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\title{Ground state and functional integral representations of the CCR algebra \\ with free
evolution}
\author{J. L\"{o}ffelholz,
%G. Morchio, F. Strocchi}
\\ Berufsakademie, Leipzig \and
G. Morchio,
\\ Dipartimento di Fisica, Universita' di Pisa and INFN, Pisa \and
F. Strocchi
\\ Scuola Normale Superiore and INFN, Pisa}
\date{}
\maketitle
\begin{abstract}
The ground state representations on the CCR algebra with free
evolution are classified and shown to be either non regular or
indefinite. In both cases one meets mathematical structures which
appear as prototypes of phenomena typical of gauge quantum field
theory. The functional integral representation in the positive non
egular case is discussed in terms of a generalized stochastic
process satisfying the Markov property. In the indefinite case the
ground state is faithful and its GNS representation is
characterized in terms of a KMS operator. In the corresponding
euclidean formulation, one has a generalization of the
Osterwalder-Schrader reconstruction and the indefinite Nelson
space, defined by the Schwinger functions, has a unique Krein
structure allowing for the construction of Nelson projections,
which satisfy the Markov property. The Schwinger functions can be
represented in terms of a functional measure and complex
variables.
\end{abstract}
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\section{Introduction}
Most of the wisdom on quantum field theory and more generally on
quantum systems with infinite degrees of freedom relies on the
existence of a vacuum or ground state, so that the formulation and
the control of the model is obtained in terms of the ground state
correlation functions. In particular the analytic continuation to
imaginary time and the functional integral approach exploits this
basic property.~\cite{STW, GJ}
For a large class of models involving infrared singular canonical
variables or fields the existence of a ground state is linked to
non regular representations of such variables
~\cite{AMS1,AMS2,LMS1,LMS2} if positivity is required, whereas
regular, i.e. weakly continuous, representations are available if
one allows indefinite metric ~\cite{MPS,MPS1, S}.
In order to clarify the mathematical structures emerging in these
cases, including the euclidean functional integral formulation, we
investigate the simple case of the ground state representations of
the Heisenberg algebra with free time evolution.
Even if the model may look trivial, it reproduces the basic
properties of the temporal gauge in quantum electrodynamics
~\cite{CT}, where the longitudinal variables $div {\bf A}, \,\,div
{\bf E}$ correspond to a collection of free quantum mechanical
canonical variables ~\cite{LMS3}.
The analysis of the model sheds light on the occurrence of unusual
features, like the interplay between the time translation
invariance, the energy spectral condition, the positivity and the
regularity of the vacuum state etc. ~\cite{LMS3}.
The model also provides non trivial information on the functional
integral formulation of the temporal gauge and in general of
models with infrared singular variables.
In Sect.2 we show that the ground state representations of the
Heisenberg algebra, which correspond to the the standard field
quantization of the temporal gauge, violate positivity and the
energy spectral condition and share the basic features of the
KMS representations in statistical mechanics ~\cite{BR, H}. The
ground state turns out to be faithful on the Heisenberg algebra
and the corresponding GNS representation, with cyclic ground state
vector $\Psio$, is given by the tensor product of a Fock and
anti-Fock representations ~\cite{MSV, MS} of the canonical
variables $Q_\pm \eqq (q \pm p')/\sqrt{2},\,\,\,P_\pm \eqq (\pm p
+ q')/\sqrt{2}$, with $q'\eqq i S q S,\,\,\, p'\eqq -i S p S $
and $S$ the antiunitary KMS operator defined by $ S A \Psio = A^*
\Psio$.
In Sect.3 we show that the ground state positive representations
of the Weyl algebra are non regular and satisfy the energy
spectral condition. The corresponding functional integral
formulation and the relation with stochastic processes is
discussed in Sect.4, where we show that one has
Osterwalder-Schrader positivity and that the Markov property holds
~\cite{GJ,N}.
Actually, the euclidean correlation functions can be expressed as
expectations of the euclidean fields $U(\a, \t) = e^{i\a x(\t)}$
with a functional measure which is the product of the conditional
Wiener measure $d W_{0, x}(x(\t))$ on trajectories $x(\t)$
starting at $x$ at time $\t=0$, times the ergodic mean $d \nu(x)$,
which defines a measure on the Gelfand spectrum of the Bohr
algebra ~\cite{LMS1} generated by the $U(\a, \t)$. The
Osterwalder-Schrader reconstruction theorem provides the (non
regular) quantum mechanical representation, with a ground state,
which is cyclic with respect to the euclidean algebra at time
zero.
In Sect. 5 we analyze the euclidean correlation functions of the
indefinite representation of the Heisenberg algebra. The
Ostervalder-Schrader positivity does not hold but an
Osterwalder-Schrader reconstruction can be done, yielding the
indefinite quantum mechanical representation.
Nelson positivity fails and we have an indefinite inner product
space with a Krein structure very similar to that of the massless
scalar field in two space time dimensions ~\cite{MPS}. A
generalization of Nelson strategy ~\cite{N} can be performed and
projection operators $E_{\pm}, E_0$ can be defined, yielding the
same operator formulation of the Markov property as in the
positive case.
The euclidean (indefinite) correlation functions have a functional
integral representation in terms of the conditional Wiener measure
$d W^{x, v} \eqq d W_{0,0}(x(\t) - x - v |\t|) $ and the quantum
mechanical expectation on the variables $x = x(0)$, and $ v =
\lim_{\t \ra + \infty} x(\t)/\t$.
The correlation functions $< x(\t_1) ...x(\t_n)>$ can also be
obtained as a functional integral of complex variables
$z_\a(\t_1)...z_\a(\t_n)$ , $$z_\a(\t) \eqq (1 - \a^2|\t|)(x -
\a^{-2} v)/2 + i ( 1 + \a^{2}\,|\t|) (x + \a^{-2}\,v)/2 + x(\t) -
x - v |\t|,$$ with a measure which is the product of the
conditional Wiener measure $d W_{0,0}(x(\t) - x - v |\t|) $ times
the gaussian measure $d \g_{\a^{-1}}(x) \,d \g_\a(v)$, where $d
\g_\a(.)$ denotes the gaussian measure with variance $\a^2/2$.
In view of the strong analogies between the simple model discussed
in this note and the massless scalar field in two space time
dimensions, the above analysis can be extended to the quantum
field theory model to provide a functional integral representation
of its euclidean correlation functions.
