Content-Type: multipart/mixed; boundary="-------------0306131202949" This is a multi-part message in MIME format. ---------------0306131202949 Content-Type: text/plain; name="03-281.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-281.keywords" Quantum field theory ---------------0306131202949 Content-Type: application/x-tex; name="wmb.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="wmb.tex" \documentclass[12pt]{revtex4} %\documentclass[ams]{revtex4} %\documentclass[prl]{revtex4} %\documentclass[pr]{revtex4} %\documentclass[jmp]{revtex4} %\documentclass[aip]{revtex4} %\documentclass[12pt]{article} \pagestyle{myheadings} \markright{Draft, not for circulation \hfill} \setlength{\oddsidemargin}{.01in} \setlength{\evensidemargin}{.1in} \setlength{\textwidth}{6.8in} \setlength{\textheight}{8.5in} \setlength{\topmargin}{0.0000in} \setlength{\leftmargin}{-0.3in} \textwidth=15.0 true cm \textheight=22.0 true cm \renewcommand{\thefootnote}{\alph{footnote}} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{amssymb} \usepackage{epsfig} \begin{document} %\baselineskip=0.8cm \oddsidemargin=0.8cm \newtheorem{axiom}{Theorem}[section] \newtheorem{definition}{Definition}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{lemma}{Lemma}[section] \def\bib{\bibitem} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\ba{\begin{eqnarray}} \def\eea{\end{eqnarray}} \def\df{\stackrel{\rm def}{=} } \renewcommand{\theequation}{\arabic{section}.\arabic{equation}} \def\R{\mathbb{R}} \def\d{{\cal D}} \title{Existence of the Bogoliubov $S(g)$ Operator for the $(:\phi^4:)_2$ Quantum Field Theory} \author{\bf W. F. Wreszinski, Luiz Alberto Manzoni and Oscar Bolina} \affiliation{Instituto de F\'{\i}sica\\ Universidade de S\~ao Paulo\\ Caixa Postal 66318 \\ 05315-970 -- S\~ao Paulo, SP\\ Brasil/Brazil} \date{\today} \maketitle \vskip 1 cm \noindent \begin{center} {\bf Abstract} \end{center} \noindent We prove the existence of the Bogoliubov $S(g)$ operator for the $(:\phi^4:)_2$ quantum field theory. The construction is nonperturbative and relies on a theorem of Kisy\'nski. It implies almost automatically the properties of unitarity and causality for disjoint supports in the time variable. \vskip .5 cm \noindent \vfill \hrule width2truein \smallskip {\baselineskip=10pt \noindent {\small {\bf E-mails:} wreszins@fma.if.usp.br, lmanzoni@fma.if.usp.br, bolina@fma.if.usp.br} \par } \section{Introduction and Summary} \label{intro} \noindent Recent progress in perturbative quantum field theory for the St\"uckelberg-Bogoliu{\-}bov-Epstein-Glaser $S(g)$ operator \cite{BLT75, EGl73} in nonabelian gauge theories \cite{Sch01} (see also \cite{DFr99}), revived interest in a long-standing problem: is it possible to construct $S(g)$ {\it nonperturbatively} in quantum field theory? This question is of obvious relevance to theories where the (dimensionless) coupling constant is large ($\gtrsim 1$) -- e.g. strong interactions -- for which perturbation theory is not expected to be asymptotic. \newline For certain superrenormalizable theories -- the $(:P(\phi ):)_2$ theories -- there exists, for weak coupling, a construction of the true (LSZ-Haag-Ruelle) scattering operator, due to Osterwalder and S\'eneor \cite{OSe76} and Eckmann, Epstein and Fr\"ohlich \cite{EEF76}, one of the crowning achievements of constructive quantum field theory. The method of proof was, however, perturbative: the perturbation series for the scattering operator was shown to be asymptotic. \newline In contrast to the true scattering operator, $S(g)$ is, in perturbation theory, the generating functional for the time-ordered products of Wick polinomials. However, on the basis of \cite{EGl76} one might expect that, in the present massive case, defining % $$g_{\varepsilon}(x)\equiv g(\varepsilon x)\; ;\hspace{1.0cm}g\in {\cal S} (\mathbb{R}^2)$$ \noindent the (adiabatic) limit % \be S\Psi \equiv \lim_{\varepsilon \rightarrow 0}\; S(g_{\varepsilon})\Psi \label{adiab} \ee \noindent exists, $\forall \;\Psi \in {\cal D}$, where ${\cal D}$ is a Poincar\'e-invariant dense set in Fock space ${\cal F}$. Thus we expect that the physical $S$-matrix elements are obtainable as % \be (\Phi, S\Psi)\equiv \lim_{\varepsilon \rightarrow 0}\; \; (\Phi , S(g_{\varepsilon})\Psi)\; , \label{adiab2} \ee \noindent with $\Phi \in {\cal F}$, $\Psi \in {\cal D}$, where $g(0)>0$ should be identified with the coupling constant. In \cite{DFr99} an algebraic construction of the adiabatic limit was performed for perturbative QED. \newline A natural nonperturbative approach to construct $S(g)$ for the $(:\phi^4:)_2$ theory (and hopefully for any super-renormalizable QFT) consists in proving the existence of a (unique) solution of the evolution (propagator) equation ($\hslash =1$) % \be i\frac{\partial U(t,s)}{\partial t}\Psi = \tilde{H}(t)U(t,s)\Psi\; , \label{eqprop} \ee % with % \be \tilde{H}(t) \equiv H_g(t) + M {\bf 1}\; , \ee % where $M$ is a constant introduced in order to make $\tilde{H}(t)$ a positive operator (see section \ref{existence}) and % \be H_g(t) \equiv H_0 + V_g(t)\; . \label{hg} \ee % In (\ref{eqprop}) $U(t,s)$ is a two-parameter family of unitary operators on (symmetric) Fock space ${\cal F}$. $H_0$ is the free field Hamiltonian corresponding to a zero-time scalar field $\phi (x,0)$ of mass $m$ \cite{GJa70, GJa72}, and, formally, for % \be g\in {\cal D}(\mathbb{R}^2) \label{g} \ee \noindent let % \be V_g(t)=\int\; dx\; g(x,t):\phi^4(x,0):\; . \label{vg} \ee % Above, ${\cal D}$ denotes the Schwartz space of infinitely differentiable functions of compact support. The operators in (\ref{eqprop}) are expected to satisfy the propagator conditions: % \ba &&U(t,s)U(s,r)=U(t,r)\; ,\hspace{1.0cm}-\infty $ supp $g_2$ \item[] and/or \item[$(ii.b)$] supp $g_1 \sim$ supp $g_2$ ({\it causality}) \end{itemize} \noindent where ``$\sim$'' means ``spacelike to'', i.e., $(x-y)^2= (t_1-t_2)^2-(x_1-x_2)^2<0$, $\forall \; (t_1,x_1) \in {\rm supp}\; g_1 $ and $\forall \; (t_2,x_2) \in {\rm supp}\; g_2 $; \item[$(iii)$] There exists a unitary representation $U(a, \Lambda )$ of the Poincar\'e group on ${\cal F}$ -- the scalar field representation of mass $m$ --such that % $$U(a, \Lambda ) S(g)U(a, \Lambda )^{-1} = S(\{ a, \Lambda \} g)\; ,$$ % \noindent where % $$(\{ a, \Lambda \} g)(x) =g(\Lambda^{-1}(x-a))\; $$ \\ ({\it Lorentz covariance}). \end{itemize} The basic problem to prove (\ref{eqprop})-(\ref{udcon}) is the fact that $D(H_g(t))$ is, for each $g \in {\cal D}(\mathbb{R}^2)$, time-dependent. In section \ref{existence} we state the basic existence theorem we employ, which is due to Kisy\'nski \cite{Kis64} (see also \cite{Sim71}). In section \ref{prova} we prove our central existence theorem for $S(g)$, as well as properties $(i)$ and $(ii.a)$. In section \ref{kis} we provide a brief summary of the remarkable results of \cite{Kis64}, establishing a concrete link between them and our conditions in section \ref{prova}. We leave the conclusion and open problems to section \ref{conclusion}. Appendix A summarizes some of the basics elements of the construction of \cite{Kis64} and \cite{Yos80} for the convenience of the reader. %%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% \section{The Basic Existence Theorem} \label{existence} \setcounter{equation}{0} \noindent The Hamiltonian of the $(:\phi^4:)_2$ theory \cite{GJa68} is given by (\ref{hg}), where % \be H_0=\int \omega (k)a^*(k)a(k) dk\; , \label{H0} \ee with \be \omega (k)=\left( k^2+m^2\right)^{\frac{1}{2}}\; , \label{omega} \ee \noindent is the free field Hamiltonian on symmetric Fock space ${\cal F}$, with \be [a(k),a^*(k')]=\delta (k-k')\; . \label{com} \ee The self-interaction $V_g$ is given by (\ref{vg}), with the $t=0$ scalar free field of mass $m$: \be \phi (x)=\frac{1}{(4\pi )^{\frac{1}{2}}}\int e^{-ikx}\left[ a^*(k)+a(-k)\right]\omega (k)^{-\frac{1}{2}}dk \label{phit0} \ee % Thus $V_g$ may be written \cite{GJa68} % \ba V_g(t)&=& \sum_{j=0}^4 {4 \choose j} \int a^*(k_1)\cdots a^*(k_j)a(-k_{j+1})\cdots a(-k_4) \nonumber \\ \\ &\times&\tilde{g}({\scriptstyle \sum\limits_{i=1}^4 } k_i , t)\prod_{i=1}^4\omega (k_i)^{-\frac{1}{2}} dk_i\;\nonumber , \nonumber \label{HI} \eea where % \be \tilde{g}(k,t) \equiv \int dx \; e^{ikx} g(x,t) \label{tg} \ee \noindent The number operator $N$ is defined by % \be N=\int dk \;a^*(k)a(k) \; , \label{N} \ee By \cite{GJa68} (Lemma 2.2) % \be \left\| (N + {\bf 1})^{-\frac{j}{2}} \; V_g(t) \; (N + {\bf 1})^{-\frac{4 - j}{2}} \right\| \leq \; {\rm const.} \; \left\| W \right\|_{L^2} \; , \hspace{1.0cm} |j| \leq 4 \label{gjdes} \ee % where % \be\label{w} W(k,t)\equiv \tilde{g}({\scriptstyle \sum\limits_{i=1}^4} k_i , t)\prod_{i=1}^4\omega (k_i)^{-\frac{1}{2}} \ee The above mentioned lemma just uses the Fock space definitions of the creation and annihilation operators and the Schwartz inequality. We need two theorems due to Glimm and Jaffe, which we state as adapted to our case: % \begin{axiom}\label{sa}{\rm \cite{GJa68}} {\bf (a)} $H(t)$ is self-adjoint on the domain \be D(H(t))=D(H_0) \cap D(V_g(t))\; ,\label{DHtotal} \ee \noindent where $D(V_g(t))$ is the domain of the unique self-adjoint closure of $V_{g}(t)$ on the domain \be\label{dom} D_{0}=\bigcup_{n=0}^{\infty} D(H^{n}_{0}). \ee {\bf (b)} $H(t)$ is essentially self-adjoint on $D_{0}$. \end{axiom} % \begin{axiom}\label{posit} {\rm \cite{GJa72}} For each $g \in {\cal D}(\mathbb{R}^{2})$, there exists $0 < M_{g} < \infty$ such that \be\label{hinf} H_g(t) \geq -M_{g} {\bf 1} \ee as a bilinear form on $D_{0} \times D_{0}$. \end{axiom} By theorem \ref{posit} and {\bf (b)} of theorem \ref{sa}, $H(t)$ is a semi-bounded self-adjoint operator, and thus defining \be\label{mmc} M=M_{g}+c, \ee for some $c > 0$, then \be\label{hmc} \tilde{H}(t)=H_g(t)+M{\bf 1} \geq c {\bf 1} \ee % is a positive self-adjoint operator. Let ${\cal F}_{+2}=D(H_{0})$ endowed with the Hilbert space structure given by \be\label{f+2} f_{+2}(x,y)=\langle \left (H_{0}+1 \right )x, \left (H_{0}+1 \right )y \rangle \ee % and denote $\sqrt{f_{+2}(x,x)}$ by $||x ||_{+2}$. By the Riesz lemma we may associate ${\cal F}_{+2}$ and the space ${\cal F}_{-2}$ of continuous conjugate linear functions on ${\cal F}_{+2}$. While we consider ${\cal F}$ isomorphic to its conjugate dual space ${\cal F}^{*}$, the isomorphism being the identity, the isomorphism of ${\cal F}_{+2}$ with ${\cal F}_{-2}$ is given by the operator $\left (H_{0}+1 \right )^{2}$, because % \[ ||v||_{-2}=\sup{ \left \{ |\langle w, v \rangle | ~:~ ||w ||_{+2} \leq 1 \right \} }. \] % Since $f_{+2}(x,y)=\langle x, \left (H_{0}+1 \right )^{2}y \rangle$, we have % \begin{eqnarray} \left\| \left (H_{0}+1 \right )^{2}y \right\|_{-2}&=& \sup{ \left \{ | \langle w, \left (H_{0}+1 \right )^{2}y \rangle | ~:~ ||w ||_{+2}=\sqrt{\langle w, \left (H_{0}+1 \right )^{2}w \rangle} \leq 1 \right \} } \nonumber\\ &=&\left\| \left (H_{0}+1 \right )^{2}y \right\| =||y||_{+2}, \nonumber \end{eqnarray} % from which also, for $y \in {\cal F}$: % \be\label{n-2} ||y||_{-2}=||\left (H_{0}+1 \right )^{-1}y ||, \ee % which explains the notation ${\cal F}_{-2}$. Clearly $||x || \leq || x ||_{+2}$ for $x \in {\cal F}_{+2}$, and by (\ref{n-2}), $||y ||_{-2} \geq || y ||$ for $y \in {\cal F}$. Thus, under the above conditions: % \be\label{escala} {\cal F}_{+2} \subset {\cal F} \subset {\cal F}_{-2}. \ee A bounded operator {\bf B} from ${\cal F}_{+2}$ to ${\cal F}_{-2}$ is thus such that, for some constant $c$, % \be\label{B} ||{\bf B} \psi ||_{-2} \leq c || \psi ||_{+2}\;\;\;\;\;\;\;\; \psi \in {\cal F}_{+2}, \ee % or, by (\ref{f+2}) and (\ref{n-2}), % \be\label{HB} ||\left (H_{0}+1 \right )^{-1}{\bf B} \psi|| \leq c || \left (H_{0}+1 \right )\psi || \;\;\;\;\;\;\;\; \psi \in {\cal F}_{+2}, \ee % or % \be\label{HBH} ||\left (H_{0}+1 \right )^{-1}{\bf B}\left (H_{0}+1 \right )^{-1} \phi|| \leq c || \phi || \;\;\;\;\;\;\;\; \phi \in {\cal F}. \ee % Now, by (\ref{hmc}), we may define ${\tilde{H}}(t)^{1/2}$, and, by (\ref{gjdes}) for $x \in {\cal F}_{+2}$, the closed sesquilinear form % \be \label{forma} S(x,y)= \langle \tilde{H}(t)^{1/2}x, \tilde{H}(t)^{1/2}y \rangle \ee % which is, by the form representation theorem \cite{Far75}, the form of the operator ${\tilde{H}}(t)$. In section \ref{prova} we show the explicit connection of (\ref{forma}) to the basic theorem of Kisy\'nski \cite{Kis64}, which we state in the form of theorems II.23 and II.24 of \cite{Sim71}, with slight changes: \begin{axiom}\label{texist} Let (\ref{escala}) hold and $\tilde{H}(t)$ $(-T \leq t \leq S)$ be a one-parameter family of strictly positive (i.e. satisfying (\ref{hmc})) self-adjoint operators on ${\cal F}$. Suppose that $\tilde{H}(t): {\cal F}_{+2} \rightarrow {\cal F}_{-2}$ are bounded and twice differentiable, with a continuous second derivative, in the $|| \cdot ||_{-2,2}-$norm (\ref{B}). Then there exists a two-parameter family $U(t,s)$ of unitary propagators satisfying (\ref{eqprop}), (\ref{propcon}) and (\ref{udcon}). \end{axiom} %%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% \section{The central existence theorem}\label{prova} \setcounter{equation}{0} \noindent We now use theorem \ref{texist} in order to prove our main % \begin{axiom}\label{main} The $\left (: \phi^{4} : \right )_{2}$ theory, as defined by (\ref{hg}), (\ref{g}), (\ref{vg}), (\ref{H0}) and (\ref{omega}) satisfies a stronger condition than the hypothesis of theorem \ref{texist}: $H_{g}( \cdot )$ is infinitely differentiable as an operator from ${\cal F}_{+2}$ to ${\cal F}_{-2}$. \end{axiom} % In order to prove theorem \ref{main} we first show a useful auxiliary result. % \begin{lemma}\label{wg} Let $W$ be defined by (\ref{w}). Then there exists $r > 1$ such that % \be\label{w2} || W( \cdot , t) ||_{2} \leq || g(\cdot , t) ||_{r} \ee % where % \be\label{gr} || g(\cdot , t) ||_{r}=\left (\int_{-\infty}^{+\infty} dk | \tilde{g}(k,t)|^{r} \right )^{1/r}. \ee \end{lemma} {\bf Proof.} We have % \ba || W(\cdot , t) ||^{2}_{2}&=&\int_{-\infty}^{+\infty} dk_{1} \omega(k_{1})^{-1} \cdot \int_{-\infty}^{+\infty} dk_{2} \omega(k_{2})^{-1} \nonumber \\ \label{w22} \\ &\cdot & \int_{-\infty}^{+\infty} dk_{3} \omega(k_{3})^{-1} \cdot \int_{-\infty}^{+\infty} dk\, ' | \tilde{g}(k\, ',t)|^{2} \omega \left( k\, '-{\scriptstyle \sum\limits_{i=1}^3 } k_i \right)^{-1} \nonumber \eea % by the change of variable $k\, '={\scriptstyle \sum\limits_{i=1}^3 } k_i$. Introducing further the variables $K_{1}, K_{2}, K_{3}$ such that % \begin{eqnarray*} K_1 &=& k_1+k_2+k_3 \\ K_2 &=& k_1+k_2 \\ K_3 &=& k_1 \; \end{eqnarray*} % so that $k_{3}=K_{1}-K_{2}$ and $k_{2}=K_{2}-K_{3}$, we write (\ref{w22}) as % \be\label{wom} || W(\cdot , t) ||^{2}_{2}=\left( \omega^{-1} * \left( \omega^{-1} * \left( \omega^{-1} * \left( \omega^{-1} * | \tilde{g} |^{2} \right) \right) \right) \right) (0), \ee % where the convolution is defined as usual by % \[ \left (f*g \right )(k)=\int_{-\infty}^{+\infty} dk_{1} f(k-k_{1})g(k_{1}). \] % Consider, now, the quantity associated to the right-hand side of (\ref{w22}): % \ba I(q, t)&\equiv& \int_{-\infty}^{+\infty} dk_{1}\; \omega(k_{1}-q)^{-1} \cdot \int_{-\infty}^{+\infty} dk_{2}\; \omega(k_{2})^{-1} \nonumber \\ \label{I} \\ &\cdot & \int_{-\infty}^{+\infty} dk_{3} \;\omega(k_{3})^{-1} \cdot \int_{-\infty}^{+\infty} dk\, ' \; | \tilde{g}(k\, ',t)|^{2} \omega \left( k\, '- {\scriptstyle \sum\limits_{i=1}^3 } k_i \right )^{-1} \nonumber \eea % Since $g\in {\cal D}(\mathbb{R})$ this function is differentiable, hence continuous, in $q$ for any compact subset containing the origin, which implies that $I(0,t) \leq \|I(\cdot ,t)\|_{\infty}$ (where $\| \cdot \|_{\infty}$-norm is with respect to the $q$-variable). \newline We now apply Young's inequality \cite{Lieb} % \[ || f*g ||_{r} \leq C_{rpq}||f||_{p} ||g ||_{q} \] % with $C_{rpq}$ a constant and % \[ \frac{1}{p}+\frac{1}{q}=1+\frac{1}{r} \] % to (\ref{wom}), starting with $r=\infty$. Above, % \[ ||f||_{p}=\left ( \int_{-\infty}^{+\infty} dk | f(k) |^{p} \right )^{1/p}. \] % We thus obtain % \[ \left\| W(\cdot , t) \right\|^{2}_{2} \leq C_{2 r_1 r_2}\left\| \omega^{-1} \right\|_{r_{1}}\; \left\| \left( \omega^{-1} * \left( \omega^{-1} * \left( \omega^{-1} * | \tilde{g} |^{2} \right) \right) \right) \right\|_{r_{2}} \] % with $r^{-1}_{1}+r^{-1}_{2}=1$, and so on, up to (indicating all the constants resulting from the Young's inequality by $C'$) % \be\label{wr8} || W(\cdot , t) ||^{2}_{2} \leq C' \left\| \omega^{-1} \right\|_{r_{1}} \; \left\| \omega^{-1} \right\|_{r_{3}}\; \left\| \omega^{-1} \right\|_{r_{5}} \; \left\| \omega^{-1} \right\|_{r_{7}} \; \left\| |\tilde{g} |^{2} \right\|_{r_{8}} \ee % with $r^{-1}_{3}+r^{-1}_{4}=1+r^{-1}_{2}$, $r^{-1}_{5}+r^{-1}_{6}=1+r^{-1}_{4}$, $r^{-1}_{7}+r^{-1}_{8}=1+r^{-1}_{6}$. We require $r_{i} > 1$, for $i=1,3,5,7$, so that $|| \omega^{-1} ||_{r_{i}} < \infty$, the choice $r_{1}=r_{2}=2$, $r_{3}=r_{4}=\frac{4}{3}$, $r_{5}=r_{6}=\frac{8}{7}$, $r_{7}=r_{8}=\frac{16}{15}$ is, for instance, possible. By (\ref{wr8}) % \be\label{wgr} || W(\cdot , t) ||^{2}_{2} \leq C \left\| |\tilde{g} |^{2} \right\|_{r} \ee % with % \be\label{r} r > 1 \; . \ee % Above % \be\label{gtr} \left\| \; |\tilde{g} |^{2}\; \right\|_{r}= \left( \int_{-\infty}^{+\infty} dk \left| \tilde{g} (k,t) \right|^{2r} \right)^{1/r}. \ee % obtaining finally, (\ref{wgr}). \hfill $\square$ \newline {\bf Proof of \ref{main}} By (\ref{gjdes}), % \be\label{NVN} \left\| \left(N+{\bf 1} \right)^{-1}V_{g}(t) \left(N+{\bf 1} \right)^{-1} \right\| \leq {\rm const.} || W ||_{L^{2}} \ee % and, by (\ref{omega}), $\omega(k) \geq m {\bf 1}$; hence % \[ \left\| \left( H_{0}+{\bf 1} \right)^{-1}\left( N+{\bf 1} \right)\right\| \leq d_{1} \;\;\;\;\;\;\;\;\;\;\;\; \left\| \left( N+{\bf 1} \right)\left( H_{0}+{\bf 1} \right)^{-1}\right\| \leq d_{2}, \] % for constants $d_{1}$ e $d_{2}$. Hence, by (\ref{NVN}) and (\ref{w2}), % \be\label{HVH} \left\| \left( H_{0}+{\bf 1} \right)^{-1}V_{g}(t) \left( H_{0}+{\bf 1} \right) \right\| \leq {\rm const.} || g( \cdot, t )||_{r} \ee % with $r > 1$: a fortiori this holds for $H_{g}(\cdot )$ by (\ref{hg}), hence % \be\label{HHH} \left\| \left( H_{0}+{\bf 1} \right)^{-1}H_{g}(t) \left( H_{0}+{\bf 1} \right) \right\| \leq {\rm const.} || g( \cdot, t )||_{r}. \ee By (\ref{HBH}) and theorem \ref{texist} we need only prove that the l.h.s. of (\ref{HHH}) is three times differentiable. We shall prove that % \be \label{hgo} \left\| \left( H_{0}+{\bf 1} \right)^{-1}\! \left( \! \frac{H_{g}(t+h)-H_{g}(t)}{h} -{H '}_{g}(t)\! \right)\! \left( H_{0}+{\bf 1} \right)^{-1} \right\| \longrightarrow 0 \;\;\;\; {\rm as} \;\;\;\; h \rightarrow 0, \ee % where % \be \label{hg'} H'_{g}(t)=H_{0}+V_{g'}(t) \ee % with % \be \label{vg'} V_{g'}(t)=\int dx : \phi^{4}(x,0) : g'(x,t) \ee % and % \[ g'(x,t)\equiv\frac{\partial g(x,t)}{\partial t}. \] % We now prove (\ref{hgo}). By (\ref{HHH}) % \begin{eqnarray} & &\left\| \left( H_{0}+{\bf 1} \right)^{-1}\! \left( \! \frac{H_{g}(t+h)-H_{g}(t)}{h} -{H '}_{g}(t)\! \right)\! \left( H_{0}+{\bf 1} \right)^{-1} \right\| \nonumber\\ & & \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \leq {\rm const.} \! \int_{-\infty}^{\infty}\!\!\!\!\! dk \left | \int \!\! dx \, e^{-ikx}\! \left( \! \frac{g(x,t+h)-g(x,t)}{h} -g'(x,t)\! \right) \right |^{2r}\!\!\!. \label{h2r} \end{eqnarray} % We now write the integral on the right-hand side of (\ref{h2r}) as % \[ \int_{-\infty}^{\infty} dk \cdots = \int_{-\infty}^{1} dk \cdots + \int_{-1}^{1} dk \cdots + \int_{1}^{\infty} dk \cdots \] % and estimate only the last integral above; the others are similar. Thus % \begin{eqnarray}\label{J} J&\equiv&\int_{1}^{\infty} dk \left | \int dx e^{-ikx} \left ( \frac{g(x,t+h)-g(x,t)}{h} -g'(x,t) \right ) \right |^{2r} \nonumber\\ & \leq & \int_{1}^{\infty} \frac{dk}{k^{2}} \left | \int dx e^{-ikx} \left ( \frac{D^{2}_{x}g(x,t+h)-D^{2}_{x} g(x,t)}{h} -D^{2}_{x}g'(x,t) \right ) \right |^{2r} \end{eqnarray} % where we have used two partial integrations and $D_{x}\equiv \frac{\partial }{\partial x}$. Let now % \be\label{V} V(x,t)\equiv D^{2}_{x}g(x,t). \ee % Now $V$ is also an infinitely differentiable function of compact support and % \be\label{Vtay} V(x, t+h)=V(x,t)+hV^{'}(x,t)+\frac{h^{2}}{2!}V^{''}(x,t+t^{*}_{h}(x)) \ee % by Taylor's formula with remainder, where $0 < t^{*}_{h}(x) < h$. Putting (\ref{Vtay}) into (\ref{J}) we get % \[ J \leq c\frac{h}{2} \left ( \int_{-\infty}^{\infty} dx \left | V^{''}(x, t+t^{*}_{h}(x)) \right | \right )^{1/2} \leq c \; h ( \sup_{x,t} | V^{''}(x,t) | ). \] % The other estimates are similar and yield (\ref{hgo}). We now notice that the bounds (\ref{HHH}) \underline{continue to hold} for $H'_{g}(t)$ with $|| g(\cdot , t) ||_{r}$ replaced by $|| g'(\cdot , t) ||_{r}$ on the right-hand side of (\ref{HHH}). Thus the same proof applies to $H'_{g}(t)$, $H''_{g}(t)$, ... and in fact $H_{g}(t)$ is infinitely differentiable as an operator from ${\cal F}_{+2}$ to ${\cal F}_{-2}$. \hfill $\square$ \begin {proposition}\label{caus} The $S(g)$ matrix for the $(:\phi^4)_2$ theory, as defined in (\ref{sg}), is unitary and it satisfies the causality condition for disjoint supports [condition $(ii.a)$ -- section \ref{intro}]. \end{proposition} {\it Proof.