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% entorno proof

\begin{document}

\title{Dispersion Relations in the Stochastic Limit of Quantum Theory\footnote{26th Conference on Quantum Probability and Infinite Dimensional Analysis. Levico Terme, 20-26 February, 2005.}}

%\date{February 24, 2005}

\author{L. Accardi}
\address{Centro Vito Volterra. Universit\`a degli Studi di Roma \lq\lq Tor Vergata". 00133, Rome, Italy. \\e-mail: {\tt accardi@volterra.mat.uniroma2.it}}

\author{F.G. Cubillo}
\address{Departamento de An\'alisis Matem\'atico. Universidad de Valladolid. 47005, Valladolid, Spain. \\e-mail: {\tt fgcubill@am.uva.es}.}

\maketitle

%\begin{abstract}
\abstracts{
We apply new techniques based on the distributional theory of Fourier transforms to study, in the stochastic limit of quantum theory, the convergence of the rescaled creation and annihilation densities, which lead to the master fields, and the form of the drift term of the stochastic Schr\"odinger equation obtained in such limit.  This approach permit us to dispense with the ``analytical condition" and other restrictions usually considered and also to establish the dependence of the stochastic golden rules on certain properties of the dispersion function of the quantum field. 
[cf. articoli-cubi0330.da ]
%\end{abstract}
}
\section{Introduction}\label{sec1}

The {\it stochastic golden rules}~[\refcite{ALV,AK}], which arise in the stochastic limit of quantum theory as natural generalizations of the Fermi golden rule, provide a natural tool to associate a stochastic flow, driven by a white noise equation, to any discrete system interacting with a quantum field.
The stochastic limit captures the dominating contributions to the dynamics arising from the cumulative effects, on a large time scale, of small interactions; the physical idea is that, looked from the slow time scale of the system, the field looks like a very chaotic object: a {\it quantum white noise}, i.e a $\delta$-correlated (in time) quantum field also called {\it master field}. The new evolution is an approximation of the original one which preserves much nontrivial information on the original complex system related to its decay and shift properties. 

In this work we study, from an analytical point of view, the convergence of the rescaled creation and annihilation densities, which lead to the master fields, and the form of the drift term of the stochastic Schr\"odinger equation obtained in such limit, which contains the quantum mechanical fluctuation-dissipation relations.  This approach permit us to dispense with the {\it analytical condition} and other restrictions usually considered --see Section \ref{sec2}-- and also to establish the dependence of the stochastic golden rules on certain properties of the dispersion function of the quantum field. 
To be precise, we shall see that, for the region $\Gamma_1$ where the dispersion function is regular and not constant, every Bohr frequency of the system in its range gives rise to an independent master field, which is a quantum white noise concentrated over the corresponding resonant surface, whereas both the rest of Bohr frequencies and the open regions $\Gamma_{\alpha_j}$, where the dispersion function is constant, give rise to zero master fields, except for the resonant case, see Theorem \ref{thcovaa+}.
In a similar way we will show that the regions $\Gamma_{\alpha_j}$ do not contribute to the drift term whenever the resonant case is not present, whereas for the region $\Gamma_1$ we obtain the usual expression, see Theorem \ref{th5}. The contribution of the singular regions of dispersion has not been completely determined yet.