%%%%%%%%%%%%%%%%%2222222222222222222222222222222222222%%%%%%%%%%%
\newpage
\section{Ground state representations of the Heisenberg algebra
with free evolution}
At the algebraic level the quantum free particle is described by
the Heisenberg *-algebra $\Ha$ generated by $q, p$ (for simplicity
we consider the one dimensional case), satisfying the canonical
commutation relations (CCR) \be[\, q ,\, p \,] = i \ee and the
invariance under the *-operation (selfadjointness): $q = q^*$, $
p = p^*$. The time evolution is defined by the following one
parameter group of *-automorphisms $\alpha_t$, $t \in \reali$ \be
\alpha_t (q) = q + p \,t \equiv q(t) \ , \ \ \ \ \ \ \alpha_t (p)
= p \ . \ee It is natural to ask whether there are time
translationally invariant linear functionals $\o$ on $\Ha$, $ \o
(\alpha_t (A)) = \o (A)$, and to investigate the generalized GNS
representations defined by them.
\begin{Proposition} A state $\o$ on the Heisenberg algebra
invariant under time translations, i.e. such that $\forall A \in
\Ha$, $ \o (\alpha_t (A)) = \o (A)$, has the following properties:
\ni
i) it is not positive;
\ni ii) the Fourier transform of $\o (A \alpha_t (B))$, $A, B \in
\Ha$ is not a measure, in general;
\ni
iii) its two point function has the following form
\be
\o(q^2) = c, \, \ \ \ \ \o(p^2) = 0, \ , \ \ \ \ \o (q p) = - \o(p
q) = i/2 \ , \ee where $c$ is a constant;
\ni iv) furthermore, if $\o$ is Gaussian, i.e. its truncated
correlation functions vanish, the GNS representation $({\cal
H}_\o,\, \pi_\o)$ defined by it has a non trivial commutant
$\pi_\o(\A_H)^\prime$.
\ni v) if $\o$ is Gaussian and $c = 0$, then $\o$ is invariant
under scale transformations $$ q \ra \l\,q, \,\,\,\,\,p \ra
\l^{-1}\,p, \,\,\,\,\l \in \Rbf,$$ and such transformations are
implemented by isometric operators in ${\cal H}_\o$.
\end{Proposition}
\Pf \,\,\, Property i) follows from $$ \o(p^2) = \o (d
\alpha_t(q) / d t \;\, p) = d/dt \; \o (\alpha_t (q p)) = 0. $$
Property iii) follows from the condition of time translation
invariance of the two point function $\o (q(t) \,q(s))$ and the
CCR. The Fourier transform of the two point function $$W(t)\eqq
\o(q \,\alpha_t(q))= c + i \,t/2 $$ is given by $$\tilde{W}(\o)
=\sqrt{2\pi}\,(c \,\d(\o) - \d'(\o)/2)$$ has support at the origin
and it is not a measure.
\vspace{2mm} If $\o$ is Gaussian, the matrix elements of the
operator $ U(t) $ which implements the time translations on $ D
\equiv \pi_\o (\Ha) \psz$ can be explicitly computed and the
(weak) derivative $ d U(t)/ d t $ exists and defines a hermitean
operator $ H $ on $ D $. Without loss of generality, we can
redefine $ H $ so that $ H \psz = 0$. Now, the CCR imply
\be
H = p^2 /2 + h \ , \ \ \ \ h \in \pioHp \ee and, if $\pioHp = \{
\lambda 1 \} \ , \lambda \in \complessi$, the equations $(\psz , H
\psz) = 0$, $\o(p^2)=0 $ imply $h = 0$. This leads to a
contradiction $$ 0 = \o (q^2 H) = \o (q p)^2 = -1/4.$$
\vspace{2mm} The existence of a non-trivial commutant is
reminiscent of the structure which characterizes the theory of KMS
states and more generally the Tomita-Takesaki theory. Actually,
one can show
\begin{Proposition} The Gaussian states $\o$ of Proposition 2.1 are
faithful, i.e.,
\be
\o (B A) = 0, \ \ \ \forall B \in \Ha \ \ \ \Rightarrow \ \ A =
0. \ee
\end{Proposition}
\Pf \,\,\,\, If, for some polynomial $P(q,p)$, $\o (B P(q,p)) =
0$, $\forall B \in \Ha$, then, by time translation invariance of
$\o$, the same holds for $\alpha_t (P) = P(q+p t, p)$, for all
$t$. This implies that the coefficient of the highest power of $t$
vanishes, i.e., $P(p,p) \psz =0$. Then, $\o(q^k P(p,p)) = 0$,
$\forall k$, which implies $P=0$.
\vspace{2mm}By exploiting the property of $\o$ being faithful, one
has
\begin{Proposition} If $\o$ is a state on the Heisenberg algebra
$\Ha$, with the properties i-iv) of the above Proposition with
$c=0$, then $ \pioHp $ is the (concrete) algebra $\Hap$ generated
by pseudo-canonical variables $q', p'$ satisfying
\be
[\, q' ,\, p' \,] = -i \ee and the GNS representation defined by
$\o$ is the direct product of two Fock representations for the
pairs of canonical variables
\be
Q_\pm \equiv (q \pm p')/ \sqrt 2 \ , \ \ \
P_\pm \equiv (\pm p + q')/ \sqrt 2
\ee
satisfying
\be
[Q_\pm , P_\pm] = \pm i \ , \ \ \ \ [Q_\mp , P_\pm] = 0 \ , \ee
namely, by introducing the destruction operators operators
\be
A \equiv (Q_{+} + i P_{+}) / \sqrt 2 \ , \ \ \ \ B \equiv (Q_{-} +
i P_{-}) / \sqrt 2 \ , \ee which satisfy
\be
[A, A^*] = 1 \ , \ \ \ [B, B^*] = -1 \ , \ \ \ [A,B]= [A, B^*] = 0 \ ,
\ee
one has that the vector $\psz$ which represents the state $\o$
satisfies the Fock conditions
\be
A \psz = 0 \ , \ \ \ \ B \psz = 0 \ .
\ee
\end{Proposition}
\Pf\,\,\,\, Since $\o$ is faithful, in ${\cal H}_\o$ one can
introduce the analogue of the Tomita-Takesaki antilinear operator
$S$
\be
S A \psz \equiv A^* \psz \ \ \ \ \ \forall A \in \pioH. \ee As in
the standard case, quite generally one has
\be
S \psz = \psz\ , \ \ \ S^2 = 1 \ , \ \ \ S A S \in \pioHp \ , \ \ \
\forall A\in \pioH \ .
\ee
Similarly, the adjoint $S^*$ satisfies
\be
S^* \psz = \psz\ , \ \ \ S^{* \, 2} = 1 \ ,
\ \ \ S^* A' S^* \in \pioH \ , \ \ \ \forall A' \in \pioHp \ . \ee
In fact, e.g., one has $$ (A \psz, S^* \psz) = \overline{ ( S A
\psz , \psz)} = (A \psz, \psz). $$ By using the properties i) -
iv) of the state $\o$, one easily checks that
\be
(S q^j S)^* = (-1)^j S q^j S \ ; \ \ \ \ \
(S p^j S)^* = (-1)^j S p^j S \ .
