rr$, where \[ \tau_{V}(t)(B)= \e^{\i tL_{V}}B\e^{-\i tL_{V}}, \: B\in \cF_{V} . \] \begin{proposition} \ \label{st3.6} \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & $\hat{I}\tau_{V}(t)(B)\hat{I}^{-1}= \e^{\i tH_{V}}\hat{I}B\hat{I}^{-1}\e^{-\i tH_{V}}$ for $B\in \cF_{V}$ and $t\in \rr$; \cr {\rm (ii)} & Assume that either $V\in L^{2+\epsilon}(Q, \Sigma_{0}, \mu)$ for $\epsilon>0$ or that $V\in L^{2}(Q, \Sigma_{0}, \mu)$ and $V\geq 0$. It follows that $\hat{I}\cF_{V}\hat{I}^{-1}= \cF$. \cr} \end{proposition} \proof {\rm (i)} follows from Proposition \ref{st3.5} {\rm (iii)}. To prove {\rm (ii)} we recall from Theorem \ref{pt3} that, under the assumptions of the proposition, $L+V$ is essentially selfadjoint on $\cD(L)\cap \cD(V)$ and $H_{V}= \overline{L+V}$. Hence, by Trotter's formula, \[ \e^{\i tH_{V}}= \slim_{n\to\infty}(\e^{\i tL/n}\e^{\i tV/n})^{n}. \] Thus \[ \e^{\i tH_{V}}A\e^{-\i tH_{V}}=\wlim_{n\to +\infty}(\e^{\i tL/n}\e^{\i tV/n})^{n}A(\e^{-\i tV/n}\e^{-\i tL/n})^{n}. \] Since $\e^{\i sV}\in \cU\subset \cF$, $A\in \cF$ implies that $\e^{\i sV}A\e^{-\i sV}\in \cF$. Moreover, $\e^{\i sL}A\e^{-i sL}\in \cF$ by definition. So $\e^{\i tH_{V}}A\e^{-\i tH_{V}}\in \cF$, if $A\in\cU$, and hence \[ \hat{I}\cF_{V}\hat{I}^{-1}\subset \cF. \] According to Tomita's theorem (see, e.g., [BR]) $\cF'=J\cF J$ and $\cF_{V}'=J_{V}\cF_{V}J_{V}$. Using Proposition \ref{st3.5} {\rm (iv)}. Thus \[ (\hat{I}\cF_{V}\hat{I}^{-1})'= \hat{I}\cF_{V}'\hat{I}^{-1}= \hat{I}J_{V}\cF_{V}J_{V}\hat{I}^{-1}= J\hat{I}\cF_{V}\hat{I}^{-1}J\subset J\cF J= \cF'. \] Taking commutants we obtain \[ \cF=\cF''\subset (\hat{I}\cF_{V}\hat{I}^{-1})''= \hat{I}\cF_{V}\hat{I}^{-1}. \] Hence $\cF= \hat{I}\cF_{V}\hat{I}^{-1}$\qed . The results in this section are summarized in the following theorem. \begin{theoreme} Let $(\cF, \cU, \tau , \omega)$ be a stochastically positive $\beta$-KMS system. Let $\cH, \Omega, L$ be the associated GNS Hilbert spaces, GNS vector and Liouvillean. Let $V$ be a selfadjoint operator on $\cH$, affiliated to $\cU$, such that \[ \matrix { V,\: \e^{-\beta V}\in L^{1}(Q, \Sigma_{0}, \mu) \hbox{ and} & \hbox{ either } & V\in L^{2+\epsilon}(Q, \Sigma_{0}, \mu), \:\epsilon>0, \cr & \hbox{ or } & V\in L^{2}(Q, \Sigma_{0}, \mu) \hbox{ and } V\geq 0.} \] % Then \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent =0pt \hsize=12cm \strut # \strut} \cr {\rm (i)} & $L+V$ is essentially selfadjoint on $\cD(L)\cap \cD(V)$; \cr {\rm (ii)} & $\Omega\in \cD(\e^{-\frac{\beta}{2}H_{V}})$, where $H_{V}=\overline{L+V}$; \cr {\rm (iii)} & $(\cF, \cU, \tau_{V}, \omega_{V})$ is a stochastically positive $\beta$-KMS system for \\ $\tau_{V, t}(A)= \e^{\i tH_{V}}A\e^{-\i tH_{V}}$, $\omega_{V}(A)= \|\e^{-\frac{\beta}{2}H_{V}}\Omega\|^{-2}(\e^{-\frac{\beta}{2}H_{V}}\Omega, A\e^{-\frac{\beta}{2}H_{V}}\Omega)$, $A\in \cF$. \cr} \end{theoreme} \vskip .3cm \noindent {\bf Perturbed Liouvillean} \vskip .2cm \noindent In the next theorem, we identify the Liouvillean for the perturbed system. \begin{theoreme} \label{liouvillean} Assume that $V$ is a selfadjoint operator affiliated to $\cU$ such that \beq \e^{-\beta V}\in L^{1}(Q, \Sigma_0, \mu) \eeq and \beq \label{hyp} \begin{array}{l} V\in L^{p}(Q, \Sigma_0, \mu),\: \e^{-\frac{\beta}{2}V}\in L^{q}(Q, \Sigma_0, \mu) \, \hbox{ for }\, p^{-1}+ q^{-1}= \12,\:2< p, \, \, q< \infty,\\ \hbox{or }V\in L^{2}(Q, \Sigma_{0}, \mu) \hbox{ and }V\geq 0. \end{array} \eeq % Let $L_{V}$ be the Liouvillean associated to the $\beta$-KMS system $(\cF, \tau_{V }, \omega_{V})$. Then $H_{V}-JVJ$ is essentially selfadjoint on $\cD(H_{V})\cap \cD(JVJ)$ and $L_{V}= \overline{H_{V}-JVJ}$. \end{theoreme} \begin{lemma} \label{l1} For $A\in\cU$ one has $JA\Omega_{V}=\|\e^{-\frac{\beta}{2}H_{V}}\Omega\|^{-1}\e^{-\frac{\beta}{2}H_{V}}A^{*}\Omega$. \end{lemma} \proof Let us set $c= \|\e^{-\frac{\beta}{2}H_{V}}\Omega\|^{-1}$. Then $A\Omega_{V}= c\cV(A F_{[0, \beta/2]})$. Moreover, $JA\Omega_{V}= c\cV( U (\beta/2) A^{*} F_{[0, \beta/2]})$, since $F_{[0,\beta/2]}$ is invariant under $R_{\beta/4}$. Since $A^{*}$ belongs to the space $\cM_{\beta/2}=L^{\infty}( Q, \Sigma_{0}, \mu)$ defined in Section 7.2, $\cV(A^{*})= A\Omega\in \cD(\e^{-\frac{\beta}{2}H_{V}})$ and \[ c\e^{-\frac{\beta}{2}H_{V}}A^{*}\Omega= c\cV(U (\beta /2) A^{*} F_{[0, \beta/2]})= JA\Omega_{V} \, \, \qed . \] \begin{lemma}\label{l2} Let $f_{1}$ be a real function in $L^{2}(Q, \Sigma_0, \mu)$ such that $f_{1} F_{[0, \beta/2]}\in L^{2}(Q, \Sigma_{[0, \beta/2]},\mu)$. Then $\Omega_{V} $ and $\Omega$ are vectors in $ \cD(f_{1})$. The vector $f_{1}\Omega$ is in $ \cD\bigl(\e^{-\frac{\beta}{2}H_{V}}\bigr)$ and satisfies $Jf_{1}\Omega_{V}= \|\e^{-\frac{\beta}{2}H_{V}}\Omega\|^{-1}\e^{-\frac{\beta}{2}H_{V}}f_{1}\Omega$. \end{lemma} \proof Since $f_{1}\in L^{2}(Q, \Sigma_0, \mu)$, we have $\Omega\in \cD(f_{1})$. Now $f_{1} F_{[0, \beta/2]}\in L^{2}(Q, \Sigma_{[0, \beta/2]},\mu)$, thus~$\Omega_{V}\in \cD(f_{1})$. % Let $f_{n}= f_{1}\one_{\{|f_{1}|\leq n\}}$. By dominated convergence $f_{n} F_{[0, \beta/2]}\to f_{1} F_{[0, \beta/2]}$ in $L^{2}(Q, \Sigma_{[0, \beta/2]}, \mu)$, \ie \[ f_{1}\Omega_{V}= \cV(f_{1} F_{[0, \beta/2]})=\lim_{n\to \infty}\cV(f_{n} F_{[0, \beta/2]}) =\lim_{n\to\infty}f_{n}\Omega_{V}. \] Applying Lemma \ref{l1} to $A=f_{n}$ we obtain, for $u\in \cD(\e^{-\frac{\beta}{2}H_{V}})$, \[ \begin{array}{rl} &(\e^{-\frac{\beta}{2}H_{V}}u, f_{1}\Omega)= \lim_{n\to \infty}(\e^{-\frac{\beta}{2}H_{V}}u, f_{n}\Omega)\\[2mm] = &\lim_{n\to \infty}(u, \e^{-\frac{\beta}{2}H_{V}}f_{n}\Omega)=\lim_{n\to \infty}(u, Jf_{n}\Omega_{V})= (u, Jf_{1}\Omega_{V}). \end{array} \] This shows that $f_{1}\Omega\in \cD(\e^{-\frac{\beta}{2}H_{V}})$ and $\e^{-\frac{\beta}{2}H_{V}}f_{1}\Omega= Jf_{1}\Omega_{V}$ \qed . \begin{lemma} \label{l3} Assume that $V$ is a selfadjoint operator, affiliated to $\cU$, which satisfies (\ref{hyp}). Then \[ \Omega_{V}\in \cD(H_{V})\cap \cD(V) \hbox{ and } (H_{V}- JVJ)\Omega_{V}= (H_{V}- JV)\Omega_{V}=0. \] \end{lemma} % \proof We first verify that $V$ satisfies the hypotheses of Lemma \ref{l2}, \ie that \beq \label{check} V \e^{-\int_{0}^{\beta/2}U(t)V \d t}\in L^{2}(Q, \Sigma_{[0, \beta/2]},\mu). \eeq Let $2\leq p, q\leq \infty$ be as in (\ref{hyp}). If $p=2$, then $V\geq 0$ a.e., thus (\ref{check}) is clearly satisfied. If~$q<\infty$, then, applying H\"{o}lder's inequality, it suffices to prove that \[ V \in L^{p}(Q, \Sigma, \mu) \hbox{ and } \e^{-\int_{0}^{\beta/2} U(t) V \d t} \in L^{q}(Q,\Sigma, \mu).