Topology Classes of Flat U(1) Bundles and Diffeomorphic Covariant Representations of the Heisenberg Algebra

Abstract

The general construction of self-adjoint configuration space representations of the Heisenberg algebra over an arbitrary manifold is considered. All such inequivalent representations are parametrised in terms of the topology classes of flat U(1) bundles over the configuration space manifold. In the case of Riemannian manifolds, these representations are also manifestly diffeomorphic covariant. The general discussion, illustrated by some simple examples in non relativistic quantum mechanics, is of particular relevance to systems whose configuration space is parametrised by curvilinear coordinates or is not simply connected, which thus include for instance the modular spaces of theories of non abelian gauge fields and gravity.

Revisiting the Fradkin–Vilkovisky theorem

“…Incidentally, were it not for such subtle points, the usual statement of the FV theorem would be correct, so that the BFV-PI would always be vanishing, irrespective of the choice of gauge fixing, clearly an undesirable situation since the correct quantum evolution operator could then not be reproduced. This is explicitly illustrated by the fact that using the compactification regularisation and in the limit β → 0, the BFV-PI vanishes for the choices (17) and (84), and this independently of the functions F (λ) or the parameter ρ. Indeed for these two choices, it is precisely the parameter β which controls any contribution from the gauge fixing "fermion" to the BFV-PI.…”

“…Given the choice of gauge fixing function in (17) and the expression for the associated Hamiltonian H eff in (31), it is clear that (35) factorizes into two contributions, whether the conditions π f = 0 = π i required for BRST invariance of the external states are enforced or not,…”

“…Furthermore, the representation of the Heisenberg algebra [λ,π] = ih, which is to be used on this space with the nontrivial mapping class group π 1 (S 1 ) = Z Z, is that of vanishing U(1) holonomy 2 [17]. Consequently, this sector of Hilbert space is now spanned by a discrete set ofπ-eigenstates for all integer values m,…”

“…Given the choice of gauge fixing function in (17) and the expression for the associated HamiltonianĤ eff in (31), it is clear that (58) as well as its extension for whatever values for m f and m i factorizes as…”

2T Physics, Scale Invariance and Topological Vector Fields

“…These are in some cases rather subtle symmetry transformations because their finite character is related with topological aspects of the underlying manifold. And this relation is important because the topological aspects of a manifold are related to the non trivial diffeomorphic-covariant representations of the Heisenberg algebra over the manifold [1]. We can then use the finite symmetry transformations on a certain topologically non-trivial manifold to investigate the general structure of quantum mechanics.…”