\ee
We may then introduce the following hermitean operators
\be
q' \equiv i S q S \ , \ \ \ \ \ p' \equiv - i S p S \ . \ee
They satisfy the following pseudo-canonical commutation relations
$$ [q',p'] = S [q,p] S = -i . $$ Furthermore, by the definition of
the antilinear operator S, we have $$ (q + i q') \psz = (q - S q
S) \psz = 0 \ , \ \ \ \ (p - i p') \psz = (p - S p S) \psz = 0. $$
The above equations imply the Fock conditions for the destruction
operators $ A, B $ defined above. They also imply that the
operator
\be
H \equiv (p^2 + {p'}^2)/2 = (p + i p') (p - i p')/2 \ee
annihilates $\psz$ and can be taken as the Hamiltonian, since it
has the right commutation relations with $q, p$. By the same kind
of argument, $\psz$ is annihilated also by $ q^2 + (q')^ 2 $.
Finally, if an operator $ C : D \mapsto D $ commutes with $\pioH$,
then $C$ is identified by its action on $\psz$. Now, $\forall \psi
\in D$, there exists a polynomial $P(q',p')$ such that $ P(q',p')
\psz = \psi$, since $$
D = S D = S \pioH \psz = S \pioH S \psz = \Hap \psz$$
and therefore $\pioHp = \Hap$.
\vspace{3mm}The above characterized representation of $\Ha$ in an
indefinite inner product space can be given a Krein structure.
\begin{Proposition} The indefinite inner product
representation space $D = \pioH) \psz$ can be given a Krein
structure by introducing a metric operator $\eta$ with $\eta \psz
= \psz$ and
\be
\eta A \eta = A \ , \ \ \ \ \ \eta B \eta = - B \ ,
\ee
equivalently,
\be
\eta q \eta = p' \ , \ \ \ \ \ \eta p \eta = q' \ .
\ee
\end{Proposition}
\Pf \,\,\,\, It follows easily from the positivity of the two
point function $ (\,B \psz , \eta\, B \psz)$.
\vspace{2mm} The general case $ c \neq 0$ can be reduced to the
one discussed above by putting
\be
q = c_+ Q_+ + c_- Q_-,\,\,\,\,\,
p = d_+ P_+ - d_- P_-, \ \ \ \ c_\pm , \ d_\pm \in \reali
\ee
with
\be
c_+ d_+ + c_- d_- = 1 \ \ \ \ c_+^2 - c_-^2 = 2 c \ .
\ee A possible choice is
\be
d_+ = d_- = (c_+ + c_-)/2 = 1/\sqrt{2} \ , \ \ \ \ c_+ - c_- =
2 c \ee
The so obtained structure provides an example of the
Tomita-Takesaki theory in indefinite inner product spaces;
surprisingly, the commutant is described by pseudo-canonical
variables, as it happens for the time component of the
electromagnetic potential in the Gupta-Bleuler formulation of the
free electromagnetic field. The above representation of $ B, B^*$
is the same as the anti-Fock representation of the CCR discussed
in ~\cite{MSV,MS}
Because of the indefinite inner product, the analog of the modular
operator $ \triangle \equiv S^* S $ is not positive and it is
fact given by $$ S^* S p^k q^j \psz = S^* q^j p^k S^* \psz =
(-1)^{j+k} p^k q^j \psz. $$ One may then introduce a square root
of $\triangle $ $$
\triangle^{1/2} p^k q^j \psz \equiv i^k (-i)^j p^k q^j \psz.
$$ It is not difficult to check that the so defined $
\triangle^{1/2} $ is hermitean $$
( \triangle^{1/2} p^l q^m \psz , \,p^k q^j \psz ) =
(-i)^l i^m \, \o ( q^m p^l p^k q^j ) =$$
$$i^k (-i)^j \, \o ( q^m p^l p^k q^j ) =
(\, p^l q^m \psz , \, \triangle^{1/2} p^k q^j \psz ) \ ,
$$ since the above correlation functions vanish unless $ k + l =
m + j$.
In analogy with the standard case, one may then introduce the
analog of the modular conjugation $ J $ defined by
\be
J \equiv S \triangle^{-1/2}
\ee
Since also $ \triangle^{-1/2} $ (defined by changing
$i$ into $ -i$ in the definition of $ \triangle^{1/2} $)
is hermitean, one has
\be
J J^* = J^* J = 1 \ ,
\ee
i.e. $ J $ is an anti-unitary operator (with respect to the indefinite
inner product of $ D $). As in the standard case one has
$ J \triangle^{1/2} J = \triangle^{-1/2} $.
\vspace{2mm} The above representation of the Heisenberg algebra in
a Krein space allows for the construction of the Weyl algebra
(equivalently of the Heisenberg group) as the algebra generated by
the (pseudo) unitary operators $ U(\alpha) \equiv \exp i \alpha
q$, $ V(\beta) \equiv \exp i \beta p $; therefore, in this way
one gets a ground state (regular) representation of the Weyl
algebra for the free particle.
\begin{Proposition} The formal series corresponding to the
exponentials $ U(\alpha) $, $ V(\beta) $, $\alpha, \beta \in
\reali $ converge strongly in the Krein topology defined in
Proposition 2.4 and define (pseudo) unitary operators in the
corresponding Krein space. The so obtained Weyl operators $
U(\alpha)$, $ V(\beta) $ generate a Weyl algebra $\Wa$ and in this
way one gets a regular representation in a Krein space, defined by
a state $\o$ invariant under the free time evolution and
characterized by the following expectations
\be
\o ( e^{i \alpha q} \, e^{i \beta p} ) = e^{-i \alpha \beta /2 }
\ \ \ \ \ \forall \alpha, \beta \in \reali
\ee
\end{Proposition}
\Pf \,\,\,\, By expressing $q$ and $p$ in terms of the
destruction and creation operators $ A, B $ defined in
Proposition 2.3, one gets (by Fock space methods) the analog of
the standard Fock estimate $$
|| q^n \psz ||_K \leq 2^n \sqrt {n!},
$$ where $ || \ ||_K$ denotes the Krein-Hilbert norm, and a
similar estimate for $ p^n \psz $. As in the standard case, this
yields the strong convergence of the series and the existence of
the (pseudo) unitary operators $ U(\alpha) $, $ V(\beta)$. The
above expectations follow from the correlation functions
\be
\o(q^n p^m) = \delta_{n,m} (i/2)^n \, n!