\] Applying (\ref{bound}) we find \[ \|\e^{-\int_{0}^{\beta/2} U(t) V \d t}\|_{L^{q}(Q, \Sigma, \mu)}\leq \|\e^{-\frac{\beta}{2}V}\|_{q}<\infty. \] Let $u\in \cD\bigl(\e^{-\frac{\beta}{2}H_{V}}\bigr)\cap \cD(H_{V})\cap \cD\bigl(H_{V}\e^{-\frac{\beta}{2}H_{V}}\bigr)$ and set $c:= \|\e^{-\frac{\beta}{2}H_{V}}\Omega\|^{-1}$. Then \[ (H_{V}u, \Omega_{V})=c(\e^{-\frac{\beta}{2}H_{V}}H_{V}u, \Omega)=c(\e^{-\frac{\beta}{2}H_{V}}u, H_{V}\Omega)=c(\e^{-\frac{\beta}{2}H_{V}}u, V\Omega), \] since $\Omega\in \cD(V)\cap \cD(L)$ and $H_{V}\Omega= L\Omega+ V\Omega=V\Omega$. Applying Lemma \ref{l2} to $f_{1}=V$ we obtain \[ c(\e^{-\frac{\beta}{2}H_{V}}u, V\Omega)= c(u, \e^{-\frac{\beta}{2}H_{V}}V\Omega)= (u, JV\Omega_{V}). \] This implies, together with $J\Omega_{V}=\Omega_{V}$, that $\Omega_{V}\in \cD(H_{V})$ and $H_{V}\Omega_{V}=JV\Omega_{V}=JVJ\Omega_{V}$ \qed . \medskip \noindent {\bf Proof of Theorem \ref{liouvillean}.} Let $\cF_{1}$ be the set of $A\in \cF$ such that $t\mapsto \tau_{V,t}(A)$ is $C^{1}$ for the strong topology and let $A\in \cF_{1}$. Since $H_{V}$ implements the dynamics $\tau_{V,t}$, we see that~$A\in C^{1}(H_{V})$. By \cite{ABG}, this implies that $A\colon \cD(H_{V})\to \cD(H_{V})$. Since $\Omega_{V}\in \cD(H_{V})$, the vector $A\Omega_{V}\in \cD(H_{V})$. Since $JVJ$ is affiliated to $\cF'$, Lemma \ref{l3} implies \[ \matrix{ L_{V}A\Omega_{V} & = \i^{-1}\frac{\d}{\d t}\tau_{V,t}(A)\Omega_{V}\:_{|t=0}= H_{V}A\Omega_{V}- AH_{V}\Omega_{V} \hfill \cr &= H_{V}A\Omega_{V}- AJVJ\Omega_{V}= H_{V}A\Omega_{V}- JVJA\Omega_{V}. \hfill } \] This yields $L_{V}u= H_{V}u-JVJu$ for $u\in \cF_{1}\Omega_{V}$. By Proposition \ref{propliou}, we know that $\cF_{1}\Omega_{V}$ is a core for $L_{V}$. This implies that $L_{V}$ is the closure of $H_{V}-JVJ$ on $\cF_{1}\Omega_{V}$ and hence also the closure of $H_{V}-JVJ$ on $\cD(H_{V})\cap \cD(JVJ)$ \qed . \subsection{Markov property for perturbed of path spaces} In this subsection we show that the Markov property of a path space is preserved by the perturbations described in Subsection \ref{fkn}. \begin{proposition} \label{st3.7} Let $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu)$ be a generalized path space satisfying the Mar\--kov property and let $\{F_{[a, b]}\}$ be a FKN kernel. Then $(Q, \Sigma, \Sigma_{0}, U(t), R, \mu_F)$ satisfies the Markov property. \end{proposition} \proof Let $(Q, \Sigma, \mu)$ be a probability space, $F\in L^{1}(Q, \Sigma, \mu)$ with $F>0$ $\mu$-a.e.\ and set $\d\mu_{F}= (\int F\d\mu)^{-1}F\d\mu$. If $B\subset\Sigma$ is a $\sigma$-algebra and $f$ is $\Sigma$-measurable, then we denote by $E_{B}(f)$, (resp.\ $E_{B}^{F}(f)$) the conditional expectation of $f$ w.r.t.\ $B$ for the measure $\mu$ (resp.\ $\mu_{F}$). Then (see \cite[Sect. 2.4]{Loeve}) \beq E_{B}(fg)= E_{B}(f)g, \: E_{B}^{F}(fg)= E_{B}^{F}(f)g\: \: \mu\hbox{-a.e.\ if } g\hbox{ is } B\hbox{-measurable} \label{ste3.4} \eeq and \beq \label{ste3.3} E_{B}^{F}(f)= \frac{E_{B}(Ff)}{E_{B}(F)} \, \, \, \mu\hbox{-a.e.} \eeq To simplify the notation, let us set $E_{0}= E_{ \{Ê0\} }$ if $ \beta= + \infty$ and $E_{0}=E_{\{0, \beta/2\}}$ if $\beta < \infty$. Set $F_{+}= F_{[0,\beta/2]}$ and $F_{-}=F_{[-\beta/2, 0]}$, so that $F=F_{-}F_{+}$. Set $E_{+}^{(F)}= E_{[0, \beta/2]}^{(F)}$ and $E_{-}^{(F)}= E_{[-\beta/2, 0]}^{(F)}$. Finally set $E_{0}^{(F)}= E_{ \{Ê0\} }^{(F)}$ if $ \beta= + \infty$ and $E_{0}^{(F)}=E_{\{0, \beta/2\}}^{(F)}$ if $\beta < \infty$. Let now $f$ be $\Sigma$-measurable. Then \[ E_{+}^{F}(f)= \frac{E_{+}(Ff)}{E_{+}(F)}=\frac{E_{+}(F_{-}F_{+}f)}{E_{+}(F_{-}F_{+})}=\frac{E_{+}(F_{-}f)}{E_{+}(F_{-})}, \] using (\ref{ste3.3}), (\ref{ste3.4}) and the fact that $F_{+}$ is $\Sigma_{[0, \beta/2]}$-measurable. Next \[ \frac{E_{+}(F_{-}f)}{E_{+}(F_{-})}=\frac{E_{+}(F_{-}f)}{E_{+}E_{-}(F_{-})}=\frac{E_{+}(F_{-}f)}{E_{0}(F_{-})}, \] by the Markov property for $(Q, \Sigma, \mu)$ and the fact that $F_{-}$ is $\Sigma_{[-\beta/2, 0]}$-measurable. Since $E_{0}(F_{-})$ is $\Sigma_{[-\beta/2, 0]}$-measurable, we have, by (\ref{ste3.3}) and (\ref{ste3.4}), \[ E_{-}^{F}E_{+}^{F}(f)= \frac{E_{-}(FE_{+}(F_{-}f))}{E_{0}(F_{-})E_{-}(F)}= \frac{E_{-}(F_{-}F_{+}E_{+}(F_{-}f))}{E_{0}(F_{-})E_{-}(F_{-}F_{+})}= \frac{E_{-}(F_{+}E_{+}(F_{-}f))}{E_{0}(F_{-})E_{-}(F_{+})}, \] since $F_{-}$ is $\Sigma_{[-\beta/2, 0]}$-measurable. Now \[ \frac{E_{-}(F_{+}E_{+}(F_{-}f))}{E_{0}(F_{-})E_{-}(F_{+})}= \frac{E_{0}(Ff)}{E_{0}(F_{+})E_{0}(F_{-})}, \] by the Markov property for $(Q, \Sigma, \mu)$ and the fact that $F_{+}$ is $\Sigma_{[0, \beta/2]}$-measurable. Finally \[ \begin{array}{rl} E_{0}(F_{-})E_{0}(F_{+})= &E_{+}E_{-}(F_{-})E_{0}(F_{+})= E_{+}(F_{-}E_{0}(F_{+}))\\=& E_{+}(F_{-}E_{-}(F_{+}))= E_{+}E_{-}(F_{-}F_{+})= E_{0}(F). \end{array} \] This yields $E_{-}^{F}E_{+}^{F}(f)= E_{0}^{F}(f)$ $\mu$-a.e.