Considered is a family of irreducible Weyl representations of canonical commutation relations with infinite degrees of freedom on the abstract boson Fock space over a complex Hilbert space. Theorems on equivalence or inequivalence of the representations are reportedcdot. As a simple application, the well known inequivalence of the time-zero field and conjugate momentum of different masses in a quantum scalar field theory is rederived with space dimension dgeq 1 arbitrary. Also a generalization of representations of the time-zero field and conjugate momentum is presented. Comparison is made with a quantum scalar field on a bounded space of mathbb{R}^{d}. In the case of a bounded space with d=1, 2, 3, the representations of different masses turn out to be mutually equivalent.
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.
By means of infinite-dimensional nuclear spaces, we generalize important results on the representation of the Weyl commutation relations. For this purpose, we construct a new nuclear Lie group generalizing the groups introduced by Parthasarathy [An Introduction to Quantum Stochastic Calculus (Birkhäuser, 1992)] and Gelfand–Vilenkin [Generalized Functions (Academic Press, 1964)] (see Ref. 15). Then we give an explicit construction of Weyl representations generated from a non-Fock representation. Moreover, we characterize all these Weyl representations in quantum white noise setting.
This is an expositary article telling a short story made from the leaves of quantum probability with the following ingredients: (i) A special projective, unitary, irreducible and factorizable representation of the euclidean group of a Hilbert space known as the Weyl representation. \item The infinitesimal version of the Weyl representation includes the Heisenberg canonical commutation relations (CCR) of quantum theory. It also yields the three fundamental operator fields known as the creation, conservation and annihilation fields. (ii) The three fundamental fields, with the inclusion of time, lead to quantum stochastic integration and a calculus with an Ito's formula for products of differentials. (iii) Appropriate linear combinations of the fundamental operator processes yield all the L{é}vy processes of classical probability theory along with the bonus of Ito's formula for products of their differentials.
We consider a family of irreducible Weyl representations of canonical commutation relations with infinite degrees of freedom on the abstract boson Fock space over a complex Hilbert space. Theorems on equivalence or inequivalence of the representations are established. As a simple application of one of these theorems, the well-known inequivalence of the time-zero field and conjugate momentum for different masses in a quantum scalar field theory is rederived with space dimension [Formula: see text] arbitrary. Also a generalization of representations of the time-zero field and conjugate momentum is presented. Comparison is made with a quantum scalar field in a bounded region in [Formula: see text]. It is shown that, in the case of a bounded space region with [Formula: see text], the representations for different masses turn out to be mutually equivalent.