\ee
It is worthwhile to mention that the operators $ U(\alpha)$, $
V(\beta)$ do not commute with the metric operator $\eta$, and
therefore they are not unitary with respect to the positive
(Hilbert) scalar product $ ( \cdot , \eta \cdot ) $ defined by
$\eta$. Since $\o$ is time translation invariant, the time
evolution automorphisms $\alpha_t$, $t \in \reali$ are
implemented by a one parameter group $\Ut $ of (pseudo) unitary
operators, and information about their spectral properties is
given by the Fourier transforms of the correlation functions $\o
(A \,\alpha_t(B))$.
\begin{Proposition} In the representation of the Weyl algebra in
the Krein space discussed above, the Fourier transform of the
correlation functions $ \o (A \,\alpha_t(B))$ are tempered
distributions with the following properties:
\ni
i) for $ A, B \in \Ha $ they have support at the origin,
\ni
i) for $ A, B \in \Wa $ their support is the entire real line.
\end{Proposition}
\Pf \,\,\, In fact, by the definition of the time evolution and
the invariance of $\o$ one has $$
\o (p^k q^j \, \alpha_t( p^l q^m)) = P_m (t),
$$ with $ P_m(t) $ a polynomial of degree $m$. On the other hand
$$
\o (U(\alpha) \,\alpha_t ( U(\beta ))) = e^{-i \alpha \beta t/2}.
$$ In the latter case, the support of the Fourier transform is
contained in the positive real axis for the diagonal expectations,
$\alpha = \beta$, but not in general.
%%%%%%%%%%%%%%%%%%%%33333333333333333333333333333%%%%%%%%%%%%%%%%%
\section{Ground state positive representations of the Weyl algebra}
The lack of positivity of the time translationally invariant state
can be avoided by looking for representations of the Weyl algebra
which do not yield representations of the Heisenberg algebra. The
analog of eq.(2.2) is now \be{\at(W(\a,\b))= W(\a,\b +\a t).}\ee
In this case one can also achieve the positivity of the energy
spectrum, in the sense that the Fourier transforms of the
correlation functions $ \O (A \alpha_t(B))$ are tempered
distributions with support contained in $\reali^+$
\begin{Proposition} A time translationally invariant positive
state $\O$ on the Weyl algebra $\Wa$ satisfying the positivity of
the energy spectrum is identified by having the following
expectations of $ W(\alpha, \beta) \equiv U(\alpha) V(\beta) \exp
(i \alpha \beta /2)$
\be
\O ( W(\alpha, \beta) ) = 0, \ \ \ \ if \,\,\,\, \alpha \neq 0 \ ; \ \ \ \
\O ( W(0 , \beta) ) = 1 \ .
\ee
\end{Proposition}
\def \at {\alpha_t}
\Pf\,\,\,\,\, Time translation invariance implies that the above
expectation is independent of $\beta$ if $\alpha \neq 0$. On the
other hand, \be{\Omega(W(\a,0)\,\at(W(\g,0)) = \Omega(W(\a + \g,\g
t))\,e^{-i \a \g t/2},}\ee so that, for $\a = \g$, $$\Omega(W(\a +
\g, \g t)) = \Omega(W(\a+\g, 0))$$ and positivity of the energy
requires that it vanishes. For $\a = - \g$, the Fourier transform
of eq.(3.3) can be supported in $\Rbf^+, \, \forall \a \in \Rbf$,
only if the Fourier transform of $\Omega(W(0, \a t))$ is supported
at the origin. Positivity of the state $\Omega$ then implies that
$\Omega(W(0, \a t)) = 1$. Thus, eqs.(3.2) hold. Conversely,
eqs.(3.2) define a positive state(as limit of ground states of
harmonic oscillators ~\cite{AMS1}). Moreover, eqs.(3.2) imply
$$\Omega(W(\a, \b)\, \at(W(\g, \d))) = \d_{-\a, \g}\,e^{- i \a(\d
+ \b)/2}\,e^{i \a^2 t/2}$$ and therefore positivity of the energy
follows.
\vspace{2mm} In conclusion, by the GNS construction, one has a
nonregular representation of the Weyl algebra in a Hilbert space
$\H$, with cyclic vector $\Psi_{\Omega}$. Since eqs.(3.2) imply
\be{V(\b) \,\Psi_{\Omega} = \Psi_{\Omega},}\ee $\Psi_\Omega$ is
also cyclic with respect to the algebra generated by the
$U(\a)$'s. The occurrence of non regular representations should
not be regarded as too bizarre, since they can be related to
reasonable physical descriptions. In fact, if we consider a free
particle in a bounded volume $V$, it is reasonable to consider the
algebra of canonical variables $\AV$ generated by $\exp (i \beta
p)$, $\b \in \reali$ and by the (continuous) functions of $q$,
$f_V(q)$, with support contained in $V$.
There is a natural embedding of $\AV$ into the Weyl algebra $\Wa$,
i.e., if $$ f_V (x) = \sum c_n \, e^{i k_n x} \ , \ \ \ \ \ x \in
V , $$ then its periodic extension $$ f (q) = \sum c_n \, e^{i
k_n q} \ , \ \ \ \ \ q \in \reali, $$ defines a corresponding
element of $\Wa$. Then, if $\psV (x) = const$ denotes the ground
state in the volume $V$ (with periodic or Neumann boundary
conditions for the Hamiltonian), one has $$ (\psV , f_V (x) \,
e^{i\,\b\,p}\, \psV) = \O(f(q) \,e^{i\,\b\,p}), $$ where $\O$ is
the nonregular state characterized in the above Proposition. The
nonregular representation provides therefore a volume independent
mathematical description of the above concrete situation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%4444444444444444444444444%%%%%%%%%%%%%%%%
\section{Euclidean formulation and stochastic processes. Positive
case}In order to discuss the stochastic processes associated to
the quantum free particle, it is convenient to derive the
corresponding euclidean formulation.
For the representation characterized in the previous section, we
have for the two point (Wightman) function for the Weyl operators
$U(\a)$ \be{ \O( U(-\a)\,e^{i H t}\,U(\a')) = \d_{\a,-\a'}\, e^{i
\a^2\,t/2}}\ee with Fourier transform $$\tilde{W}(\o) = \sqrt{2
\pi} \,\d(\o - \a^2/2)\,\d_{\a, - \a'}.$$ The n-point
(Wightman)functions of the Weyl operators $U(\a)$ are obtained by
induction from $$\at(U(\a)\,V(\b))= U(\a)\,V(\b + \a t)\,e^{i \a^2
t/2}$$ using eq.(3.4), and are given by
\def \a {\alpha}
$$\Omega(\a_{t_1}(U(\alpha_1))...\alpha_{t_n}(U(\alpha_n))) =
\Omega(\alpha_{t_1}(U(\alpha_1)\,\alpha_{t_2-t_1}(U(\alpha_2)...