\section{Preliminaries}\label{sec2}


In what follows we shall consider quantum systems describing the interaction of
a discrete spectrum system $S$ with free Hamiltonian 
$$
H_S:=\sum_r \varepsilon_r P_{\varepsilon_r}
$$
and {\it Bohr frequencies} ${\omega=\varepsilon_r-\varepsilon_{r'}}$, $(\varepsilon_{r}, \varepsilon_{r'}\in \hbox{Spec}\,H_S)$,
and a bosonic quantum field as reservoir $R$ with free Hamiltonian (on Fock space) 
$$
H_R:=\int dk\,\om(k)a^+(k)a(k),
$$
where $\om(k)$ is the {\it dispersion function}, $a^\pm(k)$ are the creation and annihilation densities, and the reference vector is
mean zero Gaussian and gauge invariant, with covariance of the form 
\begin{equation}\label{crca}
\langle\left(\begin{array}{cc} a^{+}(k) a({k'})& 0 \\
            0             &a({k'}) a^{+}(k)\end{array}\right)\rangle=
\left(\begin{array}{cc} N(k)  & 0 \\
            0     & N(k) +1 \end{array}\right) \delta (k-k').
\end{equation}
We will assume that the total Hamiltonian has the form  
$$
H^{(\lambda)}:=H_0+\lambda H_I=H_S+H_R+\lambda H_I,
$$
where $\l$ is a real coupling parameter and the interaction Hamiltonian $H_I$ is of {\it dipole type}, i.e.\footnote{The asterisk $^\ast$ denotes the Hermitian conjugate for operators and the complex conjugate for scalars. For distributional densities we use the symbol $^+$ instead $^\ast$.}
$$
H_I=\sum_j \left(D_j^*\otimes A(g_j)+D_j\otimes A^*(g_j)\right),
$$
where $D_j$ are system operators and
$$
A^\ast(g_j):=\int dk\, g_j(k) a^+(k),\quad
A(g_j):=\int dk\, g^\ast(k) a(k),
$$ 
being the functions $g_j$ the {\it cutoff} or {\it form factors}.
Often we will simplify the notations by omitting the symbol $\otimes$.


In the {\it stochastic limit approach} we consider the
time rescaling $t\to t/\lambda^2$ in the solution
$U^{(\lambda)}_t=e^{itH_0} e^{-itH^{( \lambda)}}$
of the Schr\"o\-din\-ger equation in interaction picture:
$$
{ \frac{\partial}{\partial t} }  U^{( \lambda)}_t
=-i { \lambda  } H_I (t) \ U_t^{(\lambda )},
$$
$$
H_I(t)=e^{it H_0}H_Ie^{-itH_0},
$$
and study the limits, in a topology to be specified,
of the rescaled interaction Hamiltonian and of the rescaled propagator:
$$
\lim_{\lambda\to0}{\frac{1}{\lambda}}\,H_I\left({\frac{t}{\lambda^2}}\right)
=:h_t,
$$
$$
\lim_{\lambda\to0}U^{(\lambda)}_{t/\lambda^2}=: U_t.
$$

In {\it canonical form} this reduces to
find the limit of the  {\it rescaled creation and annihilation densities}
\begin{equation}\label{rcad}
a^\pm_{\lambda,\omega}(t,k):=
{1\over\lambda}\,e^{\mp i{t\over\lambda^2}\,(\omega(k)-\omega)}a^\pm(k),
\end{equation}
obtaining the white noise Schr\"odinger equation
${\partial\over\partial t} U_t=-ih_tU_t$, whose normally ordered form is the quantum stochastic differential equation
\begin{equation}\label{1.19}
dU_t=(-idH(t)-Gdt)U_t,
\end{equation}
where
$$
dH(t):=
\lim_{\lambda\to0}\int^{t+dt}_t d\tau\,{1\over\lambda}\,
H_I\left({\tau\over \lambda^2}\right)
$$
is called the {\it martingale term} and
\begin{equation}\label{drift}
Gdt:=\lim_{\lambda\to0}{1\over\lambda^2}\,
\int^{t+dt}_t dt_1 \int^{t_1}_t dt_2\,
\langle H_I\left({t_1\over \lambda^2}\right)
H_I\left({t_2\over\lambda^2}\right)\rangle
\end{equation}
is known as the {\it drift term}.


Among the usual assumptions to achieve this program we have the following:
\begin{itemize}
\item
the cut--off functions $g_j$ are Schwartz functions;
\item
the dispersion function $\omega(k)$ and the cut--off
functions $g_j$ are related by the following {\it analytical condition}:
$$
\int_{\Bbb R}dt|\langle g_i,e^{it\omega(p)}g_j\rangle|=\int_{\Bbb R}
dt\,\left|\int_{\Bbb R^d}dk\,e^{it\omega(k)} g^\ast_i(k)g_j(k)\right|<+
\infty;
$$
\item
the $(d-1)$--dimensional Lebesgue measure of the surface
$\{k:\omega(k)=0\}$ is equal to zero
(this implies, in particular $\delta(\omega(k))=0$).
\end{itemize}

In this work we apply new techniques, based on the distributional theory of Fourier transforms 
~[\refcite{GS,Maz,SCH,T}], which permit us to dispense with the above conditions  and to establish the dependence of the stochastic golden rules on certain properties of the dispersion function $\om(k)$.