\ and completes the proof \qed . \section{Free Klein-Gordon fields at positive temperature} \init \label{sec2} In this section we recall some results about the complex Klein-Gordon field and show that it provides an example of a charge symmetric K\"{a}hler structure. The classical Klein-Gordon equation describing a charged particle of mass $m$ is \[ \p_{t}^{2}\Phi- \p_{x}^{2}\Phi + m^{2}\Phi=0, \: (t,x)\in \rr^{d+1},\: \] where $\Phi\colon \rr^{d+1} \to \cc $ is a complex valued function. For later use we recall the discrete symmetries of the Klein-Gordon equation, namely the {\em parity} $\bp$, {\em time reversal} $\theta$ and {\em charge conjugation} $\bc$: \[ \bp \Phi(t,x):= \Phi(t, -x), \: \theta \Phi(t,x)= \overline{\Phi}(-t,x)\hbox{ and }\bc \Phi(t,x)= \overline{\Phi}(t,x). \] In particular, real solutions of the Klein-Gordon equation without external field describe neutral scalar particles. In the sequel only time-reversal and charge conjugation will play a role. \subsection{The complex Klein-Gordon field} \label{ckg} Let us now describe the abstract Klein-Gordon equation that we will consider in the sequel. \vskip .3cm \noindent {\bf Abstract Klein-Gordon equation} \vskip .2cm \noindent Let $\ch$ be a Hilbert space. We denote by $\i$ the complex structure on $\ch$ and by $\hf_{\ch}$ the scalar product on $\ch$. We assume that $\ch$ is equipped with a conjugation denoted by $\Phi \to \overline{\Phi}$. Let \beq \epsilon\geq m >0 \label{e2.01} \eeq be a real selfadjoint operator on $\ch$, \ie such that $\overline{\epsilon \Phi}= \epsilon \overline{\Phi}$. For $0\leq s\leq 1$ we denote by $\ch_{s}$ the Hilbert space $\cD(\epsilon^{s})$ with complex structure $\i$ and scalar product $v, u \mapsto (v, \epsilon^{2s}u)_{\ch}$ and by $\ch_{-s}$ the completion of $(\ch, \i)$ for the norm $(v, \epsilon^{-2s }v)_{\ch}$. The space $\ch_{-s}$ can be identified with the anti-dual of $\ch_{s}$ using the sesquilinear form $\langle v, u\rangle=(v, u)_{\ch}$ for $v\in \ch_{-s}$ and $u\in \ch_{s}$. We consider the abstract Klein-Gordon equation \[ \hbox{(KG)}\:\: (\p_{t}^{2}\Phi) (t)+ \epsilon^2\Phi (t)=0, \] where $\Phi(t)$ is a function of $t\in \rr$ with values in $\ch$. This (complex) KG equation describes a classical field of scalar charged particles. The complex structure on $\ch$ yields a complex structure on the space of solutions of (KG), associated to the $U(1)$ gauge group. Following the convention of Subsection \ref{notat} this `charge' complex structure will be denoted by $\j$. It is defined by \[ (\j \Phi)(t):= \i \Phi(t) \hbox{ for }\Phi\hbox{ a solution of (KG) and }t\in \rr. \] The following quantity does not depend on $t$: \[ q(\Psi, \Phi):= \i \bigl(\Psi(t), (\p_{t}\Phi)(t) \bigr)_{\ch}- \i \bigl((\p_{t}\Psi)(t), \Phi (t)\bigr)_{\ch} . \] Hence it defines a symmetric sesquilinear form on the space of solutions of (KG). The following transformations preserve the solutions of (KG): \vskip .2cm -- {\em gauge transformations} $\Phi(t)\mapsto \e^{\, \i \alpha}\Phi(t)=(\e^{\, \j \alpha}\Phi)(t)$, $\alpha\in [0, 2\pi]$; -- {\em time-reversal} $\theta \colon \Phi(t)\mapsto \overline{\Phi}(-t)$; -- {\em charge conjugation} $\bc \colon \Phi(t)\mapsto \overline{\Phi (t)}$. \vskip .3cm \noindent {\bf Energy space} \vskip .2cm \noindent It is convenient to identify a solution of (KG) with its Cauchy data at $t=0$, \[ f:= (\Phi(0),(\p_{t}\Phi)(0))\in \ch\times \ch. \] To do so one introduces the {\em energy space } ${\cal E}:= \ch_{1}\oplus \ch$ equipped with the norm \[(f,f)_{\cal E}= (f_{1}, \epsilon^2f_{1})_{\ch}+ (f_{2}, f_{2})_{\ch}, \] where we set $f= (f_{1}, f_{2})$. Note that the complex structure $\j$ becomes $\i\oplus \i$ on ${\cal E}$. Setting $f_{t}= \bigl(\Phi (t), (\p_{t}\Phi) (t) \bigr)$ one can rewrite the Klein-Gordon equation as the first order system: \[ \j (\p_{t}f)_{t}= Lf_{t} \hbox{ for }L=\left(\begin{array}{cc} 0&\i \\ -\i \epsilon^2&0 \end{array}\right). \] It is convenient to diagonalize $L$ using the unitary map \[ \matrix{ U_{0}\colon &{\cal E} & \to & \ch\oplus \ch \cr & f & \mapsto & u= (u_{1}, u_{2}), \cr} \] where \[ U_{0}:=\frac{1}{\sqrt{2}}\left(\begin{array}{cc} \epsilon&\i\\ \epsilon &-\i \end{array}\right) \hbox{ and } U_{0}^{-1}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc} \epsilon^{-1}&\epsilon^{-1}\\ -\i &\i \end{array}\right). \] It follows that \[ U_{0}LU_{0}^{*}= \left(\begin{array}{cc} \epsilon&0\\ 0&-\epsilon \end{array}\right). \] In particular, $L$ is selfadjoint on ${\cal E}$ with domain $U^{-1}(\ch_{1}\times\ch_{1})$ and the evolution $\rr\ni t\mapsto \e^{-\j tL}$ is a strongly continuous unitary group. Therefore the space of solutions of (KG) can be identified with ${\cal E}$. On ${\cal E}$ the symmetric form $q$ is \[ q(g, f)= \i (g_{1}, f_{2})_{\ch}- \i (g_{2}, f_{1})_{\ch}. \] \vskip .2cm \noindent {\bf Charged K\"{a}hler space structure} \vskip .2cm \noindent On ${\cal E}$ we put the `energy' complex structure $\i:= \j\frac{L}{|L|}$. \begin{proposition} The space $({\cal E},\j, \i , q)$ is a charged K\"{a}hler space. \end{proposition} \proof Clearly $[\i, \j]=0$. We have to prove that \[ (g,f):= {\rm Im}q(g, \i f)+ \i {\rm Im}q(g, f) \] is a positive definite symmetric sesquilinear form on $({\cal E}, \i)$. If $U_{0}f= (u_{1}, u_{2})$ and $U_{0}g=(v_{1}, v_{2})$, then \[ \begin{array}{l} q(g, f)= -(v_{2}, \epsilon^{-1}u_{2})_{\ch}+ (v_{1}, \epsilon^{-1}u_{1})_{\ch},\\[3mm] q(g, \i f)= -(v_{2}, -\i \epsilon^{-1}u_{2})_{\ch}+ (v_{1}, \i \epsilon^{-1}u_{1})_{\ch}= \i(v_{1}, \epsilon^{-1}u_{1})_{\ch}+\i(v_{2}, \epsilon^{-1}u_{2})_{\ch},\\[3mm] \end{array} \] and consequently \beq (g,f)= (v_{1}, \epsilon^{-1}u_{1})_{\ch}+ \overline{(v_{2},\epsilon^{-1}u_{2})_{\ch}}. \label{e1.1b} \eeq \qed \begin{definition} We denote by $\bigl({\cal E}_{\rm q}, \i, \hf \bigr)$ the completion of $({\cal E}, \i)$ for the scalar product~$\hf$. \end{definition} \begin{proposition} The space ${\cal E}_{\rm q}$ is equal to the space $\ch_{\12}\oplus\ch_{-\12}$ equipped with the complex structure \[ \i=\left(\begin{array}{cc} 0&-\epsilon^{-1}\\\epsilon &0 \end{array}\right) \] and the scalar product $(g,f)= {\rm Re}(g_{1}, \epsilon f_{1})_{\ch}+ {\rm Re}(g_{2}, \epsilon^{-1}f_{2})_{\ch}+ \i \bigl({\rm Re}(g_{1}, f_{2})_{\ch}- {\rm Re}(g_{2}, f_{1})_{\ch} \bigr)$. \end{proposition} \vskip .2cm \noindent {\bf Standard form of the complex Klein-Gordon field} \vskip .2cm \noindent It is convenient to introduce the map \[ \begin{array}{l} U_{\rm q}(f_{1}, f_{2}):= \frac{1}{\sqrt{2}} \bigl( \epsilon^{\12} f_{1}+ \i \epsilon^{-\12}f_{2}, \epsilon^{\12} \overline{f}_{1} + \i \epsilon^{-\12}\overline{f}_{2}\bigr)=:(u_{1}, u_{2} ). \end{array} \] Using (\ref{e1.1b}) we obtain that $U_{\rm q}$ extends to a unitary map \[ U_{\rm q}\colon \bigl({\cal E}_{\rm q}, \i, (\cdot, \cdot)\bigr)\to (\ch, \i)\oplus (\ch, \i). \] Let us describe the various objects after conjugation by $U_{\rm q}$. We will denote by the same letter an object acting on ${\cal E}_{\rm q}$ and its conjugation by $U_{\rm q}$ acting on $\ch\oplus\ch$. \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent = 0pt \hsize=12cm \strut # \strut} \cr -- & {\em symmetric form}: after conjugation by $U_{\rm q}$ the symmetric form $q(g,f)$ becomes \[ q \bigl((v_{1}, v_{2}),(u_{1}, u_{2}) \bigr)= (v_{1}, u_{1})-(u_{2}, v_{2}). \] \cr -- & {\em `charge' complex structure}: after conjugation by $U_{\rm q}$ the complex structure $\j$ becomes \[ \j = \left(\begin{array}{cc} \i &0\\ 0 &-\i \end{array}\right). \] \cr -- & {\em Hamiltonian}: the infinitesimal generator of $\rr\ni t\mapsto \e^{-\j tL}$ on $\bigl({\cal E}_{\rm q}, \i , \hf \bigr)$ is the {\em Hamiltonian}, denoted by ${\rm h}$. After conjugation by $U_{\rm q}$, \[ {\rm h} =\left(\begin{array}{cc} \epsilon &0\\ 0 &\epsilon \end{array}\right). \] In particular ${\rm h}$ is positive. \cr -- & {\em Gauge transformations:} the infinitesimal generator of $[0, 2\pi]\ni \alpha\mapsto \e^{-\j \alpha}$ on $\bigl({\cal E}_{\rm q}, \i , \hf \bigr)$ is the {\em charge operator} ${\rm q}$. After conjugation by $U_{\rm q}$, \[ {\rm q}= \left(\begin{array}{cc} 1&0\\ 0 &-1 \end{array}\right). \] We have ${\rm q}= -\i \j$. Hence ${\rm q}$ is a charge operator in the sense of Subsection \ref{sec1.2}. \cr -- & {\em Time reversal:} we have $\theta (f_{1}, f_{2})= (\overline{f}_{1}, -\overline{f}_{2})$, and after conjugation by $U_{\rm q}$, \[ \theta (u_{1}, u_{2})= (\overline{u}_{1}, \overline{u}_{2}). \] \cr -- & {\em charge conjugation:} we have $\bc (f_{1}, f_{2})= (\overline{f}_{1}, \overline{f}_{2})$, and after conjugation by $U_{\rm q}$, \[ \bc (u_{1}, u_{2})= (u_{2}, u_{1}). \] We see that $({\cal E}_{\rm q}, \j ,\i,q, \bc)$ is a charge-symmetric K\"{a}hler space. \cr} \vskip 1cm From now on we will set $X:= \ch\oplus \ch$ with elements $x= (x^{+}, x^{-})$ and equip $X$ with the complex structures \[ \i = \left(\begin{array}{cc} \i &0\\ 0 &\i \end{array}\right) \hbox{ and } \j = \left(\begin{array}{cc} \i &0\\ 0 &-\i \end{array}\right), \] with the symmetric form and the scalar product \[ q( y, x)= ( y^{+}, x^{+})- (x^{-}, y^{-} ) \hbox{ and } (y, x):= ( y^{+}, x^{+})+ (y^{-}, x^{-}), \] the Hamiltonian and the charge operator \[ {\rm h}= \left(\begin{array}{cc} \epsilon &0\\ 0 &\epsilon \end{array}\right) \hbox{ and } {\rm q}=\left(\begin{array}{cc} \one &0\\ 0 &-\one \end{array}\right), \] and the time-reversal and the charge conjugation \[ \theta (x^{+}, x^{-})= (\overline{x^{+}}, \overline{x^{-}}) \hbox{ and } \bc (x^{+}, x^{-})= (x^{-}, x^{+}). \] From the discussion above we obtain the following theorem. \begin{theoreme} The map $U_{\rm q}\colon ({\cal E}_{\rm q}, \j ,\i,q, \bc)\to (X, \j, \i, q, \bc)$ is unitary between $\bigl({\cal E}_{\rm q}, \i, \hf\bigr)$ and $\bigl(X, \i, \hf \bigr)$, and isometric between $({\cal E}_{\rm q}, \j, q)$ and $(X, \j, q)$. It satisfies \[ U_{\rm q}aU_{\rm q}^{-1}= a \hbox{ for }a={\rm h},\: {\rm q},\:{\rm t},\: {\rm c}. \] \end{theoreme} For later use we set $\kappa:= \theta \bc$ and $X_{\kappa}:=\{x\in X| \kappa x=x\}=\{(x^{+}, \overline{x}^{+}),\: x^{+}\in \ch\}$. Note that in terms of solutions of (KG) we have $\kappa\Phi(t,x)= \Phi(-t, x)$ and an element of $X_{\kappa}$ corresponds to a solution of (KG) with Cauchy data $(u,0)$, where $u\in \ch_{\12}$. We see that $\kappa$ is a conjugation on $(X, \i, \hf)$ and hence ${\rm Im\hf}$ vanishes on $X_{\kappa}$. Since~$[\kappa , \j]=0$, the vector space $X_{\kappa}$ is a complex vector space for the complex structure~$\j$. For comparison with the physics literature, let us consider the case $\ch=L^{2}(\rr^{d}, \d x)$ and $\epsilon= (-\Delta_{x}+ m^{2})^{\12}$. Then $\ch_{-\12}$ is the Sobolev space $H^{-\12}(\rr^{d})$. In the physics literature one defines for $u\in \coinf(\rr^{d})$ the time-zero field $\phi_{\rm p}(u)$ to be the Hermitian field associated with the solution of (KG) with Cauchy data $\bigl(\frac{1}{2\pi}\epsilon^{-1}u, 0\bigr)$. After the unitary transformation $U_{\rm q}$, $\bigl(\frac{1}{2\pi}\epsilon^{-1}u, 0 \bigr)$ becomes the element \[ \frac{1}{\sqrt{2}2\pi} \bigl( \epsilon^{-\12}u, \epsilon^{-\12}\overline{u} \bigr)\in L^{2}(\rr^{d} )\oplus L^{2}(\rr^{d}), \] \ie \[ \phi_{\rm p}(u)= \frac{1}{\sqrt{2}2\pi}\phi \bigl( \epsilon^{-\12}u, \epsilon^{-\12}\overline{u}\bigr). \] In the physics litterature one also considers the {\em complex time-zero field} $\varphi_{\rm p}(u)$ defined as $\phi_{\rm p}(u)+ \i \phi_{\rm p}(\i u)$, \ie \[ \varphi_{\rm p}(u)= \frac{1}{2\pi}\varphi \bigl(\epsilon^{-\12}u, \epsilon^{-\12}\overline{u} \bigr). \] \subsection{The real Klein-Gordon field} \init \label{sec3} We now quickly discuss the real Klein-Gordon field. \vskip .3cm \noindent {\bf Abstract real Klein-Gordon equation} \vskip .2cm \noindent Let $\ch_{\rr}$ be a real Hilbert space. Let $\epsilon \geq m>0$ be a selfadjoint operator on $\ch_{\rr}$. We consider the Klein-Gordon equation: \[ \p_{t}^{2}\Phi(t) + \epsilon^2\Phi(t) =0, \] where $\Phi$ is a function of $t\in \rr$ with values in $\ch_{\rr}$. The real Klein-Gordon equation describes a classical field of scalar neutral particles. Let us denote by $\ch:=\cc\ch_{\rr}$ the complexification of $\ch_{\rr}$ with its canonical scalar product~$(\cdot, \cdot)_{\ch}$. The space $\ch$ is equipped with the canonical conjugation $\ch\ni \Phi\mapsto \overline{\Phi}$, $\Phi\in \ch$. On the space of real solutions of the Klein-Gordon equation, the charge conjugation $\bc$ acts as identity and the time-reversal $\theta $ takes the form $\theta \colon\Phi(t)\mapsto \Phi (-t)$. We will