\alpha_{t_n-t_{n-1}}(U(\alpha_n))...)$$ \be{= \delta_{\sum
\alpha_i,\,0}\,e^{i \sum_{i=2}^n (\t_i-\t_{i-1})
(\sum_{k=i}^n\alpha_k)^2/2}.}\ee The corresponding n-point
Schwinger functions are immediately obtained by analytic
continuation to ordered imaginary times $\t_1\leq\t_2 ...\leq\t_n$
: \be{\S(\a_1 \t_1, ...\a_n \t_n)=\delta_{\sum \alpha_i,\,0}\,e^{-
\sum_{i=2}^n (t_i-t_{i-1}) (\sum_{k=i}^n\alpha_k)^2/2}.}\ee and
extended by symmetry to all euclidean times $\t_1, \t_2 ...\t_n
\in \Rbf$.
>From the existence and positivity of the Hamiltonian in the
representation defined by eq.(3.2), it follows that the above
Schwinger functions can also be written as \be{\S(\a_1 \t_1,
...\a_n \t_n)= (\Psi_\Omega, U(\a_1) e^{-(\t_2-\t_1)H}U(\a_2)
...e^{-(\t_n-\t_{n-1})H} U(\a_n) \Psi_\Omega)}\ee By standard
arguments, one can introduce the corresponding Borchers algebra
and euclidean fields $U(\a,\t)$ so that the Schwinger functions
define a linear functional $E$ on the euclidean fields $$\S(\a_1
\t_1, ...\a_n \t_n)= E(U(\a_1,\t_1)...U(\a_n, \t_n)).$$
Equation (4.3) implies the Osterwalder-Schrader (OS) positivity of
the Schwinger functions, with the OS reflection operator $\theta$
defined by $$\theta U(\a, \t) = U(\a, -\t),$$ i.e. one has
$$E(\overline{\theta B}\,B) \geq 0,$$ $\forall B$ belonging to the
algebra generated by $U(\a, \t),\, \a\in \Rbf, \,\t \geq 0$, where
$\overline{U(\a,\t)} \eqq U(-\a,\t)$.
It is a non trivial fact that also Nelson positivity holds. As a
matter of fact, the above Schwinger functions can be expressed as
expectations of fields $U(\a,\t) \eqq e^{i \a x(\t)}$ with
functional measure $$d\mu(x(\t)) = dW_{0,x}(x(\t)) \,d\nu(x),$$
where $dW_{s,y}$ denotes the Wiener measure for trajectories
starting at the point $y$ at time $t=s$ and $d\nu$ denotes the
ergodic mean; in fact $d \nu$ defines a measure on the Gelfand
spectrum $ \Sigma$ of the Bohr algebra generated by $ e^{i \a x}$
~\cite{LMS1} and $$\int d \nu(x) \,d W_{0,x}(x(\t)) \,e^{i\a_1
x(\t_1)}...e^{i\a_n x(\t_n)}$$ $$= \int d \nu(x) \,e^{i \sum_i
\a_i x} \int d W_{0,x}(x(\t)) \,e^{i \sum_i \a_i (x(\t_i) - x)}$$
$$= \d_{ \sum_i \a_i,0}\,\int d W_{0,0}(y(\t)) \,e^{i \sum_i \a_i
y(\t_i)} = \S(\a_1 \t_1, ...\a_n \t_n), \,\,\,\,y(\t_i) \eqq
x(\t_i) - x.$$
The measure $d \mu$ is invariant under time translations, $$d
W_{0,y}(x(\t)) d \nu(y) = d W_{s,y}(x(\t)) d \nu(y)$$ because $d
\nu(x)$ is invariant under translations $x \ra x + a$ and
therefore stationary for the Brownian motion.
The measure $d \mu$ is invariant under translations $x(\t) \ra
x(\t) + a $; they can be given the meaning of gauge trasformations
and can be regarded as the analog of the gauge group of
traslations of step $2 \pi$ for a particle on a circle
~\cite{LMS1}. In conclusion, the euclidean correlation functions
can be obtained as the stochastic process with expectations
defined by $d\mu$.
Since the ground state $\Psi_\O$ is cyclic with respect to the
euclidean algebra at time zero in the space $\H$, obtained by the
OS recostruction theorem, such a space can be identified with
$L^2(\Sigma, d\nu)$.
One can explicitly check that the Markov property holds; if $\t_1
\leq 0 \leq \t_2$ one has $$\int d \mu\, e^{i\a x(\t_1)} \,e^{i\b
x(\t_2)}=\int d \nu(x) e^{i (\a+\b) x} \int d W_{0,x}(x(\t))
e^{i\a (x(\t_1)-x)} e^{i\b (x(\t_2)-x)}$$ $$=\int d \nu(x)
e^{i(\a+\b)x)} d W_{0,0}^-(y(\t))\, e^{i \a y(\t_1)}\, d
W_{0,0}^+(y(\t))\, e^{i \b y(\t_2)}.$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Euclidean formulation and stochastic processes. Indefinite
case}\vspace{1mm} {\bf a. Schwinger functions and
Osterwalder-Schrader reconstruction} \vspace{1mm}\newline We start
by deriving the euclidean formulation by analytic continuation.
The Laplace transforms of the two point function $W(t)$, derived
in Sect.2, gives an analytic function $\W(z_1, z_2) = W(\zeta),
\,\zeta \eqq z_2 - z_1 = \xi + i \t, \,\xi \in \Rbf, \,\t >0 $,
and its restrictions at the euclidean points $\xi=0, \t >0$ define
the Schwinger two point function $$\S(\t_1, \t_2) = S(\t) = W(i
\t)= c - \t/2, \,\,\,\t \eqq \t_2 - \t_1 \,>0.$$ The extension to
$\t \leq 0$ is given by the symmetry condition $S(-\t) =S(\t)$, so
that in conclusion $$\S(\t_1, \t_2) = S(\t) = c - |\t|/2.$$ The
n-point Schwinger functions factorize as usual for quasi free
states and define a functional $\Phi_S$ on the euclidean
Borchers'algebra, fully determined by the two point function
$$\Phi_S(\overline{f} \times g ) \eqq < f, \,g > \eqq \int
d\t_1\,d\t_2\, \overline{f}(\t_1)\,\S(\t_1, \t_2)\,g(\t_2),$$ $f,
g \in \S(\Rbf) \eqq \S$.