\section{The Dispersion Function}\label{sec3}

In what follows we shall assume that the dispersion function $\R^d\ni k\mapsto \om(k)\in\R$ is such that $\om(k)\geq 0$ for all $k\in\R^d$ and we can write
$$\R^d=\Gamma_1\cup\Gamma_2\cup \Gamma_3,$$
where:
\begin{itemize}
\item[(i)]
$\Gamma_1$ is an open set of $\R^d$ in which $\om(k)$ is a $C^\infty$-function and $\nabla \om(k) \neq 0$ for every $k\in \Gamma_1$. We shall denote by $\Gamma_1^1$ the range of the restiction of $\om(k)$ to $\Gamma_1$, i.e.
$$\Gamma_1^1:=\hbox{Rang}(\om_{|\Gamma_1}),$$
and assume that the boundary $\partial \Gamma_1^1$ of $\Gamma_1^1$ has Lebesgue measure zero.
\item[(ii)] 
$\Gamma_2=\cup\Gamma_{\alpha_j}$, being $\Gamma_{\alpha_j}$ an open subset of $\R^d$ where the dispersion function $\om(k)$ is constant and equal to $\alpha_j$, i.e.
$$\om(k)=\alpha_j,\quad \forall\,k\in\Gamma_{\alpha_j}.$$
\item[(iii)]
$\Gamma_3=\R^d\backslash (\Gamma_1\cup\Gamma_2)$, that is $\Gamma_3$ contains the boundaries of $\Gamma_1$ and $\Gamma_2$ and other possible regions of singular points of the dispersion function $\om(k)$. 
\end{itemize}


\section{Convergence of the Rescaled Densities}\label{sec4}

Let us study the convergence, {\it in the sense of correlators}, of the rescaled creation and annihilation densities given in Eq.(\ref{rcad}).
To simplify the notation we restrict our attention to the vacuum reference vector, so that $N(k)=0$ (see Eq.(\ref{crca})). The extension of the results to the general case is immediate.
Moreover, because the mean zero Gaussianity, we have only to prove the convergence, in the sense of Schwartz distributions~[\refcite{SCH}], of the covariance
$$
\<a_{\lambda,\omega}(t,k) a^+_{\lambda,\omega'}(t',k')\>=
{1\over\lambda^2}\,e^{-i{t-t'\over\lambda^2}\,(\omega(k)-\omega)+i{t'\over\lambda^2}\,(\omega-\omega')}\delta(k-k'),
$$
i.e. we must calculate, for any Schwartz test functions $\phi$, $\varphi$, $f$ and $g$, 
$$
\lim_{\lambda\to 0} \int dt\,dt'\,dk\,dk'\,\phi(t)\varphi(t')f(k)g(k') 
\<a_{\lambda,\omega}(t,k) a^+_{\lambda,\omega'}(t',k')\>.
$$

The following theorem shows that, on $\Gamma_1$, every Bohr frequency $\omega$ in the open range $\Gamma_1^1$ of the dispersion function gives rise to an independent master field, which is a quantum white noise concentrated over the resonant surface $\omega(k)-\omega=0$, and the rest of Bohr frequencies give rise to zero master fields, while, on the open regions $\Gamma_{\alpha_j}$ where the dispersion function is constant, the limit does not exist in the resonant case $\alpha_j=\omega=\omega'$ and again gives rise to zero master fields otherwise.


\begin{theorem}\label{thcovaa+}
Under the conditions for $\om(k)$ given above, in the sense of Schwartz distributions, i.e. in $\S'(\R^{2d+2})$:
%\begin{itemize}
%\item[(a)]
\begin{eqnarray}
&(a)& \mbox{Over $\Gamma_1$, if $\omega$ doesn't belong to the boundary $\partial$ $\Gamma_1^1$ of $\Gamma_1^1$,} \nonumber \\
%$$
&& \lim_{\lambda\to 0}\<a_{\lambda,\omega}(t,k) a^+_{\lambda,\omega'}(t',k')\>\Big|_{\Gamma_1} \nonumber \\
&& =\delta_{\om,\om'}2\pi\delta(t-t')\delta(k-k')\delta(\om(k)-\om)\chi_{\Gamma_1^1}(\om).
%$$
\nonumber \\
%\item[(b)]
&(b)& \mbox{Over each $\Gamma_{\alpha_j}$,} \nonumber \\
%$$
&& \lim_{\lambda\to 0}\<a_{\lambda,\omega}(t,k) a^+_{\lambda,\omega'}(t',k')\>\Big|_{\Gamma_{\alpha_j}}= 
\left\{\begin{array}{ll}
\hbox{doesn't exist}, & \hbox{ if } \alpha_j=\omega=\omega',\quad [*]
\\[1ex]
0, &  \hbox{ if } \alpha_j\neq\omega \hbox{ or } \alpha_j\neq\omega'.
\end{array}\right. \nonumber
%$$
%\end{itemize}
\end{eqnarray}
\end{theorem}