still denote by $\epsilon$ the complexification of $\epsilon$ acting on $\ch$. We can now apply the results of Subsection \ref{ckg} to the Hilbert space $\ch$. The real energy space is ${\cal E}_{\rr}:= {\cal E}\cap \ch_{\rr}\times \ch_{\rr}$. The image of ${\cal E}_{\rr}$ under the transformation~$U$ is \[ U{\cal E}_{\rr}=: {\cal S}_{\rr}= \bigl\{(u_{1}, u_{2})\in \ch\oplus\ch| u_{2}= \overline{u_{1}} \bigr\}. \] Note that $\e^{-\j tL}$ preserves ${\cal E}_{\rr}$. More general, if $F\colon \rr\to \cc$ is a bounded measurable function such that $\overline {F}(\lambda)= F(-\lambda)$ then $F(L)$ preserves ${\cal E}_{\rr}$. Therefore $\i$ preserves~${\cal E}_{\rr}$ and hence defines a complex structure on ${\cal E}_{\rr}$. The space $({\cal E}_{\rr}, \i , q)$ is a K\"{a}hler space. \begin{definition} We denote by $\bigl({\cal E}_{{\rm q}, \rr}, \i, \hf \bigr)$ the closure of $({\cal E}_{\rr}, \i)$ for the scalar product~$\hf$. \end{definition} \begin{proposition} The space ${\cal E}_{{\rm q}, \rr}$ is equal to $\ch_{\12, \rr}\oplus \ch_{-\12, \rr}$ equipped with the complex structure \[ \i=\left(\begin{array}{cc} 0&-\epsilon^{-1}\\\epsilon &0 \end{array}\right) \] and the scalar product $(g,f)= (g_{1}, \epsilon f_{1})_{\ch}+ (g_{2}, \epsilon^{-1}f_{2})_{\ch}+ \i \bigl((g_{1}, f_{2})_{\ch}- (g_{2}, f_{1})_{\ch} \bigr)$. \end{proposition} \vskip .2cm \noindent {\bf Standard form of the real Klein-Gordon field} \vskip .2cm \noindent We set \[ \matrix{ U_{\rr}\colon & {\cal E}_{\rr} & \to & \ch \cr & f & \mapsto &(\epsilon^{\12} f_{1}+ \i\epsilon^{-\12} f_{2}). \cr} \] Then $U_{\rr}$ extends to a unitary map between $\bigl({\cal E}_{{\rm q}, \rr}, \i , \hf \bigr)$ and $\ch$. Let us describe the various objects after conjugation by $U_{\rr}$: \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent = 0pt \hsize=12cm \strut # \strut} \cr $ $- & {\em Hamiltonian}: The infinitesimal generator of $\rr\ni t\mapsto \e^{-\j tL}$ on $\bigl({\cal E}_{{\rm q}, \rr}, \i , (\cdot, \cdot)\bigr)$ is the {\em Hamiltonian} denoted by ${\rm h}$. After conjugation by $U_{\rr}$, \[{\rm h}=\epsilon. \] In particular, ${\rm h}$ is positive. \cr $ $- & {\em Time reversal:} We have $\theta (f_{1}, f_{2})= (f_{1},-f_{2})$. After conjugation by $U_{\rr}$, one finds $\theta u_{1}=\overline{u}_{1}$. \cr} \vskip .3cm \noindent From the discussion above we obtain the following theorem. \begin{theoreme} There exist a map $U_{\rr}$ between $({\cal E}_{{\rm q},\rr}, \i,q, \theta)$ and $(\ch, \j, q, \theta)$ which is unitary between $\bigl({\cal E}_{\rm q,\rr}, \i, \hf \bigr)$ and $\bigl(\ch, \j, \hf \bigr)$, and satisfies \[ U_{{\rm q},\rr}aU_{{\rm q},\rr}^{-1}= a \hbox{ for }a={\rm h},\:{\rm t}. \] \end{theoreme} \vskip .3cm \noindent For later use we set $\kappa:= \theta $ and $\ch_{\kappa}:=\{h\in \ch\mid h=\overline{h}\}$. \subsection{Free Klein-Gordon fields at positive temperature} \label{poskg} We can now apply the results of Section \ref{quasi} to the real and complex Klein-Gordon fields. In the complex case we set $X=\ch\oplus\ch$, ${\rm h}=\epsilon\oplus \epsilon$, ${\rm q}=\one\oplus -\one$ and introduce for $|\mu|0, \label{polynome} \eeq and a real function $g\in L^{1}_{\rr}(\rr, \d x)\cap L^{2}(\rr, \d x)$ with $g\geq 0$. We set \[ V_{\Lambda}=\int g(\x):\!P(\phi_{\Lambda}(\x))\!:_{0}\d \x, \] where $:\: :_{0}$ denotes the Wick ordering with respect to the covariance at temperature $0$ given by $c_{0}(f,f)=\12(f,f)_{\ch}$. For technical reasons we will also need to consider similar UV cutoff interactions with the Wick ordering done with respect to the covariance at inverse temperature $\beta$ given by $c_{\beta}( f,f)=\12 (f,f)_{\rho}=\12 (f, (1+2\rho)f)$, $f \in \ch$. We set \[ V_{\Lambda, \beta}=\int g(\x):\!P(\phi_{\Lambda}(\x))\!:_{\beta}\d \x, \] where $:\: :_{\beta}$ denotes Wick ordering with respect to $c_{\beta}$. Note that $V_{\Lambda}$ and $V_{\Lambda,\beta}$ are affiliated to~$\cU$. We first collect some properties of these auxiliary interactions. \begin{lemma} \label{int.3} The family $\{ V_{\Lambda,\beta} \}$ is Cauchy in all spaces $L^{p}(Q, \Sigma_0, \mu)$ for $1\leq p<\infty$ and converges when $\Lambda\to \infty$ to a function $V_{\beta}\in L^{p}(Q, \Sigma_0, \mu)$, $1\leq p<\infty$, which satisfies $\e^{-tV_{\beta}}\in L^{1}(Q, \Sigma_0, \mu)$ for all $t>0$. We set \[ V_{\beta}=:\int g(\x):\!P(\phi(\x))\!:_{\beta}\d \x. \] \end{lemma} \proof We use the identification of $L^{2}(Q, \Sigma_0, \mu)$ with $\G(\ch_{\rho})$ presented in Lemma \ref{int.2}. Then Wick ordering with respect to $c_{\beta}$ coincides with Wick ordering with respect to the Fock vacuum on $\G(\ch_{\rho})$. By exactly the same arguments as those used in the $0$-temperature case (see e.g.\ \cite{SHK} or \cite[Sect. 6]{DG} for a recent survey) we obtain that, for $0\leq p\leq 2n$, the cuttoff interaction $V_{\Lambda, \beta}$ is a linear combination of Wick monomials of the form \[ \sum\limits_{r=0}^{p}\left(\begin{array}{c}p\\r\end{array}\right) \int w_{p,\Lambda}(k_{1},\ldots, k_{r}, k_{r+1}, \ldots, k_{p}) a^{*}(k_{1})\cdots a^{*}(k_{r}) a(-k_{r+1})\cdots a(-k_{p})\d k_{1}\cdots \d k_{p}, \] where \[ w_{p,\Lambda}(k_{1}, \cdots, k_{p})= \hat{g} \bigl(\sum_{1}^{p}k_{i} \bigr)\prod_{1}^{p}\hat{\chi}\Bigl(\frac{k_{i}}{\Lambda}\Bigr)\epsilon(k_{i})^{-\12}. \] Recalling that $1+2\rho= \frac{1+\e^{-\beta\epsilon}}{1-\e^{-\beta \epsilon}}$ we see that \[ w_{p, \Lambda}\in \otimes^{p}\ch_{\rho}=L^{2} \Bigl(\rr^{p}, \prod_{1}^{p}\frac{1+\e^{-\beta\epsilon(k_{i})}}{1-\e^{-\beta \epsilon(k_{i})}}\d k_{1}\dots, \d k_{p} \Bigr). \] The sequence $\{ w_{p, \Lambda} \}$ is Cauchy in this space. Consequently $w_{p, \Lambda}\to w_{p, \infty}$ when $\Lambda\to \infty$, where \[ w_{p,\infty}(k_{1}, \cdots, k_{p})= \hat{g} \bigl(\sum_{1}^{p}k_{i} \bigr)\prod_{1}^{p}\epsilon(k_{i})^{-\12}. \] We can now apply these Wick monomials to the Fock vacuum and conclude that $V_{\Lambda, \beta}\Omega$ converges to a vector $V_{\beta}\Omega$ in $\G(\ch_{\rho})$, or equivalently that $V_{\Lambda, \beta}$ converges to $V_{\beta}$ in $L^{2}( Q, \Sigma_0, \mu)$. Since $V_{\lambda,\beta}\Omega$ is a finite particle vector, it follows from a standard argument (see e.g.