Euclidean fields $ x(f), \,f \in \S, \,f $ real, can be introduced
as usual, with two point function $$< x(f)\, x(g) > = < f, \,g
>$$ and n-point functions given by the "gaussian"functional ($f$
real) \be{ < e^{i x (f)} > \equiv e^{ - /2} = e^{ \int
f(\t) f(s) \, |\t-s| \; d\t \; d s/4}. }\ee The
Oster\-wal\-der-Schra\-der positivity is not satisfied , since
$\S(-\t_1, \t_2) = c - |\t_1 + \t_2|/2$ is not a positive kernel
and therefore $$\Phi_S(\overline{\theta f} \times f), \,\,\,\,
(\theta f)(\t) \eqq f(- \t),$$ is not positive.
However, the Osterwalder-Schrader (O-S) reconstruction of the real
time indefinite vector space, in terms of the Schwinger functions
can be done by the standard extension of the recostruction without
positivity ~\cite{JS}. The O-S scalar product is defined by
factorization starting from $$ < f, \, g
>_{OS} \equiv < \theta f , g > = i/2
\; ( \overline {\tilde f (0)} \, \tilde g' (0) - \overline
{\tilde f' (0)} \, \tilde g (0) ) + c \overline {\tilde f (0)} \,
\tilde g (0) ) $$ for $f,g \in \S (\reali^+) \eqq \S^+ $.
The null space of the O-S scalar product, $$ N_{OS} \equiv \{ f
\in \S (\reali^+): \, _{OS} \, = 0 \ \ \forall g \in \S
(\reali^+) \} \ \ , $$ has codimension two, so that the space $K
\eqq \S (\reali^+) / N_{OS}$ is two--dimensional. \vspace{1mm}
\newline {\bf b. Nelson space} \newline The two point Schwinger
function does not satisfy positivity, i.e. $\S(\t_1, \t_2)$ is not
a positive kernel; in more detail one has
\begin{Proposition} The vector space $\S(\Rbf)$ with inner product
\be{< f, \,g >= \int d\t_1\,d\t_2\, \overline{f}(\t_1)\,\S(\t_1,
\t_2)\,g(\t_2)} \ee is an indefinite inner product space with one
negative dimension (pre-Pon\-tri\-a\-gin space): $< \,, \, >$ is
positive on $\S_0(\Rbf) \eqq \{f \in \S(\Rbf), \,\int f d\t= 0
\}$, and there is a function $\chi$ with $\int \chi d\t \neq 0$
such that $< \chi, \, \chi > < 0$.
\end{Proposition}
\Pf \,\,\,In fact,for $f,g \in \S (\reali^+) $, by ($\theta f(\t)
\eqq f(-\t)$) $$ < f, \, g
>_{OS} \equiv < \theta f , g > = i/2
\; ( \overline {\tilde f (0)} \, \tilde g' (0) - \overline
{\tilde f' (0)} \, \tilde g (0) ) + c \overline {\tilde f (0)} \,
\tilde g (0) ). $$ the Fourier transform of $S$ is
$$\tilde{S}(\o) = 2 \pi c\,\d(\o) - (d/d\o) P(1/\o),$$ where $P$
denotes the principal value. Then, if $\tilde{f}(0)=0$ $$\int
d\o\, \overline{\tilde{f}}(\o)\,\tilde{S}(\o) \tilde{f}(\o) > 0.$$
It is easy to see that $$\chi(\t) = [e^{-(\t+T)^2/L} +
e^{-(\t-T)^2/L}](2 \sqrt{\pi}\,L)^{-1}$$ has the properties of the
Proposition, for $L$ small enough and $T$ large enough. {\bf
Remark}. The above indefinite inner product space has a functional
structure very similar to the indefinite inner product space
defined by the (real time) two point function of the massless
scalar field in two space time dimensions ~\cite{MPS}. For
simplicity, in the following we consider the case $c=0$.
\begin{Proposition}The above indefinite inner product space can be
given a Krein structure by introducing e.g. the following Krein
inner product \be{[ f, \, g ] = < f, \,g > + (< f, \, \chi> -
\overline{\tilde{f}}(0)) (<\chi,\, g> - {\tilde{g}}(0)),}\ee where
$\chi$ has been chosen such that $\tilde{\chi}(0)=1,
\,\,<\chi,\,\chi>=0$
\end{Proposition}
\Pf \,\,\,\,The proof is the same as for the massless scalar
field ~\cite{MPS}.
\vspace{2mm} Even if the Krein topology that can be associated to
the above inner product space is unique, as in the case of the
massless field in two dimensions ~\cite{MPS}, there are many
possible choices for the positive scalar product which turns
$\S(\Rbf)$ into a (pre-)Krein space. In the following we shall
exploit this freedom in order to obtain a scalar product with a
more direct interpretation, in particular in terms stochastic
processes.
For this purpose, we first remark that the sequences $f_n$, with
$f_n(\t)$ smooth approximations of the Dirac delta function
$\d(\t)$, converge weakly in the topology of the inner product
(5.1); they actually define strongly compatible sequences in the
sense of ~\cite{MS}, i.e. $\lim_{n, \,m} $ exists and
$\lim_n < f_n, \,g >$ exists $\forall g \in \S(\Rbf)$. Therefore,
one can consider the weak extension of $\S(\Rbf)$ defined by them.
Such an extension is contained in the above Krein space as a
consequence of its uniqueness.
In particular such extension contains Dirac delta functions $\d_\s
\eqq \d(\t - \s)$ and one has \be{ < \d_\s, \,g > = -
{\scriptstyle{\frac{1}{2}}} \int g(s) |s - \s| d s, \,\,\,\,\,<
\d_\s, \d_\rho > = - {\scriptstyle{\frac{1}{2}}} \,|\s -
\rho|.}\ee
A further weak extension is defined by the set of strongly
compatible sequences $f_n(\t) = f(\t - n)/n, \, f \in \S(\Rbf),
\,\,\,\tilde{f}(0) = 1$, since one has $$\lim_{n \ra \pm \infty} <
f_n, \,g > = \mp \tilde{g}(0)/ 2,$$ $$\lim_{(n,m) \ra \infty}\,<
f_n, \, f_m > =0, \,\,\,\,\,\lim_{n \ra \infty} < f_n, \, \d_\s >
= - 1/2.$$ Thus, by introducing \be{ w \eqq weak-\lim_{n \ra
\infty}\, f_n(\t),}\ee we have \be{< w, \,g > = - \tilde{g}(0)/2,
\,\,\,< w, \,w > = 0, \,\,\,< \d_0, \, w > = - 1/2.}\ee
As a consequence of eq.(5.5), a convenient class of Krein scalar
products can be parametrized in the following way \be{ [ f, \,g
]_\a \eqq < f, \,g
> + 2 < f, \,\a \d_0 + \a^{-1} w
> < \a \d_0 + \a^{-1} w, \,g >, }\ee $\alpha$ playing the r\^{o}le of
a scale parameter. Thus, even if the correlation functions are
scale invariant, the Krein product is not. Summarizing, we have
\begin{Proposition} The Krein product (5.7) defines a Krein
topology equivalent to that defined by eq.(5.2). The resulting
Krein space $K_\a$ has the following decomposition \be{K_\a =
\overline{(\frac{d}{d \t})^2 \S^+ } \oplus \overline{(\frac{d}{d
\t})^2 \S^-} \oplus \{a \d_0 + b w,\, \,a, b \in \Cbf \}, }\ee
where $\S^\pm \eqq \S(\Rbf^\pm)$; the decomposition is both
orthogonal with respect to the indefinite inner product $<.\, ,
.>$ as well as with respect to the positive product $[. \,, .]$.
