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

Abstract

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] = e^iφ(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 $\mathrm{\Psi \in }\mathfrak{H}\hspace{0.17em}\mathrm{by}{\mathrm{\hspace{0.17em}\psi (f,\; g)=(V[g]W[f]\Phi }}_0\mathrm{,\hspace{0.17em}\Psi )}$ . 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.