\ \cite[Thm.\ 1.22]{Si1} or \cite[Lemma 5.12]{DG}) that $V_{\Lambda, \beta}\to V_{\beta}\in L^{p}(Q, \Sigma_0, \mu)$ for all $1\leq p<\infty$. We will now prove that $\e^{-tV_{\beta}}\in L^{1}(Q, \Sigma_0, \mu)$. We argue as in the $0$-temperature case: we first verify that $\|w_{p, \Lambda}-w_{p, \infty}\|\leq C\Lambda^{-\epsilon_{0}}$ for some $\epsilon_{0}>0$ and therefore $\|V_{\Lambda, \beta}-V_{\beta}\|_{L^{2}(Q, \Sigma_0, \mu)}\leq C\Lambda^{-\epsilon_{0}}$. Applying again \cite[Lemma 5.12]{DG} we find \beq\label{int.e0} \|V_{\Lambda, \beta}-V_{\beta}\|_{L^{p}(Q, \Sigma_0, \mu)}\leq C(p-1)^{n}\Lambda^{-\epsilon_{0}},\: p>1. \eeq Using the Wick ordering identities (\ref{wick}) we obtain as identities between functions on~$K$ (see, e.g., \cite[Lemma 6.6]{DG}): \[ : P(\phi_{\Lambda}(\x)) :_{\beta}\geq -C \bigl(\|\phi_{\Lambda}(\x)\Omega\|^{2n} +1\bigr). \] Now $\|\phi_{\Lambda}(\x)\Omega\|= C\|\epsilon^{-1}\hat{\chi}\Bigl(\frac{\cdot}{\Lambda} \Bigr)\|_{\ch_{\rho}}\leq C(\ln (\Lambda))^{\12}$. This yields \beq \label{int.e01}V_{\Lambda, \beta}\geq -C\ln (\Lambda)^{n}. \eeq Applying now \cite[Lemma V.5]{Si1} we deduce from (\ref{int.e0}) and (\ref{int.e01}) that $\e^{-tV_{\beta}}\in L^{1}(Q, \Sigma_0, \mu)$ for all $t>0$ \qed . \begin{proposition} \label{int.4} The family $\{ V_{\Lambda} \}$ is Cauchy in all spaces $L^{p}(Q, \Sigma_0, \mu)$ for $1\leq p<\infty$ and converges when $\Lambda\to \infty$ to a function $V\in L^{p}(Q, \Sigma_0, \mu)$, $1\leq p<\infty$, which satisfies $\e^{-tV}\in L^{1}(Q, \Sigma_0, \mu)$ for all $t>0$. We set \[ V=:\int g(\x):\!P(\phi(\x))\!:_{0}\d \x. \] \end{proposition} \proof With the help of the Wick reordering identity (\ref{wickreordering}) we find, for $f\in \ch_{\kappa}$, \[ \begin{array}{rl} :\!P(\phi_{\omega}(f))\!:_{0} & = \sum_{j=0}^{2n}a_{j} :\!\phi_{\omega}(f)^{n}\!:_{0}\\ &=\sum_{j=0}^{2n}\sum_{m=0}^{[j/2]}a_{j}\frac{j!}{m!(j-2m!)}: \!\phi(f)^{j-2m}\!: _{\beta}\bigl(-\12 (c_{0}-c_{\beta})(f,f) \bigr)^{m}. \end{array} \] For $f= f_{\Lambda, x}$ \[ \begin{array}{rl} r_{\Lambda}:=&(c_{\beta}-c_{0})(f_{\Lambda, x},f_{\Lambda, x})= ( f_{\Lambda, 0}, \rho f_{\Lambda, 0})\\ =&\int\e^{-\beta\epsilon(k)}\hat{\chi}\bigl(\frac{k}{\Lambda} \bigr)\d k= r_{\infty}+ O(\Lambda^{-\infty}), \end{array} \] where $r_{\infty}=\int\e^{-\beta\epsilon(k)}\d k$. On the other hand, \[ \int_{Q}|\phi_{\omega}(f_{\Lambda,\x})|^{p}\d\mu\in O \bigl(| c_{\beta}(f_{\Lambda, \x}, f_{\Lambda,\x})|^{p})\in O(\ln(\Lambda)^{p} \bigr). \] Therefore \[ :\!P(\phi_{\Lambda}(\x))\!:_{0}= :\!\tilde{P}(\phi_{\Lambda}(\x))\!:_{\beta} + O \bigl(\ln(\Lambda)^{2n}\Lambda^{-\infty} \bigr) \hbox{ uniformly for }\x\in \supp g, \] where \[ \tilde{P}(\lambda)=\sum_{j=0}^{2n}\sum_{m=0}^{[j/2]} a_{j}\frac{j!}{m!(j-2m!)} \lambda^{j-2m} \bigl(\12 r_{\infty} \bigr)^{m}. \] We see that $\tilde{P}(\lambda)-P(\lambda)$ is of degree less than $2n-1$. Applying Lemma \ref{int.3} to $\tilde{P}$ this yields \[ \lim_{\Lambda\to \infty}\int g(\x):\!P(\phi_{\Lambda}(\x))\!:_{0}\d\x= \lim_{\Lambda\to \infty}\int g(\x):\!\tilde{P}(\phi_{\Lambda}(\x))\!: _{\beta}\d\x= \int g(\x):\!\tilde{P}(\phi(\x))\!: _{\beta}\d\x, \] which completes the proof of the proposition \qed . \subsection{The spatially cutoff $\e^{\alpha \phi}\,\!_{2}$ interaction} \label{int.sub11} As in Subsection \ref{int.sub1} we set, for $|\alpha|<\sqrt{2\pi}$, \[ V_{\Lambda}=\int g(\x):\!\e^{\alpha\phi_{\Lambda}(\x)}\!:_{0}\d \x \] and \[ V_{\Lambda, \beta}=\int g(\x):\!\e^{\alpha\phi_{\Lambda}(\x)}\!: _{\beta}\d \x. \] Note that, as above, $V_{\Lambda}$ and $V_{\Lambda, \beta}$ are affiliated to $\cU$. \begin{lemma} \label{int.5} For $|\alpha|<\sqrt{2\pi}$ the family $\{ V_{\Lambda, \beta} \}$ is Cauchy in $L^{2}(Q, \Sigma_0, \mu)$ and converges when $\Lambda\to \infty$ to a positive function $V_{\beta}\in L^{2}(Q, \Sigma_0, \mu)$. We set \[ V_{\beta}=:\int g(x):\!\e^{\alpha \phi(\x)}\!:_{\beta}\d\x. \] \end{lemma} \proof The proof is completely similar to the $0$-temperature case where $\rho=0$ (see e.g.\ \cite{Si1}, \cite{HK}). For completeness we will give an outline. Note first that by (\ref{wickdef}) $: \!\e^{\alpha\phi_{\Lambda}(x)}\!:_\beta$ is a positive function on $Q$, hence the same holds for $V_{\Lambda,\beta}$ as $g\geq 0$. We now show that $V_{\Lambda, \beta}$ converges in $L^{2}(Q, \Sigma_0, \mu)$, and we will identify $V_{\Lambda, \beta}$ with $V_{\Lambda,\beta}\Omega$. We have \[ \one_{\{n\}}(N)V_{\Lambda, \beta}= \frac{\alpha^{n}}{n!}\int g(\x):\! \phi_{\Lambda}^{n}(\x)\!:\Omega \d\x= \frac{\alpha^{n}}{(4\pi)^{n/2}\sqrt{n!}}\hat{g}(\sum_{1}^{n}k_{i})\prod_{1}^{n}\hat{\chi}\Bigl(\frac{k_{i}}{\Lambda}\Bigr)\frac{1}{\epsilon(k_{i})^{\12}}. \] Hence \[ \begin{array}{rl} \|\one_{\{n\}}(N)V_{\Lambda,\beta}\|^{2}& = \frac{1}{n!}\bigl(\frac{\alpha^{2}}{4\pi}\bigr)^{n}\int |\hat{g}(\sum_{1}^{n}k_{i})|^{2}\prod_{1}^{n}\bigl|\hat{\chi} \bigl(\frac{k_{i}}{\Lambda}\bigr)\bigr|^{2}\frac{1+2\rho(k_{i})}{\epsilon(k_{i})}\d k_{1}\dots\d k_{n}\\ & \leq \frac{1}{n!}\bigl(\frac{\alpha^{2}}{4\pi}\bigr)^{n}\int |\hat{g}(\sum_{1}^{n}k_{i})|^{2}\prod_{1}^{n}\frac{1+2\rho(k_{i})}{\epsilon(k_{i})}\d k_{1}\dots\d k_{n}=: \epsilon_{n}. \end{array} \] Next we find \[ \epsilon_{n}= \frac{1}{n!} \Bigl(\frac{\alpha^{2}}{2\pi}\Bigr)^{n}\int g(\x)g({\rm y})K_{\beta}(\x-{\rm y})^{n}\d\x\d{\rm y} \] for \[ K_{\beta}(\x)=\12\int\e^{\i k\x}\frac{1+2\rho(k)}{\epsilon(k)}\d k. \] We claim now that \beq \label{int.e1} \e^{\frac{\alpha^{2}}{2\pi}|K_{\beta}(\x)|}\in L^{1}(\rr)+L^{\infty}(\rr) \hbox{ for }|\alpha|<\sqrt{2\pi}. \eeq This implies that \beq \label{int.e2} \sum_{n=0}^{\infty}\epsilon_{n}\leq \int g(\x)g({\rm y})\e^{\frac{\alpha^{2}}{2\pi}|K_{\beta}|(\x-{\rm y})}\d\x\d{\rm y}<\infty. \eeq If we set \[ K_{0}(\x)=\12\int\e^{\i k\x}\frac{1}{\epsilon(k)}\d k, \] then because of the rapid decay of $\rho(k)$ when $|k|\to \infty$, we have $K_{0}-K_{\beta}\in L^{\infty}(\rr)$, and (see \cite[equ. (2.4)]{HK}) $K_{0}(\x)\in O(1)$ in $|\x|\geq 1$, $K_{0}(\x)= -\ln(\x)+ O(1)$ in $|\x|\leq 1$. This implies (\ref{int.e1}). Now by the arguments in the proof of Lemma \ref{int.3}, we see that \[ \lim_{\Lambda \to \infty}\one_{\{n\}}(N)V_{\Lambda, \beta}=\frac{\alpha^{n}}{n!