Bargmann's Hilbert Space Research Articles

States of infinitely many oscillators

Abstract Infinite quantum systems of harmonic oscillators are investigated in the framework of the Bargmann hilbert space of analytic functions. Two models are studied in detail. The first corresponds to an electromagnetic field in a box and the second corresponds to a slightly enlarged version of the Einstein model of a crystal. Representations are constructed. The factor type of the algebra of local observables is determined in both cases. At T = 0 it is always of type I∞. At T ≠ 0 it is of type I∞ for the first example and of type III for the second example. A discussion of the total number operator and the elementary excitations in both situations is given. The time evolution of the system in the different representations is investigated.

On the application of bargmann hilbert spaces to dynamical problems: S. Schweber, Physics Department, Brandeis University, Waltham, Massachusetts

On the application of bargmann hilbert spaces to dynamical problems: S. Schweber, Physics Department, Brandeis University, Waltham, Massachusetts

Continuous-Representation Theory. IV. Structure of a Class of Function Spaces Arising from Quantum Mechanics

A rigorous development of the continuous representation of Hilbert space by bounded, continuous, multidimensional phase-space functions ψ(p, q) is presented. It is shown that these functions form a closed subspace of L2(p, q) whose elements are functions and not equivalence classes. Differential properties are investigated and it is pointed out that there are a multitude of definitions whereby ψ(p, q) possesses continuous derivatives of all orders. In one of these definitions, each ψ(p, q) is proportional to a multidimensional, entire function f(q − ip), establishing a connection between Bargmann's Hilbert space of entire functions and one example of a continuous representation. Attention is devoted to the purely functional characterization of the continuous representation by means of the reproducing kernel as a special case of Aronszajn's general theory. Properties of various operators in a continuous representation are carefully defined.