The proof of this result cast some light on the resonant case $\alpha_j=\omega=\omega'$ of item (b): Over each $\Gamma_{\alpha_j}$ the final expression in our calculations is\footnote{We use the following conventions:
The Fourier transform $f^\wedge$ and the inverse Fourier transform $f^\vee$ of a test function $f\in\S(\R^d)$ are given by 
$$f^\wedge(s):={1\over(2\pi)^{d/2}}\int dx\, f(x)e^{ix\cdot s},\quad 
f^\vee(s):={1\over(2\pi)^{d/2}}\int dx\, f(x)e^{-ix\cdot s},$$
so that $f^{\wedge\vee}=f^{\vee\wedge}=f$.
The Fourier transform $F^\wedge$ and the inverse Fourier transform $F^\vee$ of a distribution $F\in\S'(\R^d)$ are defined by the relations 
$$\<F^\wedge,f^\wedge\>=\<F,f\>,\quad
\<F^\vee,f^\vee\>= \<F,f\>,$$
being dual pair $\<\cdot,\cdot\>$ antilinear on the left and linear on the right.} 
$$
\lim_{\lambda\to 0}  {2\pi\over\lambda^2}\,
\phi^\vee\left({\alpha_j-\omega\over\lambda^2}\right)\,
\varphi^\wedge\left({\alpha_j-\omega'\over\lambda^2}\right)
\int_{\Gamma_{\alpha_j}} dk\,f(k)g(k)
$$
$$=\lim_{\lambda\to 0}  {2\pi\over\lambda^2}\,
\phi^\vee(0)\,\varphi^\wedge(0)\int_{\Gamma_{\alpha_j}} dk\,f(k)g(k),
$$
which is equal to zero when $\phi^\vee(0)=0$ or $\varphi^\wedge(0)=0$, or $\pm\infty$ otherwise. Thus the limit also exists in this case, and is equal to zero, if we restrict our attention to test functions with zero mean in time.

\medskip

What happens over $\Gamma_3$ or when $\omega\in\partial \Gamma_1^1$?
For example, for dispersion functions of the form
$$
\om(k)=|k|^\mu,\quad \mu>0,
$$
we have $\Gamma_1=\R^d\backslash\{0\}$, $\Gamma_2=\emptyset$, $\Gamma_3=\{0\}$, $\Gamma_1^1=(0,\infty)$ and $\partial \Gamma_1^1=\{0\}$, so that the frequency of interest is $\omega=0$. We obtain in this case
$$
\begin{array}{l}
\ds\left.\lim_{\lambda\to 0}\<a_{\lambda,\omega}(t,k) a^+_{\lambda,\omega'}(t',k')\>
\right|_{\om=0}%=
\\[4ex]
\hspace{5ex}
\ds=\left\{\begin{array}{cl}
0,& \hbox{ \rm if }d-\mu>0,\\
\ds\delta_{\om,\om'}{2\pi^{d/2+1}\over\Gamma(d/2)}\delta(t-t')\delta(k-k')\delta(k), 
& \hbox{ \rm if }d-\mu=0.
\end{array}\right.
\end{array}
$$
When $d-\mu<0$, our techniques do not give an answer.


\section{The Drift}\label{sec5}

As Eq.(\ref{drift}) shows, the {\it drift term} $Gdt$ in the stochastic Schr\"odinger equation given in Eq.(\ref{1.19}) is the limit of the
expectation value in the reservoir state of the second term in the iterated series
solution for the rescaled Shr\"odinger equation in interaction picture.