}\int g(\x):\!\phi(\x)^{n}\!:\Omega\d\x. \] Since $\one_{\{n\}}(N)V_{\Lambda, \beta}\to V_{n}$ in $L^{2}(Q, \Sigma_0, \mu)$ for each $n$ and $\sup_{\Lambda}\|\one_{\{n\}}(N)V_{\Lambda, \beta}\|^{2}\leq \epsilon_{n}$ with $\sum \epsilon_{n}<\infty$, we see that $V_{\Lambda, \beta}$ converges to some element $V\in L^{2}(Q, \Sigma_0, \mu)$, which is a.e.\ positive as a limit of positive functions \qed . \begin{proposition} \label{int.6} For $|\alpha|<\sqrt{2\pi}$, the family $\{ V_{\Lambda} \}$ is Cauchy in $L^{2}(Q, \Sigma_0, \mu)$ and converges to a positive function $V\in L^{2}(Q, \Sigma_0, \mu)$. We set \[ V=:\int g(x):\!\e^{\alpha \phi(\x)}\!:_{0}\d\x. \] \end{proposition} \proof By the Wick reordering identity (\ref{wickreordering2}) we have \[ :\!\e^{\alpha \phi_{\Lambda, x}}\!:_{0}= :\!\e^{\alpha \phi_{\Lambda, x}}\!:_{\beta}\e^{\frac{\alpha^{2}}{2}r_{\Lambda}}, \] Hence $V_{\Lambda}=\e^{\frac{\alpha^{2}}{2}r_{\Lambda}} V_{\Lambda, \beta}$, which implies, using Lemma \ref{int.5}, that $V_{\Lambda}$ converges in~$L^{2}(Q, \Sigma_0, \mu)$ to the positive function $\e^{\frac{\alpha^{2}}{2}r_{\infty}}V_{\beta}$ \qed . \subsection{The spatially cutoff $P(\varphi^*\varphi)_2$ interaction} \label{chargedp} We consider now the complex Klein-Gordon field in one space dimension which is described by the Weyl algebra $\fW(X)$ for $X=\ch\oplus\ch$, $\ch=L^{2}(\rr, \d k)$. We recall that the Gibbs state at inverse temperature~$\beta$ is given by $\omega(W(x))=\e^{\frac{1}{4}(x, (1+2\rho x))}$, where $\rho=(\e^{\beta {\rm h}}-1)^{-1}$ and ${\rm h}=\epsilon\oplus \epsilon$. We set \[ \varphi_{\Lambda}(\x)=\varphi_{\omega}(f_{\Lambda,\x}\oplus f_{\Lambda, \x}),\:\varphi^{*}_{\Lambda}(\x)=\varphi^{*}_{\omega}(f_{\Lambda,\x}\oplus f_{\Lambda, \x}),\:\x\in \rr. \] Note that $f_{\Lambda, \x}$ is invariant under the conjugation $h\to \overline{h}$. This implies that $\varphi_{\Lambda}(x)$ is affiliated to~$\cU$, since~$f_{\Lambda,\x}\oplus f_{\Lambda, \x}\in X_{\kappa}$. Moreover, $\varphi_{\Lambda}^{*}(\x)\varphi_{\Lambda}(\x)= \12 \bigl(\phi^{2}_{\omega}(f_{\Lambda,\x}\oplus f_{\Lambda, \x} )+\phi^{2}_{\omega}(\i f_{\Lambda,\x}\oplus -\i f_{\Lambda, \x}) \bigr)$. For $P$ a real polynomial of degree $2n$, which is bounded from below, and $g$ a positive function in~$L^{1}(\rr)\cap L^{2}(\rr)$, we set \[ V_{\Lambda}=\int g(\x):\!P(\varphi_{\Lambda}^{*}(\x)\varphi_{\Lambda}(\x))\!:_{0}\d \x, \] where $:\: :_{0}$ denotes Wick ordering with respect to the $0$-temperature covariance $c_{0}(x,x)=\12(x,x)$, and \[ V_{\Lambda, \beta}=\int g(\x):\!P(\varphi_{\Lambda}^{*}(\x)\varphi_{\Lambda}(\x))\!:_{\beta}\d \x, \] where $:\: :_{\beta}$ denotes Wick ordering with respect to the covariance at inverse temperature~$\beta$ specified by $c_{\beta}(x,x)=\12(x, (1+2\rho)x)$. The following two results can be shown by exactly the same methods as in Subsection \ref{int.sub1}. \begin{lemma} \label{int.7} The family $\{ V_{\Lambda,\beta} \}$ is Cauchy in all $L^{p}(Q, \Sigma_0, \mu)$ spaces and converges, when $\Lambda\to \infty$, to a function $V_{\beta}\in L^{p}(Q, \Sigma_0, \mu)$, $1\leq p<\infty$, which satisfies $\e^{-tV_{\beta}}\in L^{1}(Q, \Sigma_0, \mu)$ for all $t>0$. We set \[ V_{\beta}=:\int g(\x):\!P(\varphi^{*}(\x)\varphi(\x))\!:_{\beta}\d \x. \] \end{lemma} \begin{proposition} \label{int.8} The family $\{V_{\Lambda} \}$ is Cauchy in all spaces $L^{p}(Q, \Sigma_0, \mu)$ and converges, when $\Lambda\to \infty$, to a function $V\in L^{p}(Q, \Sigma_0, \mu)$, $1\leq p<\infty$, which satisfies $\e^{-tV}\in L^{1}(Q, \Sigma_0, \mu)$ for all $t>0$. We set \[ V=:\int g(\x):\!P(\varphi^{*}(\x)\varphi(\x))\!:_{0}\d \x. \] \end{proposition} \subsection{Scalar quantum fields at positive temperature with spatially cutoff interactions} \label{mainres} To construct the space-cutoff $P(\phi)_{2}$ and $\e^{\alpha\phi}\,\!_{2}$ models at positive temperature, we apply the general results of Subsection \ref{otot}. Note that by Subsections \ref{int.sub1} and \ref{int.sub11}, the interactions terms $V=\int g(\x):\!P(\phi(\x))\!: _{0}\d \x$ and $V= \int g(\x):\!\e^{\alpha \phi(\x)}\!:_{0}\d \x$ for $|\alpha|<\sqrt{2\pi}$ satisfy all the hypotheses of Subsection \ref{otot}. Consequently we obtain the following theorem: \begin{theoreme} Let $\bigl(\cW, \cW_\kappa, \tau^\circ,\omega \bigr)$ be the quasi-free $\beta$-KMS system describing the free neutral Klein-Gordon field in one space dimension at temperature $\beta^{-1}$, described in Subsection \ref{poskg}. Let $\cH, L, \Omega$ be the associated GNS objects described in Subsection \ref{sec4.3}. Let $V$ be the selfadjoint operator on $\cH$ affiliated to $\cW_{\kappa}$ equal either to $\int g(\x): \!P(\phi(\x))\!:_{0}\d \x$ or to $\int g(\x):\!