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Journal of Mathematical Physics

Kernel Integral Formulas for the Canonical Commutation Relations of Quantum Fields. II. Irreducible Representations

Gerhard C. Hegerfeldt

Published Jan 1, 1970• 3 citations
Publisher

Abstract

Continuing our investigation of the kernel or group integral for the canonical commutation relations introduced by Klauder and McKenna, we prove that a representation fulfilling any sort of kernel integral formula is irreducible. This has been conjectured by Klauder and McKenna. After collecting some auxiliary results, a complete classification of all representations is given for which a kernel integral formula in the form of a limit superior holds. It is shown that these are just the partial tensor-product representations and that the limit superior can be replaced by an ordinary limit over a fixed subsequence, thus allowing the transition from norms to scalar products. Then the basis-independent kernel integral in the form of a sup lim¯ is investigated. Here the supremum is taken over all bases of the test-function space. Under a not-very-strong irreducibility assumption, we show that this can be reduced to the vacuum functional and that there exists a fixed sequence of subspaces of the test-function space such that the sup lim¯ can be replaced by an ordinary limit which again allows a transition to scalar products. The results are strikingly similar to the case of cyclic field. This tempts us to conjecture that a representation fulfilling a kernel integral formula is both irreducible and cyclic with respect to the field just as in the case of finitely many degrees of freedom.

DOI
10.1063/1.1665050

Related papers

Kernel Integral Formulas for the Canonical Commutation Relations of Quantum Fields. I. Representations with Cyclic Field

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We investigate the kernel or group integral for the canonical commutation relations introduced by Klauder and McKenna and its generalizations. For the finite case the kernel integral formula has been proven by means of the Schrödinger representation. Motivated by the close similarity of the Schrödinger representation to the form of a general representation with cyclic field, we examine these representations with respect to kernel integral formulas. A general criterion is derived in which the dimensionality of the test function space does not enter, i.e., it is independent of the number of degrees of freedom. In this way the finite and infinite case can be treated on equal footing. The criterion contains as special cases the kernel integral formulas of Klauder and McKenna for finitely many degrees of freedom and for direct-(or tensor-) product representations of fields. For partial tensor-product representations we obtain a somewhat modified formula. After these applications, a considerably sharpened form of the criterion is derived in which only the vacuum expectation functional enters. Under a certain cyclicity assumption it is shown that the validity of a kernel integral for just some cyclic vector implies its validity for all vectors. It is further shown that the basis-independent integral defined by a supremum over all bases of the test function space U can be replaced by an ordinary limit over a kind of diagonal sequence through finite-dimensional subspaces of U. In the last section a representation is constructed which possesses a cyclic field but does not fulfil a kernel integral formula; this is an instructive illustration of a general theorem to be proved in II of this series of papers.

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On the Representation of the Canonical Commutation Relation of Bose Fields

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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.

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Continuous-Representation Theory. V. Construction of a Class of Scalar Boson Field Continuous Representations

John R. Klauder, James McKenna

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.

Journal of Mathematical Physics•Jan 1, 1965•37 citations

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John R. Klauder, J. McKenna, E. J. Woods

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Equivalence of Basis-Dependent and Basis-Independent Approach to Canonical Field Operators

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Journal of Mathematical Physics•Jan 1, 1971•5 citations

Partial-Tensor-Product Representations of the Canonical Commutation Relations: Classification

G. Reents

Partial-tensor-product representations of the canonical commutation relations are a generalization of direct- (or tensor-) product representations as discussed by Klauder, McKenna, and Woods. A classification is given of those representations up to unitary equivalence. The equivalence or inequivalence, respectively, of two representations is expressed in terms of the corresponding reference vectors.

Journal of Mathematical Physics•Jun 1, 1970•2 citations
Continuing our investigation of the kernel or group integral for the canonical commutation relations introduced by Klauder and McKenna, we prove that a representation fulfilling any sort of kernel integral formula is irreducible. This has been conjectured by Klauder and McKenna. After collecting some auxiliary results, a complete classification of all representations is given for which a kernel integral formula in the form of a limit superior holds. It is shown that these are just the partial tensor-product representations and that the limit superior can be replaced by an ordinary limit over a fixed subsequence, thus allowing the transition from norms to scalar products. Then the basis-independent kernel integral in the form of a sup lim¯ is investigated. Here the supremum is taken over all bases of the test-function space. Under a not-very-strong irreducibility assumption, we show that this can be reduced to the vacuum functional and that there exists a fixed sequence of subspaces of the test-function space such that the sup lim¯ can be replaced by an ordinary limit which again allows a transition to scalar products. The results are strikingly similar to the case of cyclic field. This tempts us to conjecture that a representation fulfilling a kernel integral formula is both irreducible and cyclic with respect to the field just as in the case of finitely many degrees of freedom.
We investigate the kernel or group integral for the canonical commutation relations introduced by Klauder and McKenna and its generalizations. For the finite case the kernel integral formula has been proven by means of the Schrödinger representation. Motivated by the close similarity of the Schrödinger representation to the form of a general representation with cyclic field, we examine these representations with respect to kernel integral formulas. A general criterion is derived in which the dimensionality of the test function space does not enter, i.e., it is independent of the number of degrees of freedom. In this way the finite and infinite case can be treated on equal footing. The criterion contains as special cases the kernel integral formulas of Klauder and McKenna for finitely many degrees of freedom and for direct-(or tensor-) product representations of fields. For partial tensor-product representations we obtain a somewhat modified formula. After these applications, a considerably sharpened form of the criterion is derived in which only the vacuum expectation functional enters. Under a certain cyclicity assumption it is shown that the validity of a kernel integral for just some cyclic vector implies its validity for all vectors. It is further shown that the basis-independent integral defined by a supremum over all bases of the test function space U can be replaced by an ordinary limit over a kind of diagonal sequence through finite-dimensional subspaces of U. In the last section a representation is constructed which possesses a cyclic field but does not fulfil a kernel integral formula; this is an instructive illustration of a general theorem to be proved in II of this series of papers.
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.
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.
We consider direct-product representations of the canonical commutation relations. An irreducible representation is defined on each of the incomplete direct-product spaces (IDPS) of von Neumann. We prove that two such representations are unitarily equivalent if and only if the corresponding IDPS are weakly equivalent, for which simple analytic tests exist. The matrix elements of these representations, coupled with a Friedrichs-Shapiro type of integral, fulfill group orthogonality relations. This classification into unitary equivalence classes also applies to direct-product representations of the canonical anticommutation relations.
The basis-independent approach to canonical commutation relations (CCR's), which allows arbitrary test-function spaces for smearing the field operators, is a generalization of the basis-dependent approach, in which the fields are smeared with an orthonormal system (or finite linear combinations thereof) to obtain an infinite set of qk and pk. Using recent results on continuity properties of representations of the CCR's, we show that every representation of the basis-independent type in a separable Hilbert space can be obtained by continuous extension of a suitable representation of the basis-dependent type where properties like irreducibility, cyclicity, etc., remain unaffected. In this sense, both approaches are equivalent, and the classification problem for CCR's is reduced from the simultaneous consideration of all representations for all possible test-function spaces to those for a single one (up to isomorphism).