} The unitarity follows directly from the existence theorems. For the proof of causality it is convenient explicitly dispose the dependence of the propagators on the function $g$. Let ${\rm supp}_t \; g_1 > {\rm supp}_t \; g_2$ and suppose ${\rm supp}_t \; g_1 \subset (r, +\infty )$ and ${\rm supp}_t \; g_2 \subset (-\infty , r)$, where ${\rm supp}_t$ stands for the support in the time variable. Then, for $t>r>s$ we have % \be\label{utrs} U^D_{(g_1+g_2)}(t,s) = U^D_{(g_1+g_2)}(t,r) U^D_{(g_1+g_2)}(r,s) \ee % but % \begin{eqnarray*} i\frac{\partial }{\partial t} U^D_{(g_1+g_2)}(t,r)\Psi &=& H^D_{(g_1+g_2)}(t) U^D_{(g_1+g_2)}(t,r)\Psi \\ &=&H^D_{g_1}(t) U^D_{(g_1+g_2)}(t,r)\Psi \end{eqnarray*} % and, by the uniqueness of the solutions of the above equation, we have $ U^D_{(g_1+g_2)}(t,r)= U^D_{g_1}(t,r)$. Analogously, we have $ U^D_{(g_1+g_2)}(r,s)= U^D_{g_2}(r,s)$. This, together with (\ref{utrs}) imply that % $$ U^D_{(g_1+g_2)}(t,s) = U^D_{g_1}(t,r) U^D_{g_2}(r,s) $$ % from this equation and the fact that $ U^D_{g_1}(t,s) = U^D_{g_1}(t,r)$ and $ U^D_{g_2}(r,s) = U^D_{g_2}(t,s)$ due to the support properties of $g_1$ and $g_2$, we finally have % \be\label{utrs1} U^D_{(g_1+g_2)}(t,s) = U^D_{g_1}(t,s) U^D_{g_2}(t,s) \ee % Then, by (\ref{utrs1}) and the definition (\ref{sg}), we obtain % $$ S(g_1+g_2) = S(g_1) S(g_2)\; , $$ \hfill $\square$ %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% \section{The Relation Between Kisy\'nski's Theory and Theorem \ref{main}} \label{kis} \setcounter{equation}{0} \noindent Let us now briefly summarize (without proof) some steps in Kisy\'nski's proof of the theorem \ref{texist}. First of all, we will state a crucial auxiliary theorem. Let $X$ be a Banach space with the norm $\| \cdot \|$ and $A(t)$, $t\in [-T_1, T_2]$ ($T_1, T_2 >0$), a family of linear operators in $X$. Consider the following conditions: \begin{itemize} \item[(a)] there exists a family $\| \cdot \|_t$, , $t\in [-T_1, T_2]$, of norms in $X$ equivalent to $\| \cdot \|$ such that $\left|\, \|\Psi \|_t- \| \Psi \|_s \, \right| \leq k \, \| \Psi \|_s \, |t-s|$ with $k= {\rm const.}$, $-T_1\leq s,t\leq T_2$ and $\Psi \in X$; \item[(b)] for all $t\in [-T_1, T_2]$ the set $D (A(t))$ is dense in $X$; \item[(c)] there\hfill exists\hfill a\hfill constant\hfill $\lambda_0\geq 0$\hfill such\hfill that\hfill $R(\lambda - \epsilon A(t)) = X$\hfill and\hfill \\ $\left\| (\lambda -\epsilon A(t))\Psi \right\|_t \geq (\lambda -\lambda_0)\| \Psi \|_t$ for $\epsilon =\pm 1$, $\lambda > \lambda_0$, $t\in [-T_1, T_2]$ and $\Psi \in D(A(t))$; \item[(d)] there exists a family $R(t)$, $t\in [-T_1, T_2]$, of invertible bounded linear operators in $X$, such that $R(t)$ is twice weakly continuously differentiable in $[-T_1, T_2]$ and $\left( R(t)\right)^{-1} D (A(t)) = Y = {\rm const.}$ $\forall t\in [-T_1, T_2]$; \item[(e)] $\left( R(t)\right)^{-1}A(t)R(t)$ is weakly continuously differentiable. \end{itemize} Above $R(A)$ stands for the range of the operator $A$. Then we have: \begin{axiom}\label{k4.4}{\rm (\cite{Kis64}, Theorem 4.4)} Let the conditions (a) - (e) be satisfied. Then there exists a two-parametrics family of propagators $U(t,s)$, $-T_1\leq s,t \leq T_2$, such that % $$ \Psi (t) \equiv U(t,s) \Psi (s) \; , \hspace{1.0cm} \Psi (s) \in D (A(s)) \; , $$ \noindent is the unique solution of the problem % \be \label{eq} \frac{d}{dt} \Psi (t)= A(t) \Psi (t) \ee \noindent with initial data $\Psi (s)$. The bounded propagators $U(t,s)$ are strongly continuous on $-T_1\leq s,t \leq T_2$ and satisfy: \begin{eqnarray} &&U(t,t) =1\; , \hspace{3.7cm} \forall \; t\in [-T_1, T_2]\; ; \label{U1} \\ &&U(t,s)U(s,r) =U(t,r)\; , \hspace{1.5cm} {\rm for} -T_1\leq r,s,t\leq T_2\; ; \label{UUU} \\ &&U(t,s) D (A(s)) = D (A(t))\; , \hspace{1.0cm} {\rm for} -T_1\leq s,t\leq T_2\; ; \label{UDD} \end{eqnarray} \noindent besides, $\forall \; s \in [-T_1, T_2]$ and $\Psi \in D (A(s))$ the function $U(t,s)\Psi$ is continuously differentiable (in the sense of the norm) in $X$, satisfying: % \be\label{eqU} \frac{d}{dt} U(t,s) \Psi = A(t) U(t,s) \Psi \; . \ee \end{axiom} The method of proof of this theorem is to reduce the problem to the case where we have an operator with constant domain by making use of the properties of $R(t)$ [for an outline of Kisy\'nski's solution of the problem (\ref{eq}) with $D(A(t)) = {\rm const.}$ see Appendix A]. \newline Let us now consider Kisy\'nski's approach to the abstract Schr\"odinger equation % \be\label{sch} \frac{d}{dt} \Psi (t)= -i A(t) \Psi (t)\; , \hspace{1.0cm} -T_1\leq t \leq T_2 \ee % where $\Psi \in {\cal H}$, with ${\cal H}$ a Hilbert space and $A(t)$ an operator in ${\cal H}$ defined as follows. Consider the condition: \begin{itemize} \item[($i$)] Let ${\cal H}$ be a Hilbert space, ${\cal H}_+$ a dense subset of ${\cal H}$ and, $\forall \, t \in [-T_1,T_2]$, let $\langle \cdot , \cdot \rangle_t^+$ be a scalar product defined on ${\cal H}_+$ which makes it a Hilbert space ${\cal H}_+^t$ algebraically and topologically contained in ${\cal H}$. Assume that $\langle \cdot , \cdot \rangle_t^+$ is $n$ times ($n\geq 1$) continuously differentiable on $[-T_1,T_2]$. \end{itemize} If the condition ($i$) is satisfied we have \begin{lemma} {\rm (\cite{Kis64}, Lemma 7.2)} The equality % \be \langle \Phi , \Psi \rangle_t^+ = \langle \Phi , Q(t)\Psi \rangle_{-T_1}^+ \; , \hspace{1.0cm} \Phi, \Psi \in {\cal H}_+ \; , \;\; t \in [-T_1,T_2] \ee % defines a bounded $n$ times weakly continuously differentiable operator $Q(t)$ on ${\cal H}_+^{-T_1}$. For all fixed $ t \in [-T_1,T_2]$, $Q(t)$ is hermitian with $\inf Q(t) >0$ in ${\cal H}_+^{-T_1}$. % \end{lemma} % Other consequences of the condition ($i$) are that we can define another operator $J_{-T_1}(t)$ by means of the equality (\cite{Kis64}, Lemma 7.4) % \be \langle \Phi , \Psi \rangle = \langle \Phi , J_{-T_1}(t)\Psi \rangle_t^+\; \hspace{1.0cm} \Phi \in {\cal H}_+\, ,\; \Psi \in {\cal H} \ee % with $J_{-T_1}(t)$ a positive hermitian operator in ${\cal L}({\cal H})$ such that $J_{-T_1} (t) {\cal H}_+$ is a dense subset of ${\cal H}_+^t$. Then, defining % \be \left\| \Psi \right\|_t^- \equiv \left\| J_{T_1}(t) \Psi \right\|_t^+\; , \hspace{1.0cm} \Psi \in {\cal H}\; , \ee % it follows that the completion ${\cal H}_t^-={\cal H}_{T_1}^-\equiv {\cal H}^-$ of ${\cal H}$ in the norm $\| \cdot \|_t^-$ contains ${\cal H}$ algebraically and topologically (\cite{Kis64}, Lemma 7.5). \newline Finally, we can define an operator $A(t)$ by means of the form $\langle \cdot, \cdot \rangle_t^+$ according to the following lemma: % \begin{lemma} {\rm (\cite{Kis64}, Lemma 7.7)} For all $ t \in [-T_1,T_2]$ % \be D(A(t)) = \left\{ \Psi \in {\cal H}^+: \; \sup_{\scriptscriptstyle \Phi\in {\cal H}^+\, ,\, \|\Phi\| \leq 1} \{ \left| \langle \Phi, \Psi\rangle_t^+\right| \} <+\infty \right\} \ee % \be \langle \Phi, A(t)\Psi \rangle \equiv \langle \Phi, \Psi\rangle_t^+\; , \hspace{1.0cm} \Psi \in D(A(t)) \ee % define an inversible self-adjoint positive operator $A(t)$ in ${\cal H}$, with % \be D(A(t))= \left( Q(t) \right)^{-1} D(A(-T_1)) \ee % and % \be A(t) = \left( J_{-T_1}(t) \right)^{-1} = A(-T_1)Q(t)\; . \ee % \end{lemma} % Then the operator $A(t)$ is showed to satisfy the Schr\"odinger equation (\ref{sch}) and the propagators of the problem (\ref{sch}) satisfy the properties enumerated in the theorem \ref{texist} (\cite{Kis64}, Theorem 8.1). In order to proof his Theorem 8.1 for the operator $A(t)$, as defined above, Kisy\'nski made use of the theorem \ref{k4.4} identifying $R(t)= \left( Q(t)\right)^{-1}$. Let us now show that the $(:\phi^4:)_2$ theory satisfies the necessary conditions for the theorem \ref{texist}. In fact, all we need to show is that the condition ($i$) is satisfied. However in benefit of clarity we will explicitly display the main operators introduced in Kisy\'nski's proof and some of its properties. \newline As defined in section \ref{existence}, ${\cal F}$ is the symmetric Fock space and ${\cal F}_{+2}=D(H_0)$ is a dense subset of ${\cal F}$. Then, taking the closure ${\cal F}_{+2}^t$ of ${\cal F}_{+2}$ in the norm induced by the scalar product $\langle\cdot ,\cdot\rangle_t^+$, which is related to the operator $\tilde{H}(t)$ [see equation (\ref{hmc})] by means of the form (\ref{forma}), i.e., % \be\label{pesc} \langle \Phi, \Psi\rangle_t^+ \equiv S(\Phi, \Psi) = \langle \tilde{H}(t)^{1/2}\Phi, \tilde{H}(t)^{1/2}\Psi \rangle \ee % we can show the following: \begin{proposition} ${\cal F}_{+2}^t$ is a Hilbert space such that % \be\label{ftf} {\cal F}^{t}_{+2} \subset {\cal F} \ee % algebraically and topologically. \end{proposition} % \begin{itemize} \item[] {\bf Proof.} That ${\cal F}_{+2}^t$ is a Hilbert space follows immediately from the fact that the form defined in (\ref{pesc}) is closed (see, e.g., \cite{Far75}). The property that ${\cal F}^{t}_{+2} \subset {\cal F}$ algebraically is trivial. So, it remains to show that (\ref{ftf}) holds topologically. This is achieved by showing that for $\{ \Psi_n\}_{n=1}^{\infty} \in {\cal F}_{+2}$ and $\{ \Psi_n\} \in {\cal F}_{+2}$ such that % \be\label{conv} \| f_{n}-f \| \longrightarrow 0 \ee we have \[ \| f_{n}-f \|^{t}_{+} \longrightarrow 0. \] % To show this, set \begin{eqnarray} \left( \| f_{n}-f \|^{t}_{+}\right)^2 &=&\langle \left (f_{n}-f \right ), \left (f_{n}-f \right ) \rangle ^{t}_{+} \nonumber\\ &=& \langle \left (f_{n}-f \right ), \tilde{H}(t) \left (f_{n}-f \right ) \rangle \nonumber\\ &=&\langle \left (H_{0}+{\bf 1} \right ) \left (f_{n}-f \right ), \left (H_{0}+{\bf 1} \right )^{-1}\tilde{H}(t) \left( H_{0}+{\bf 1}\right)^{-1} \nonumber \\ &\times& \left( H_{0}+{\bf 1}\right) \left (f_{n}-f \right ) \rangle \nonumber \end{eqnarray} % The Schwartz inequality applied to the last term above yields \[ \| f_{n}-f \|^{t}_{+} \leq \| \left (H_{0}+{\bf 1} \right )^{-1} \tilde{H}(t) \left (H_{0}+{\bf 1} \right )^{-1} \| \; \| \left (H_{0}+{\bf 1} \right ) \left (f_{n}-f \right ) \|^2 \] % The first term on the right-hand side is bounded due to (\ref{HHH}). The second term on the right-hand side converges since $H_{0}+{\bf 1}$ is a self-adjoint operator (hence closed) and, by hypothesis, (\ref{conv}) holds. Then the proof of the proposition is complete. \hfill $\square$ \end{itemize} In addition, it follows straightforwardly from (\ref{pesc}) and theorem \ref{main} that $\langle\cdot ,\cdot\rangle_t^+$ is $n$ times (infinitely, in fact) continuously differentiable. Then it is proved that the condition ($i$) is satisfied and the theorem \ref{texist} follows as proved in \cite{Kis64} and summarized above. \newline Now we turn to explicitly show the properties of $Q(t)$ in our case. From (\ref{pesc}) and the definition \[ \langle \Phi, \Psi \rangle^{t}_{+} \equiv \langle \Phi, Q(t) \Psi \rangle^+_{-T_1} \] % we obtain that $Q(t)$ is the operator % \be\label{Q} Q(t)=\left ( \tilde{H}(-T_1) \right )^{-1}\tilde{H}(t) \ee \begin{proposition} $Q(t)$, as defined in (\ref{Q}), is a (strictly) positive hermitian operator in ${\cal F}_{+2}$ and it is infinitely weakly differentiable. \end{proposition} \begin{itemize} \item[] {\bf Proof.