The notion of a Gaussian as the exponential of a quadratic is rather familiar. Such functions are of considerable importance in a number of contexts, for example within quantum theory. Thus, in the Schrödinger representation of the canonical commutation relations they alone minimize uncertainty and they appear as ground states for harmonic oscillators. Also in the complex-wave representation of a free boson field they arise as transforms of the Fock vacuum under certain Bogoliubov automorphisms.
IntroductionLet
at and
at∗ be respectively annihilation and creation operators at apoint
t∈R .In the theory of white noise operators [23][24], it is known that
at is acontinuous linear operator from
Γu(Ec) , the (Fock) space of test functions over acomplexified nuclear space
Ec , into itself, and at is acontinuous linear operator from
Γu(EC)∗ , the dual space of
Γu(EC) , into itself.In paticular,
at+at∗ and
at+at∗+at∗at+I are cffied the quantum Gaussian white noise and the quantum Poisson white noise, respectively (cf.[11] [24]).These are known to be elemental generalzed qunatum stochastic processes.The main purpose of this work is to examine relationships between classical and quantum white noises from the point of the classical white noise theory of Gaussian [19][20] and Poisson types [13].Then it will be clarified that quantum Gaussian and Poisson white noises are the Fock space realzations of classical Gaussian and Poisson white noises, respectively.We shall discuss them in Section 4. As amatter of fact, one can see that such facts can be proved not only through
J -transform given by (2.4) and the holomorphy, but also through Sx-transform gllven by (3.3) depending on the exponential function
ϕξX .The function
ϕξX determines aunitary isomorphism between Boson Fock space and
L2(E∗,μx) ,
X=G,
P , where
μG and
μP are a Gaussian measure and aPoisson measure on
E∗ , respectively.As is pointed out in Remark 4.3, the choice of
ϕξX (kernel function of
SX -transform)is the essential part to represent the above different quantum noises acting on the same space
Γu(Ec)∗ .In Appendix, some connections between
[6][7] and [10]
wiU be noted.
Stationary quantum stochastic process j is introduced as a *-homomorphism embedding an involutive graded algebra
K~=⊕i=1∞Ki into a ring of (abelian) cohomologies of the one-parameter group
α consisting of *-automorphisms of certain operator algebra in a Hilbert space such that every x from
Ki is translated into an additive
i−α-cocycle j(x). It is shown that (noncommutative) multiplicative markovian cocycle defines a perturbation of the stationary quantum stochastic process in the sense of such definition. The
E0-semigroup
β~ on the von Neumann algebra
N associated with the markovian perturbation of K-flow j posseses the restriction
β~∣N0,N0⊂N, which is conjugate to the flow of Powers shifts
β associated with j. It yields for
β~ an analogue of the Wold decomposition for classical stochastic process on completely nondeterministic and deterministic parts. The examples of quantum stationary stochastic processes on the algebras of canonical commutation, anticommutation and square of white noise relations are considered. In the model situation of the space
L2(R) all markovian cocycles of the group of shifts are described up to unitary equivalence of perturbations.
We construct a new class of representations of the canonical commutation relations, which generalizes previously known classes. We perturb the infinitesimal generator of the initial Fock representation (in other words, the free quantum field) by a function of the field which is square-integrable with respect to the associated Gaussian measure. We characterize perturbations which lead to representations of the canonical commutation relations. We provide conditions entailing the irreducibility of such representations, show explicitly that our class of representations subsumes previously studied classes, and give necessary and sufficient conditions for our representations to be unitarily equivalent, or just quasi-equivalent, with Fock, coherent or quasifree representations. 1991 Mathematics Subject Classification: 81S05, 46L60, 81T05.
The theory of one-mode type Interacting Fock Space IFS allows us to construct the quantum decomposition associated with stochastic processes on ℜ with moments of any order. The problem to extend this result to processes without moments of any order is still open but the Araki–Woods–Parthasarathy–Schmidt characterization of Levy processes in terms of boson Fock spaces, canonically associated with the Levy–Khintchine functions of these processes, provides a quantum decomposition for them which is based on boson creations, annihilation and preservation operators rather than on their IFS counterparts. In order to compare the two quantum decompositions in their common domain of application i.e., the Levy processes with moments of all orders the first step is to give a precise formulation of the quantum decomposition for these processes and the analytical conditions of its validity. We show that these conditions distinguish three different notions of quantum decomposition of a Levy process on ℜ according to the existence of second or only first moments, or no moments at all. For the last class a multiplicative renormalization procedure is needed.