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数理解析研究所講究録

Inequivalent Weyl Representations of Canonical Commutation Relations in an Abstract Bose Field Theory (Mathematical Aspects of Quantum Fields and Related Topics)

Asao Arai

Published Dec 1, 2016• 0 citations
OpenAlex

Abstract

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.

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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.
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.
This paper studies a class of representations (called quadratic) of the canonical commutation relations over symplectic spaces of arbitrary dimension, which naturally generalizes coherent and symplectic (i.e. quasifree) representations and which has previously been heuristically employed in the special case of finite degrees of freedom in the physics literature. An explicit characterization of canonical quadratic transformations in terms of a 'standard form' is given, and it is shown that they can be exponentiated to give representations of the Weyl algebra. Necessary and sufficient conditions are presented for the unitary equivalence of these representations with the Fock representation. Possible applications to quantum optics and quantum field theory are briefly indicated.
A large class of continuous representations of separable Hilbert spaces is constructed with the aid of representations of the canonical commutation relations (CCR) for a scalar boson field φ(f) and its canonical conjugate π(g). A representation of the CCR for a scalar boson field consists of two operator-valued functions V[g] and W[f], defined for all f, g in Schwartz's space 𝒮 of real test functions, where V[g] and W[f] are unitary operators defined on some separable Hilbert space ℌ, and which satisfy the commutation relations V[g]W[f] = e−i(f,g) W[f]V[g]. These unitary operators are related to the field and its momenta by V[g] = e−iπ(g), W[f] = eiφ(f). We explicitly construct a family of such representations with the help of von Neumann's theory of infinite direct products of Hilbert spaces, the pertinent parts of which are reviewed. A continuous representation ℌ of the Hilbert space ℭ is composed of a linear vector space of complex, bounded, continuous functionals defined on 𝒮 × 𝒮. These functionals are defined for all Ψ∈H by ψ(f, g)=(V[g]W[f]Φ0, Ψ). In this definition, Φ0 is a fixed unit vector in ℌ. The properties of the functions in ℭ depend on the choice of the representation of the CCR and on the choice of Φ0. When ℭ is constructed with the aid of an irreducible representation of the CCR, an inner product can be defined for all pairs of functionals in ℭ by an intuitively meaningful, rigorously defined integral in the sense of Friedrichs and Shapiro. With this inner product, ℭ is a complete Hilbert space congruent with ℌ. As in all continuous representations, a reproducing kernel exists and determines the functions in the continuous representation. One such space is closely related to a space of analytic functionals introduced by Segal and Bargmann. The representation of various operators as kernels and as functional derivatives is discussed. Finally, the construction of a vast number of unitary invariants for a representation of the CCR is used to establish the unitary inequivalence of uncountably many of the representations that we construct.
A presentation of the canonical commutation relation of Bose fields is given in a way which is independent of the choice of the bases of the test functions and covariant with respect to the Euclidean transformation of the coordinate system. It is shown that the representation is characterized by an integral on the conjugate space L* of the space L of the test functions and a real function on Σ⊗L* where Σ is the group of the transformations f→u-1f+φ; fεL*, φεL and u is a Euclidean transformation of L*. The conditions for the irreducibility of a representation and the unitary equivalence of the representations and the existence of unique vacuum state are given. An example of the inequivalent Euclidean covariant irreducible representations containing unique vacuum state is given.
The existence of inequivalent representations of the canonical commutation relations which describe a nonrelativistic infinite free Bose gas of uniform density is investigated, with a view to possible applications to statistical mechanics. The functional E(f, g)=(Ψ, eiφ(f) eiπ(g)Ψ) is used to describe the inequivalent representations. This functional is calculated for the free Bose gas in a box of volume V, and the limit is then taken as V → ∞. In this way we construct cyclic representations describing an infinite system of particles with a density distribution ρ(k) in momentum space. For a given ρ(k) the operator algebra generated by the φ(f), π(g) is reducible. For the ground-state representation (all particles in the zero-momentum state), the representation is a direct integral of irreducible representations (analogous to BCS theory). For finite temperatures the situation is complicated by the occurrence of representations which are not type I. The physical significance of the reducibility of the representations is discussed. It is argued that the thermal ensemble for the infinite system is a pure state at zero temperature, although there is some ambiguity as to which operators should belong to the algebra of observables. For finite temperatures, the thermal ensemble seems to be a mixture. The case of an interacting Bose gas is considered briefly.