In the following theorem we show that the open region $\Gamma_2$ does not contribute to the drift term whenever the resonant case $\alpha_k=\omega$ is not present, whereas for the region $\Gamma_1$ we obtain the usual expression for the drift. The contribution of the singular region $\Gamma_3$ to the drift has not been determined yet.

\begin{theorem}\label{th5}
Under the conditions for $\om(k)$ given above we have:

(i) If $\Gamma_2$ is not empty and no Bohr frequency $\om$ of the system coincides with one of the values $\alpha_k$, then the contribution of the region $\Gamma_2$ to the drift term is zero, whereas if any of the Borh frequencies $\om$ of the system coincides with one of the values $\alpha_k$, then $G$ does not exist.

(ii) Otherwise
$$
\begin{array}{rl}
\ds G=\sum_{ij} \sum_{\omega} &
\ds \Big((g_i|g_j)^-_{\omega}E_{\omega}^*\left(D_i\right)E_{\omega}\left(D_j\right)
+{(g_i|g_j)^+_{\omega}}^*E_{\omega}\left(D_i\right)E_{\omega}^*\left(D_j\right) %+
\\
& 
+\hbox{\it The part corresponding to the singular region } \Gamma_3\Big),
\end{array}
$$
where, for each Bohr frequency $\omega$, the $E_{\omega}(D_j)$ are system operators defined by
$$
E_{\omega}(D_j):=
\sum_{\varepsilon_{r}\in F_{\omega}}
P_{\varepsilon_r-\omega} D_j P_{\varepsilon_{r}},
$$
$$
F_{\omega}:=\{\varepsilon_{r}\in \hbox{Spec}\,H_S:
\varepsilon_{r}-\omega\in\hbox{ Spec }H_S\},
$$
and the explicit form of the constants $(g_i|g_j)^{\pm}_{\omega}$ is
$$
\begin{array}{rl}
\ds(g_i|g_j)^{-}_{\omega}= &
\ds\pi\chi_{\Gamma_1^1}(\om)\int_{\Gamma_1} dk\, g^*_i(k)g_j(k) (N(k)+1) \delta(\omega(k)-\omega) %-
\\
& \ds-i\,\hbox{P.P.}\,\int_{\Gamma_1} dk\, g^*_i(k)g_j(k)
{(N(k)+1)\over\omega(k)-\omega},
\end{array}
$$
$$
\begin{array}{rl}
\ds (g_i|g_j)^{+}_{\omega}= &
\ds \pi\chi_{\Gamma_1^1}(\om)\int_{\Gamma_1} dk\, g^*_i(k)g_j(k) N(k) \delta(\omega(k)-\omega) %-
\\
& \ds -i\,\hbox{P.P.}\,\int_{\Gamma_1} dk\, g^*_i(k)g_j(k)
{N(k)\over\omega(k)-\omega}.
\end{array}
$$
\end{theorem}

The constants $(g_i|g_j)^{\pm}_{\omega}$ are called {\it generalized susceptivities} and have an important physical interpretation. In some sense they contain all the physical information on the original Hamiltonian system and can be considered as the prototype of quantum mechanical fluctuation-dissipation relations, cf.~[\refcite{ALV}].

\bigskip

\noindent
{\bf Acknowledgements}
\medskip

\noindent
F.G. Cubillo is grateful to L. Accardi and Centro Vito Volterra for support and kind hospitality.
 

\begin{thebibliography}{99}

\bibitem{ALV}
L. Accardi, Y.G. Lu, I. Volovich, {\it Quantum Theory and Its Stochastic Limit}, Springer-Verlag, Berlin, 2002.

\bibitem{AK}
L. Accardi, S.V. Kozyrev, {\it Quantum Interacting Particle Systems}.
In {\it Quantum Interacting Particle Systems}, World Scientific, Singapore, 2002, pp. 1--193.

\bibitem{GS}
I.M. Gelfand, G.E. Shilov, {\it Les Distributions}, Dunod, Paris, 1962.

\bibitem{Maz}
V.G. Maz'ja, {\it Sobolev Spaces}, Springer-Verlag, Berlin, 1985.

\bibitem{SCH}
L. Schwartz, {\it M\'ethodes Math\'ematiques pour les Sciences Physiques}, Hermann, Paris, 1966.

\bibitem{T}
E.C. Titchmarsh,  {\it Introduction to the Theory of Fourier Integrals}, Oxford University Press, Oxford, 1948.

\end{thebibliography}


\end{document}
