On the geometry of quantum indistinguishability

Abstract

An algebraic approach to the study of quantum mechanics on configuration spaces with a finite fundamental group is presented. It uses, in an essential way, the Gelfand–Naimark and Serre–Swan equivalences and thus allows one to represent geometric properties of such systems in algebraic terms. As an application, the problem of quantum indistinguishability is reformulated in the light of the proposed approach. Previous attempts aiming at a proof of the spin-statistics theorem in non-relativistic quantum mechanics are explicitly recast in the global language inherent to the presented techniques. This leads to a critical discussion of single-valuedness of wavefunctions for systems of indistinguishable particles. Potential applications of the methods presented in this paper to problems related to quantization, geometric phases and phase transitions in spin systems are proposed.

On the Geometry of the Berry-Robbins Approach to Spin-Statistics

“…With the above information, it is not difficult to obtain the following results which were derived and discussed in more detail in [6,14] (see also [7], for the general case).…”

“…In the present discussion we restrict ourselves to the two particle case, i.e. N = 2, referring the reader to [7] for the case of general N . For N = 2 the effective non-constrained configuration space is given by the sphere Q ≡ Q 2 ∼ = S 2 = { r}.…”

“…In the present discussion we restrict ourselves to the two particle case, i.e. N = 2, referring the reader to [7] for the case of general N. For N = 2 the effective non-constrained configuration space is given by the sphere Q ≡ Q 2 ∼ = S 2 = {r}. The exchange of the particles 1 and 2 corresponds to r → −r and the underlying permutation group is now given by G = Z 2 .…”

“…In [14], Paschke proposed to study the spin-statistics connection using the tools of noncommutative geometry. The idea has been pursued further [6,7], in the hope of eventually establishing a link with quantum field theory. The idea that this might be possible is based on the fact that the algebraic language of noncommutative geometry has many features in common with that of quantum field theory [13].…”

Some Aspects of Operator Algebras in Quantum Physics

“…This notes represent the written version of lectures I gave in mini-courses at Universidade de Brasília (April 3-6, 2013), Universidad Central de Venezuela (May [23][24][25][26][27]2016) and at the Villa de Leyva Summer School "Geometric, Topological and Algebraic Methods for Quantum Field Theory" (July [15][16][17][18][19][20][21][22][23][24][25][26][27]2013). They were mainly intended as an introduction to some aspects of operator algebras, emphasizing the prominent role they play in quantum physics.…”