The metric operator $\eta_\a$ satisfies $ \eta_\a^2 = \id$ and has
a unique negative eigenvector $\a\,\d_0 + \a^{-1}\,w$.
In particular, eqs.(5.7), (5.8) imply that $\forall f, g \in
(d/d\t)^2 \S^\pm$, i.e. $f = F'', \, g = G'', \, F, G \in \S^\pm$
one has $$ [f, \,g] = < f, \,g > = (F', \,G')_{L^2}.$$
\end{Proposition}
As usual one can construct the Fock space $\Gamma(K)$ over $K$.
Euclidean fields $x(f)$, $f$ real, act on $\Gamma(K)$ and by
standard arguments they can be extended from $\S$ to $K$; the
existence of $\d_0$ and $w$ are equivalent to the existence of the
fields $x(\t)$ at fixed times, $x \eqq x(0)$, and of the
(positive time) ergodic mean of the velocity, namely $v \eqq
\lim_{\t \ra +\infty} x(\t)/ \t$.
Eq. (5.7) defines the following positive bilinear functional on
the fields at fixed time, \be{ [ x(\t) \, x(s) ]_\alpha = -(1/2)
\, (|\t-s| - |\t| - |s| - \alpha^{-2} - \alpha^2 \,|\t| \,
|s|).}\ee The OS scalar product extends by continuity to the
Krein closure of $\S^+$ and $K_{OS}$ can be identified with the
subspace of the Krein closure $ \overline \S (\reali^+) $
generated by $x \eqq x(0) = x(\d_0)$ and $v \eqq x(w)$. In fact,
by eqs. (5.4), (5.6), $$ < f, \,g >_{OS} \, = \,
< \tilde f (0) \, x - i \tilde f' (0) \, v \ , \,
{\tilde g (0)} \, x - i \tilde g' (0) \, v > \ \ .$$
By comparison with eqs. (2.3), one has the correspondence $ x \sim
q \Psi_0, \,\,\, v \sim i p \Psi_0 $. Clearly, the one particle
space determines the whole (indefinite) OS reconstruction space,
which coincides with the real time space of Prop. 2.1, with the
correspondence $$ P(x) \, Q(-i v) \sim P(q) \, Q(p) \, \Psi_0. $$
\vspace{1mm}\newline {\bf c. Markov property}
\vspace{1mm}\newline The structure displayed by the model is
actually an example of a general structure which substantially
generalizes Nelson strategy to cases in which positivity fails and
in particular allows for a generalization of the formulation of
the Markov property in terms of projections.
For simplicity, we discuss the problem at the level of the two
point function, which is assumed to define a non degenerate inner
product $< . \,, .
>$ on $\S$, satisfying the following properties:
\vspace{1mm}\newline i) $< \theta f, \, \theta g > = < f, \,g
>, \,\,\,\, (\theta f)(\t) \eqq f(-\t),$
\vspace{1mm}\newline ii) there exists an operator $D$ on $\S$,
such that, $\forall f, g \in \S$ $$< f, \,D g > = ( f , g)_{L^2},
\,\,\,D\S^\pm \subseteq \S^\pm,$$ iii) there is a Krein structure
$[ .\,, . ]$, namely a Hilbert majorant of $< .\, , . >$, such
that the corresponding closure of $\S$ decomposes as \be{
\overline{\S} = \overline{\S^-} <+> \overline{D\S^+},}\ee with
$<+>$ denoting a $< , >$ orthogonal sum.
\def \bSp {\overline{\S^+}}
\def \bSm {\overline{\S^-}}
\def \bSpm {\overline{\S^\pm}}
\def \bDSpm {\overline{D\S^\pm}}
\def \bDSm {\overline{D\S^-}}
\vspace{1mm}Then, by i) one also has \be{ \overline{\S} =
\overline{\S^+} <+> \overline{D\S^-}.}\ee Furthermore, since $
\bDSpm \subseteq \bSpm$, $\forall f \in \S$, by eq.(5.10) one has
$$f = f_- + f_{0+}, \,\,\,\,f_- \in \bSm, \,\,\,\,\,f_{0+} \in
\overline{D\S^+}$$ and by eq.(5.11) $f_- = (f_-)_+ + (f_-)_{0-},
\, \,\,(f_-)_+ \in \bSp, \,\, (f_-)_{0-} \in \bDSm$. Hence, $$ f =
f_{0+} + (f_-)_{0-} + (f_-)_+, \,\,\,\, (f_-)_+ \in \bSp \cap
\bSm.$$ Moreover, if $f \in \bSp\cap\bSm$, then $f_{0+} =0 =
(f_-)_{0-}$, i.e. $f=(f_-)_+$. In conclusion, one has \be{
\overline{\S} = \overline{D\S^-} <+> \overline{D\S^+} <+> \V,
\,\,\,\,\,\,\,\,\,\V = \overline{\S^-} \cap \overline{\S^-}.}\ee
Since the inner product space $\S$ defined by the two point
function can always be taken non degenerate, also the Krein
closure $\bar{\S}$ is non degenerate so that the decompositions
(5.10-5.12) are unique; in fact, non uniqueness is possible only
if the spaces occurring in them have a non zero intersection, but
this would imply degeneracy. Thus, any vector $f \in \bar{\S}$ has
unique decompositions according to eqs.(5.10-5.12), which means
that correspondingly there are everywhere defined idempotent
operators, which are hermitean with respect to $ < ,
>$. In particular, there are $E_\pm, \,E_0$ defined by $E_\pm
\overline{\S} = \overline{\S^\pm}$, and $E_0 \bar{\S} = \V$ and
eqs.(5.10-5.12) give \be{E_+ E_- = E_- E_+ = E_0.}\ee Thus, we
get the same operator formulation of the Markov property as in the
positive case.
For quasi free states, the above construction extends in the usual
way to the n-point Schwinger functions, in terms of the sum of the
symmetric tensor products of the above indefinite spaces.