} It follows directly from the properties of the scalar product $\langle\cdot ,\cdot\rangle_t^+$ that $Q(t)$ is infinitely weakly differentiable. \newline For $\Phi$, $\Psi \in {\cal F}_{+2}$, we have \begin{eqnarray} \left( \langle \Phi, Q(t) \Psi \rangle^{+}_{-T_1} \right)^{*} &=& \langle Q(t)\Psi, \Phi \rangle^{+}_{-T_1} \nonumber \\ &=& \langle \tilde{H}(-T_1)^{-1}\tilde{H}(t)\Psi, \tilde{H}(-T_1) \Phi \rangle = \langle \tilde{H}(t)\Psi, \Phi \rangle, \end{eqnarray} % where we have used (\ref{Q}) . We then have that % \begin{eqnarray} \left (\langle \Phi, Q(t) \Psi \rangle^{+}_{-T_1} \right )^{*}&=& \langle \Psi, \tilde{H}(-T_1)\left (\tilde{H}(-T_1) \right )^{-1}\tilde{H}(t)\Phi \rangle \nonumber\\ &=& \langle \Psi, \tilde{H}(-T_1) Q(t)\Phi \rangle=\langle \Psi, Q(t) \Phi \rangle^{+}_{-T_1}, \end{eqnarray} % which proves that $Q(t)$ is hermitian. \newline In order to prove that $Q(t)$ is strictly positive on ${\cal F}_{+2}$, we must remember that, since ${\cal F}_{+2}^t \subset {\cal F}_{+2}$ $\forall \; t$ algebraically and topologically, it follows that the norms $\|\cdot \|_{-T_1}^+ $ and $\| \cdot \|_t^+$ are equivalent, i.e., there exists $a_t\geq 1$ such that $a_t^{-1}\| \cdot\|^+ \leq \| \cdot\|_t^+ \leq a_t\|\cdot \|_{-T_1}^+$. Then, for $\Psi \in {\cal F}_{+2}$, % \ba \langle \Psi , Q(t)\Psi \rangle_{-T_1}^+ &=& ( \|\Psi\|_+^t)^2 \nonumber \\ &\geq & a_t^{-2} ( \|\Psi\|_+^{-T_1})^2 \eea % from which follows that $\inf Q(t) >0$ and the proof is complete. \hfill $\square$ \end{itemize} % %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% \section{Conclusion: Open Problems} \label{conclusion} \setcounter{equation}{0} \noindent The problem of the nonperturbative construction of $S(g)$ for the $\left (: \phi^{4} : \right )_{2}$ quantum field theory was addressed in \cite{Wre72} using Yosida's approach, which requires that the domain of $H_{g}(t)$ be time-independent. For test functions $g(x,t)=h_{1}(x) \cdot f_{1}(t)$, i.e., of the product form, this condition is satisfied, but already for a sum of two products, e.g., $g(x,t)= h_{1}(x) \cdot f_{1}(t)+h_{2}(x) \cdot f_{2}(t)$, with $f_{1}$ and $f_{2}$ having disjoint supports, this is no longer true, and thus the results of \cite{Wre72} are incomplete. The present approach does not suffer from this inconvenience, and $g$ is allowed to be an arbitrary infinitely differentiable function of compact support. Moreover, the use of a scale of spaces makes the theory very flexible, being applicable to more singular super-renormalizable theories, as well as to four-dimensional theories with an ultra-violet cutoff. It is a very challenging problem to discover a possibility of ``renormalization" of the exponentials of the type (A.7) in the latter, in analogy to the interesting approach of Barata \cite{Bar00} and Gentile \cite{Gen02} to the study of certain two-level systems. \newline There are, however, open problems even to finish this program for the present $\left (: \phi^{4} : \right )_{2}$ theory: proof of causality for space-like supports ({\it ii. b)} and proof of Lorentz covariance ({\it iii}). For this purpose, the method outlined in \cite{Wre72} seems natural: the above properties would follow from a proof of Faris's product formula \cite{Far67} under the assumptions of Theorem IV 1. We hope to return to this problem in the future. %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% \section*{\large Acknowledgements} \noindent W.F.W. was supported in part by CNPq. L.A.M. was supported by FAPESP under grant 99/04079-1. O.B. greatly appreciates the financial support by Fapesp under grant 01/08485-6. %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% \renewcommand{\thesection}{\Alph{section}} \setcounter{section}{1} \setcounter{axiom}{0} \section*{Appendix A} \label{apa} \newcounter{apend} \setcounter{apend}{1} \renewcommand{\theequation}{\Alph{apend}.\arabic{equation}} \setcounter{equation}{0} \noindent Let us consider the problem (\ref{eq}) for the case in which $D(A(t))= {\rm const.}$. The notation is as in the first part of section \ref{kis}. \newline Consider the following conditions (in that follows $t\in [-T_1,T_2]$, unless otherwise specified): % \begin{itemize} \item[(i)] there exists a family $\|\cdot \|_t$, of norms in $X$ such that $a^{-1}\| \Psi \| \leq \| \Psi\|_t\leq \|\Psi \|_s \leq a \| \Psi \|$, $a\geq 1$, for $-T_1\leq s\leq t\leq T_2$ and $\Psi \in X$; \item[(ii)] $Y$ is a dense subset of $X$ with $D(A(t))=Y$; \item[(iii)] for all $\lambda > 0$ and $\Psi \in Y$ we have $R(\lambda - A(t)) =X$ and $\|(\lambda - A(t))\Psi \|_t\geq \lambda \| \Psi \|_t$; \item[(iv)] $A(t)$ is weakly continuously differentiable. \end{itemize} \begin{axiom} {\rm (\cite{Kis64}, theorem 3.0)} Let the conditions $({\rm i})$ --$({\rm iv})$ be satisfied. Then, there exists an unique solution of the problem (\ref{eq}) and the corresponding propagator $U(t,s)$ is strongly continuous in $-T_1\leq s\leq t\leq T_2$ and satisfies the properties (\ref{U1}), (\ref{UUU}), (\ref{UDD}) and (\ref{eqU}). \end{axiom} Now we shall explain some aspects of Kisy\'nski's proof of this theorem. Consider the family of equations \be \label{eqan} \frac{d}{dt}\Phi (t) =A_n(t)\Phi (t)\; , \hspace{1.0cm}\Phi(0)=\Phi_0 \; ,\hspace{1.0cm} n=1,2,\cdots \; , \ee % with % \be \label{an} A_n(t)=nA(t)\left(n-A(t)\right)^{-1}\; . \ee % The set $Y$ supplied with the norm $|\! |\! |\cdot |\! |\! |_t = \|((1-A(t)) \cdot \|$ is a Banach space algebraically and topologically contained in $X$. Then, from (i) and (ii) it follows that $A(t)\in {\cal L} (Y, X)$ is a