Abstract
We study particular classes of states on the Weyl algebra mathcal {W} associated with a symplectic vector space S and on the von Neumann algebras generated in representations of mathcal {W}. Applications in quantum physics require an implementation of constraint equations, e.g., due to gauge conditions, and can be based on the so-called Dirac states. The states can be characterized by nonlinear functions on S, and it turns out that those corresponding to non-trivial Dirac states are typically discontinuous. We discuss general aspects of this interplay between functions on S and states, but also develop an analysis for a particular example class of non-trivial Dirac states. In the last part, we focus on the specific situation with S = L^2(mathbb {R}^n) or test functions on mathbb {R}^n and relate properties of states on mathcal {W} with those of generalized functions on mathbb {R}^n or with harmonic analysis aspects of corresponding Borel measures on Schwartz functions and on temperate distributions.
Highlights
Introduction and review of theC∗-algebraic foundationGrundling and Hurst have developed a program to implement constraints in the C∗-algebraic framework for quantum systems
The application to the C∗-algebra of the canonical commutation relations or Weyl algebra is of particular importance regarding the interplay with the so-called regular representations [19] that guarantee the existence of corresponding unbounded field operators
With μ = ω0 as in Example 2.4 and μ its restriction to O, we obtain a counterexample, because W (0, z2)−1 with z2 = 0 is not mapped to the zero operator in the GNS representation πω0 .) If we consider instead a situation where we start with a Dirac state ω on W, we may at least conclude that the operators πω(W (y)) with y ∈ L act trivially on a certain subspace of Hω generated from O
Summary
G. Hörmann attempts at rigorous C∗-algebraic constructions of basic features of quantum electrodynamics have been started in [15], with improvements supplied by Narnhofer and Thirring in [31]. 1.1, we recall the definition of the Weyl or CCR algebra associated with a symplectic space (S, β), while Sect. 1.2 provides a review of Dirac states.
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