Similarly, the euclidean $x(f)$ has an extension to $f \in
\bar{\S}$.
In the example discussed in this note, $D = d^2/ d \t ^2$ with the
Krein structure given by eq.(5.7). The decomposition (5.12)
reduces to (5.8) and it is also orthogonal in the positive scalar
product. The lesson from the model is that the space $\V$ can be
larger than the standard time zero space; in the model it contains
a time translationally invariant variable, actually the ergodic
limit of the velocity. \vspace{2mm}\newline{\bf d. Functional
integral representation} \vspace{1mm}\newline In order to
represent euclidean correlation functions, eq. (5.1), in terms of
a measure we start by noting that, for $f(\t) = dF(\t)/d\t$, $F
\in \S$, the functional (5.1) coincides with the functional
defined by the Wiener measure. For functions which are not
derivatives, positivity is lost.
Nevertheless, a functional integral representation of the
functional (5.1) can be given in terms of complex variables in the
gaussian measure spaces defined by the Krein scalar product, eq.
(5.7).
A gaussian measure $d\mu_\alpha$ is in fact defined on $\S'$ by
the expectation ($f$ real) \be{\int d \mu_\a \,e^{i x(f)} = [
e^{i x(f)} ]_\alpha \equiv e^{- [ f , f ]_\alpha/2}.}\ee The
corresponding process can be explicitly analyzed as follows: let
$y(\t)$ denote the Wiener process, $x$ and $v$ independent
gaussian variables, with variances \be{ [x^2] = \alpha^{-2}/2 \ \
\ , \ \ \ \ [v^2] = \alpha^2 / 2 .}\ee It is immediately seen
that the process \be{ x + v |\t| + y(\t) ,}\ee defined as the
product measure space of $x$, $v$ and $y(\t)$ has the variance
given by eq. (5.9), and can therefore be identified with the
process defined by eq. (5.15), i.e. \be{ d\mu_\alpha (x(\t)) \ = \
d \g_{\alpha^{-1}} (x) \ \ d\g_{\alpha} (v) \ \ d W_{0,0}
(y(\t))}\ee with $ x(\t) = x + v |\t| + y(\t)$, and $ d\g_\alpha
(x) $ the gaussian measure of variance $\alpha^2/2$.
An explicit analysis of the gaussian measure defined by the Krein
scalar product (generating kernel, Markov properties etc.)
immediately follows from the representation of eq.(5.17). In
particular, we have the representation (written for simplicity for
$0 < \t_1 < \ldots < \t_n$) $$ [f_1(x(\t_1))...f_n(x(\t_n))]_\a =
\int d\mu_\alpha (x(\t)) \ f_1(x(\t_1)) \, \ldots \,
f_n(x(\t_n)) = $$
$$ = \int d\g_{\alpha^{-1}} (x) \ d\g_{\alpha} (v) \ K(y_n -
y_{n-1}, \t_n -\t_{n-1}) \, \ldots \, K(y_2 - y_1, \t_2 - \t_1) \
$$ $$ K( y_1 - x, \t_1 ) \
f_n(y_n + v \t_n)) \, \ldots \,
f_1(y_1 + v \t_1)) \, d y_1\,...d y_n, $$
with $K(x,\t)$ the kernel defining Brownian motion.
Thus, the Markov property for $d \mu_\a$ holds in the variables
$\xi(\t) \eqq x + y(\t)$, $v(\t)$, with $v(\t)$ constant in $\t$.
The measure given by eq.(5.17) clearly defines a measure on
trajectories $x(\t),\,v(\t)$, concentrated on $v(\t) = v$ and
satisfies the Markov property in the two variables $x(\t),
\,v(\t)$.
The decomposition of eq.(5.16) for $x(f), \,f $ real, corresponds
to the decomposition of $f$ according to eq.(5.12) $$f = \d_0 \int
f(\t) d \t + w \int f(\t) |\t| d \t + h.$$
Clearly, the process defined by $d \mu_\a$ corresponds to Brownian
motion, with gaussian distribution of variance $\alpha^{-2}/2$ for
the position at $\t=0$, with an additional gaussian variable, with
variance $\alpha^2/2$, describing non-zero mean velocities, with
opposite and constant values for $\t > 0$ and $\t<0$. The limit
$\alpha \to 0$ exists in the sense of positive functionals on the
Bohr algebra and coincides with the measure on its spectrum
discussed in Sect.4.
The gaussian functional defined by eq.(5.1) can be represented in
the measure space $(x(\t) , \, d\mu_\alpha)$, eqs. (5.16), (5.17),
by introducing complex variables $z(\t)$ in order to have \be{ <
e^{i x (f)} > =
\int d\mu_\alpha (x(\tau)) \;
e^{\, i \int z_\a (\t) f(\t) \, d\t} .}\ee The simplest choice
is $$z_\a(f) = x(J_\a f), \,\,\,J_\a(\a\,\d_0 + \a^{-1}\,w) = i
(\a \,\d_0 + \a^{-1}\,w),$$ $$J_\a f = f, \,\,\forall f \in (\a
\,\d_0 + \a^{-1}\,w)^{\perp}. $$ $J_\a$ satisfies $$[ J_\a,
\,\eta_\a ] = 0,\,\,\, J_\a^2 = \eta_\a,\,\,\,< J_\a f, \,J_\a g>
= < f, \, g
>.$$
Then, by using the extension of eq.(5.14) to complex $f$, since $
\overline{J_\a g} = \eta_\a \,J_\a \overline{g}$, we have ($g$
real) $$\int d \mu\,e^{i x(J_\a g)} = e^{-[ \overline{J_\a g},
\,J_\a g]/2} = e^{ - < g, \,g>/2}.$$
In conclusion, the euclidean correlation functions of the model
have the following functional integral representation $$< x(\t_1),
...x(\t_n)> = \int d \g_\a(x, v)\, d W_{0,0}(x(\t) - x - v |\t|)\,
z_\a(\t_1)...z_\a(\t_n) = $$ $$ < \int d W_{0,0}(x(\t)- x - v
|\t|) \,x(\t_1)...x(\t_n)>,$$ where $$z_\a(\t) \eqq x(\t) - x - v
|\t| + i (1 + \a^2|\t|)\,(x + \a^{-2} v)/2 + (1 - \a^{-2}|\t|)\,(x
- \a^{-2} v)/2,$$ and $ < . >$ denotes the (quantum mechanical)
expectation, eq.(5.1), on the variables $x, v$; thus, the
expectations at all times are uniquely given by the conditional
Wiener measure $$d W^{x, v}(x(\t)) \eqq d W_{0,0}(x(\t) - x - v
|\t|)$$ and by the (quantum mechanical) expectations of $x, v$.
\newpage
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\end{document}