Weyl Algebra, States, Representations, Field Operators

Andrzei Postewnik

Evgenii Igorevich Zelenov

The standard C*-formalism of representations of commutations relations for systems of infinitely many degrees of freedom over a real numbers field is extended to the case of a p-adic numbers field. It is proved that any positive function on the Weyl algebra over a p-adic symplectic space is regular and restriction of the generating function of a state of the Weyl algebra on any nongenerate finite-dimensional subspace of p-adic symplectic space is continuous. Some nontrivial class of nonequivalent representations of commutations relations is described and analogies of Fock representations and representations with number operators are constructed. The criteria for the existence of free-field unitary quantization of symplectic transformations is given.

Peter Bongaarts

L. Bracci, Luigi E. Picasso

We define the Weyl algebra suitable to represent the quantization postulate for one-dimensional systems whose configuration space is semibounded. It consists of a group {V(β)=exp(−iβq),β∊R} of unitary operators and a semigroup {U(α),α⩾0} of nonunitary isometries. We show that the spectrum σ(q) is a half-line [x0,∞), with an arbitrary x0>−∞, and that the irreducible representations of the Weyl algebra with the same x0 are equivalent. We also consider the case when the semigroup of translations is substituted with a semigroup of partial isometries of index 1 (particle confined to a segment of unit length). The uniqueness of the irreducible representations of the related Weyl algebra is proved also for this case by exploiting the result for the half-line.

Alfred Rieckers

Abstract This is the second part of a pair of companion papers devoted to an analysis of Einstein’s fusion picture for light. While in the foregoing paper “An Actual Discusion of Einstein’s Early Fusion Picture for Photons and Classical Maxwell Fields” (referred to as Rieckers 2025) our conclusions ended with the historical ab initio motivation and mathematical elaboration of a multi-photon theory in Fock space, we now start from the non–separable C*-Weyl algebra gained by abstracting the Fock–represented Weyl algebra. To the rich state space of this antiliminary observable algebra we apply notions of a convex state space approach to identify sub–theories that cover classical field states (including the vacuum with classical zero–field), that are possibly decorated by photonic quantum noise. The main emphasis is laid on direct integrals over irreducible, disjoint representations of the C*-Weyl algebra, decomposing the effective photon theory into sectors. Certain limits of smeared central field operators suggest a mathematical realization of Einstein’s early photon notion consisting of an energy point surrounded by a local force field (respective wave function). Macroscopic classical Maxwell fields are gained by superposing the expectation values of products of those composite photon operators. The ontological status of the explicated photon notion is discussed. Connections to Einstein’s historic photon theory, summarized in Rieckers (2025), as well as to quantum optical applications are hinted at throughout the paper.