Abstract
For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra A, with a non-trivial center Z, describes observables, the other Weyl operators playing the role of intertwiners between inequivalent representations of A. In particular, this gives rise to a gauge symmetry described by the action of Z. A distinguished case is when the center of the observables arises from the fundamental group of the manifold of the positions of the quantum system. Symmetries that do not commute with the topological invariants represented by elements of Z are then spontaneously broken in each irreducible representation of the observable algebra, compatibly with an energy gap; such a breaking exhibits a mechanism radically different from Goldstone and Higgs mechanisms. This is clearly displayed by the quantum particle on a circle, the Bloch electron and the two body problem.
Highlights
The mathematical foundations of quantum mechanics rely on the Dirac–von Neumann axioms, and the equivalence between the Heisenberg formulation in terms of canonical operators and the Schrödinger formulation in terms of wave functions is provided by the Stone–von
This result is achieved by introducing the Weyl unitary operators and the corresponding Weyl algebra, defined by algebraic relations, which encode the canonical commutations relations of Heisenberg canonical variables
The use of the Weyl algebra is usually motivated by the better behavior and mathematical control of the unitary Weyl operators with respect to the Heisenberg canonical variables, which are necessarily represented by unbounded operators
Summary
The mathematical foundations of quantum mechanics rely on the Dirac–von Neumann axioms (for a critical review, see [1,2]), and the equivalence between the Heisenberg formulation in terms of canonical operators and the Schrödinger formulation in terms of wave functions is provided by the Stone–von. The conditions that yield such a classification are satisfied by all of the non-regular representations of physical interest mentioned above; they all have the above form, each with a corresponding Borel measure on (T2 )d This represents a radical departure from the standard structure of quantum mechanics, since it requires non-separable Hilbert spaces, very discontinuous expectations of one-parameter groups of unitary operators (vanishing for all non-zero values of the parameters, so that the corresponding generators do not exist), etc. From a mathematical point of view, the non-regularity of the representation is a much better price to pay, rather than living with non-normalizable state vectors The advantages of such a quantization is that the states are described by normalizable vectors of a Hilbert space; the basic quantum mechanical rules are not violated; the observable subalgebra A is regularly represented in the standard way; the canonical variables that are not gauge invariant are non-regularly represented, only their exponentials being well defined.
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