weakly continuously differentiable operator, which, by the Banach-Steinhaus theorem, implies $\| A(t)\Phi\| \leq C |\! |\! | \Phi |\! |\! |_0$ for $\Phi \in Y$ and some constant $C$ (the equivalence of the norms $ |\! |\! |\cdot |\! |\! |_t$ was used). So, by using (i) and (iii), it follows that % \be \left\| \Phi - n(n-A(t))^{-1} \Phi \right\| = \frac{1}{n} \left\| (1-A(t)/n)^{-1} \left( A(t)\Phi \right) \right\| \leq \frac{C a^2}{n} |\! |\! | \Phi |\! |\! |_0 \; , \ee % which implies that $n(n-A(t))^{-1}$ converges strongly and uniformly to $1$. Therefore, the sequence of bounded operators $A_n(t)$ converges strongly to $A(t)$. The operators $A_n(t)$ are weakly continuosly differentiable, therefore they satisfy a Lipschitz condition in the sense of the norm. Hence, it follows that $A_n(t)$ is continuous in the sense of the norm and Yosida's method \cite{Yos80} guarantee the existence and uniqueness of the evolution operators $U_n(t,s)$ of the equation (\ref{eqan}) satisfying the properties equivalent to (\ref{U1}) -- (\ref{eqU}). Besides, $U_n(t,s)$ satisfy \cite{Kis64} % \be \label{unm} \| U_n(t,s)\| \leq M \; . \ee Before to proceed we will consider the equation (\ref{eqan}) perturbed by the bounded (in $X$) weakly continuous operator $B(t)=-\frac{dA(t)}{dt}(1-A(t))^{-1}$, that is, % % \be\label{eqpert} \frac{d}{dt}\Phi (t) = \left( A_n(t)+B(t)\right) \Phi (t) \; , \hspace{1.0cm}\Phi(0)=\Phi_0 \; \ee % The evolution operator of (\ref{eqpert}), denoted $H_n(t,s)$, is given by % $$ H_n(t,s) = (1 - A(t))U_n(t,s) (1-A(s))^{-1}\; . $$ % Then, it follows that $ H_n(t,s) \in {\cal L}(X)$ is weakly continuously differentiable in $-T_1\leq s,t\leq T_2$, satisfying % \be \label{hn} \|H_n(t,s)\| \leq D \; . \ee Now, we subdivide the segment $[-T_1,T_2]$ into $K$ equal intervals. Then, the conditions ($T\equiv T_1+T_2$) % \be U_{nK}(t,s) =\exp \left\{ (t-s) A_n({\scriptstyle - T_1+\frac{i-1}{K}T}) \right\} \; , \ee % $- T_1+\frac{i-1}{K}T \leq s,t \leq - T_1+\frac{i}{K}T$, $i=1, \ldots , K$, and % \be U_{nK}(t,s)U_{nK}(s,r) =U_{nK}(t,s)\; , \hspace{1.0cm} -T_1\leq r,s,t\leq T_2\; , \ee % define a unique family of operators $U_{nK}(t,s) \in {\cal L}(X)$ continuous in the sense of the norm such that % \be \label{bounduk} \|U_{nK}(t,s)\| \leq a^2 \; . \ee % The operators $ U_{nK}(t,s)$ satisfy % $$ \frac{\partial}{\partial s}U_{nK}(t,s) = -U_{nK}(t,s) A_n({\scriptstyle - T_1+ \frac{T}{K}\left[ \frac{Ks}{T}\right] } )\; , $$ % where $[(Ks)/T]$ stands for the integer part of $(Ks)/T$. Besides, for fixed $K$, $U_{nK}(t,s)$, $n=1,2,\ldots$, is a sequence uniformly strongly convergent in $ - T_1\leq s\leq t\leq T_2$. \newline Then, by integrating $\frac{\partial}{\partial \tau}U_{nK}(t,\tau )U_n(\tau , s)$ we obtain % \be \label{unuk} U_n(t,s)-U_{nK}(t,s) = \int_s^t U_{nK}(t,\tau ) \left( A_n(\tau )- A_n({\scriptstyle - T_1+ \frac{T}{K}\left[ \frac{K\tau }{T}\right] } )\right) U_n(\tau ,s) d\tau \; . \ee % We have \cite{Kis64} % \be \label{aa} \left\| A_n(\tau )\Phi - A_n({\scriptstyle - T_1+ \frac{T}{K}\left[ \frac{K\tau }{T}\right] } ) \Phi \right\| \leq \frac{{\rm const.}}{K} |\! |\! | \Phi |\! |\! |_0 \; . \ee % Then, since % $$ U_n(t,s) = (1-A(t))^{-1} H_n(t,s)(1-A(s)) $$ % and $(1-A(s))\in {\cal L}(Y,X)$ and $(1-A(t))^{-1}\in {\cal L}(X,Y)$ are weakly differentiable, we obtain, by using (\ref{hn}), % \be \label{uf} |\! |\! | U_n(t,s) \Phi |\! |\! |_0 \leq {\rm const.} |\! |\! | \Phi |\! |\! |_0 \; , \ee % for $\Phi \in Y$ and $-T_1\leq s\leq t\leq T_2$. Then, from (\ref{bounduk}), (\ref{unuk}), (\ref{aa}) and (\ref{uf}), it follows that % \be \label{uconvu} \left\| U_n(t,s)\Phi - U_{nK}(t,s)\Phi \right\| \leq \frac{\rm L}{K} |\! |\! | \Phi |\! |\! |_0\; , \ee % with $L={\rm constant}$. \newline Now, for $\Phi \in Y$ we have % \begin{eqnarray} \left\|U_n(t,s)\Phi - U_m(t,s)\Phi \right\| &\leq & \left\|U_n(t,s)\Phi - U_{nK}(t,s)\Phi \right\| \nonumber \\ &+& \left\|U_{mK}(t,s)\Phi - U_m(t,s)\Phi \right\| \nonumber \\ &+&\left\|U_{nK}(t,s)\Phi - U_{mK}(t,s)\Phi \right\| \nonumber \\ &\leq & 2\frac{L}{K} |\! |\! |\Phi |\! |\! |_0 +\left\|U_{nK}(t,s)\Phi - U_{mK}(t,s)\Phi \right\| \label{UUUU} \end{eqnarray} % The first term in the r.h.s. may be made arbitrarily small for large $K$. After this, one choose $n$ and $m$ so large that the second term becomes arbitrarily small for all $-T_1\leq s\leq t \leq T_2$, since the sequence $U_{nK}(t,s)$ is uniformly strongly convergent. Since $Y$ is dense in $X$, and from (\ref{unm}), (\ref{UUUU}) implies that the convergence is in all of $X$, in the triangle $-T_1\leq s\leq t \leq T_2$. Then, it follows directly from the properties of $U_n(t,s)$ that $U(t,s) = s - \lim_{n\rightarrow \infty} U_n(t,s)$ is the evolution operator of (\ref{eq}) for constant domain \cite{Kis64}. \newline {\bf Remark.} The proof outlined above is valid for $ - T_1 \leq s\leq t \leq T_2$. However, by substituting the conditions (i) and (iii) above by the conditions (a) and (c) in the theorem \ref{k4.4} the proof can be extended for the square $-T_1\leq s,t \leq T_2$. \begin{thebibliography}{99} \section*{References} \bibitem{BLT75} N.N. Bogoliubov, A. N. Logunov and I. T. Todorov, {\it Introduction to Axiomatic Quantum Field Theory} (Benjamin, 1975). \bibitem{EGl73} H. Epstein and V. Glaser, {\it Ann. Inst. H. Poincar\'{e} A} {\bf 19}, 211 (1973). \bibitem{Sch01} G. 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