\e^{\alpha \phi(\x)}\!:_{0}\d \x$. Then the following statements hold true: \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent = 0pt \hsize=12cm \strut # \strut} \cr (i) & $L+V$ is essentially selfadjoint and $\Omega\in \cD(\e^{-\frac{\beta}{2}H_{V}})$, where $H_{V}:= \overline{L+V}$. \cr (ii) & Let $\tau_{V}(t)$ be the $W^{*}$-dynamics generated by $H_{V}$ and $\omega_{V}$ be the vector state induced by~$\Omega_{V}=\|\e^{-\frac{\beta}{2}H_{V}}\Omega\|^{-1}\e^{-\frac{\beta}{2}H_{V}}\Omega$. Then $\tau_{V}$ is a group of $^{*}$-automorphisms of $\cW$, continuous for the strong operator topology such that $\bigl(\cW, \cW_\kappa, \tau_{V},\omega_{V} \bigr)$ is a stochastically positive $\beta$-KMS system. \cr (iii) & The generalized path space associated to $\bigl(\cW, \cW_\kappa, \tau_{V},\omega_{V} \bigr)$ satisfies the Markov property. \cr (iv) & Let $L_{V}$, $J_{V}$ be the perturbed Liouvillean and modular conjugation associated to $\bigl(\cW, \cW_\kappa, \tau_{V},\omega_{V} \bigr)$. Then $J_{V}= J$ and $L_{V}= \overline{H_{V}-JVJ}$. \cr} \end{theoreme} Finally we state the corresponding result for the charged Klein-Gordon field: \begin{theoreme} Let $\bigl(\cW, \cW_\kappa, \tau^\circ,\omega \bigr)$ be the quasi-free $\beta$-KMS system describing the free charged Klein-Gordon field in one space dimension at temperature $\beta^{-1}$ and zero chemical potential, described in Subsection \ref{poskg}. Let $\cH, L, \Omega$ be the associated GNS objects described in Subsection \ref{sec4.3}. Let $V$ be the selfadjoint operator on $\cH$ affiliated to $\cW_{\kappa}$ equal to \\ $\int g(\x): \!P(\overline{\varphi}(\x)\varphi(\x))\!:_{0}\d \x$. Then the following statements hold true: \vskip .3cm \halign{ \indent \indent \indent # \hfil & \vtop { \parindent = 0pt \hsize=12cm \strut # \strut} \cr (i) & $L+V$ is essentially selfadjoint and $\Omega\in \cD(\e^{-\frac{\beta}{2}H_{V}})$, where $H_{V}:= \overline{L+V}$. \cr (ii) & Let $\tau_{V}(t)$ be the $W^{*}$-dynamics generated by $H_{V}$ and $\omega_{V}$ be the vector state induced by~$\Omega_{V}=\|\e^{-\frac{\beta}{2}H_{V}}\Omega\|^{-1}\e^{-\frac{\beta}{2}H_{V}}\Omega$. Then $\tau_{V}$ is a group of $^{*}$-automorphisms of $\cW$, continuous for the strong operator topology such that $\bigl(\cW, \cW_\kappa, \tau_{V},\omega_{V} \bigr)$ is a stochastically positive $\beta$-KMS system. \cr (iii) & The generalized path space associated to $\bigl(\cW, \cW_\kappa, \tau_{V},\omega_{V} \bigr)$ satisfies the Markov property. \cr (iv) & Let $L_{V}$, $J_{V}$ be the perturbed Liouvillean and modular conjugation associated to $\bigl(\cW, \cW_\kappa, \tau_{V},\omega_{V} \bigr)$. Then $J_{V}= J$ and $L_{V}= \overline{H_{V}-JVJ}$. \cr} \end{theoreme} \begin{thebibliography}{aaa} \bibitem[ABG]{ABG}Amrein, W., Boutet de Monvel, A., Georgescu, W.: {\it $C_{0}$-Groups, Commutator Methods and Spectral Theory of $N$-Body Hamiltonians}, Birkh{\"a}user, Basel-Boston-Berlin, (1996). \bibitem[Ar1]{Ar1} H.\ Araki: Relative Hamiltonian for faithful normal states of a von Neumann algebra, Publ.\ Res.\ Int.\ Math.\ Soc.\ 9 (1973) 165--209. \bibitem[Ar2]{Ar2} H.\ Araki: Positive cone, Radon-Nikodym theorems, relative Hamiltonian and the Gibbs condition in statistical mechanics. An application of Tomita-Takesaki theory, in {\it $C^{*}$-algebras and their applications to Statistical Mechanics and Quantum Field Theory} D.~Kastler Ed.\ North Holland 1976. \bibitem[BB1]{BB1} J.Bros, D. Buchholz: Asymptotic dynamics of thermal quantum fields, Nuclear Phys. B 627 (2002) 289--310. \bibitem[BB2]{BB2} J.Bros, D. Buchholz: Axiomatic analyticity properties and representations of particles in thermal quantum field theory, New problems in the general theory of fields and particles, Ann. I.H.P. 64 (1996) 495--521. \bibitem[BF]{BF} L.\ Birke, J.\ Fr\"ohlich: KMS, etc. Rev.\ Math.\ Phys.\ 14 (2002) 829--871. \bibitem[DG]{DG} J.\ Derezinski, C.\ G\'erard: Spectral scattering theory of spatially cut-off $P(\varphi)_{2}$ Hamiltonians. Comm.\ Math.\ Phys.\ 213 (2000) 39--125. \bibitem[DJP]{DJP}J.\ Derezinski, V.\ Jaksic, C.A.\ Pillet: Perturbations of $W^{*}$-dynamics, Liouvilleans and KMS states, preprint mp-arc 03--94 (2003). \bibitem[Fr1]{Fr}J.\ Fr\"ohlich: Unbounded, symmetric semigroups on a separable Hilbert space are essentially selfadjoint. Adv.\ in Appl.\ Math.\ 1 (1980) 237--256. \bibitem[Fr2]{Fr2}J.\ Fr\"ohlich: The reconstruction of quantum fields from Euclidean Green's functions at arbitrary temperatures. Helv. Phys. Acta 48 (1975) 355--363. \bibitem[GO]{OGie}R.\ Gielerak, R.\ Olkiewicz: Gentle perturbations of the free Bose gas. I.\ J.\ Statist.\ Phys.\ 80 (1995) 875--918. \bibitem[GeJ]{GeJ} C. G\'erard, C. J\"akel: in preparation. \bibitem[GJ]{GJ}J.\ Glimm, A.\ Jaffe: {\it Quantum Physics, a functional point of view}, 1981 Springer. \bibitem[H-K1]{H-K}R.\ H\o egh-Krohn: Relativistic quantum statistical mechanics in two-dimensional space-time. Comm.\ Math.\ Phys.\ 38 (1974) 195--224. \bibitem[H-K2]{HK}R.\ H\o egh-Krohn: A general class of quantum fields without cut-offs in two space-time dimensions. Comm.\ Math.\ Phys.\ 21 (1971) 244--255. \bibitem[K]{K}A.\ Klein: The semigroup characterization of Osterwalder-Schrader path spaces and the construction of Euclidean fields, J.\ Funct.\ Anal.\ 27 (1978) 277--291. \bibitem[KL1]{KL1} A.\ Klein, L.\ Landau: Stochastic Processes Associated with KMS states, J.\ Funct.\ Anal.\ 42 (1981) 368--428. \bibitem[KL2]{KL2}A.\ Klein, L.\ Landau: Periodic Gaussian Osterwalder-Schrader positive processes and the two-sided Markov property on the circle. Pacific J.\ Math.\ 94 (1981) 341--367. \bibitem[KL3]{KL3}A.\ Klein, L.\ Landau: Construction of a unique selfadjoint generator for a symmetric local semigroup. J.\ Funct.\ Anal.\ 44 (1981) 121--137. \bibitem[KL4]{KL0}A.\ Klein, L.\ Landau: Singular perturbations of positivity preserving semigroups via path space techniques, J.\ Funct.\ Anal.\ 20 (1975) 44--82. \bibitem[Lo]{Loeve} M.\ Loeve: {\it Probability Theory}, Van Nostrand 1955. \bibitem[Si1]{Si} B.\ Simon: Positivity of the Hamiltonian semigroup and the construction of Euclidean region fields. Helv.\ Phys.\ Acta 46 (1973) 686--696. \bibitem[Si2]{Si1}B.\ Simon: {\it The $P(\varphi)_{2}$ Euclidean (Quantum) Field Theory}, (1974) Princeton University Press. \bibitem[S-H.K]{SHK}Simon, B., H\o egh-Krohn, R.: Hypercontractive Semigroups and Two dimensional Self-Coupled Bose Fields, J.\ Funct.\ Anal.\ 9 (1972) 121--180. \bibitem[St]{St} O. Steinmann: Perturbative quantum field theory at positive temperatures: an axiomatic approach. Comm. Math. Phys. 170 (1995) 405--415. \end{thebibliography} \end{document} ---------------0307250648871--