The spin-statistics relation in nonrelativistic quantum mechanics and projective modules
Annales Mathématiques Blaise Pascal, Tome 11 (2004) no. 2, pp. 205-220.

In this work we consider non-relativistic quantum mechanics, obtained from a classical configuration space $𝒬$ of indistinguishable particles. Following an approach proposed in [8], wave functions are regarded as elements of suitable projective modules over $C\left(𝒬\right)$. We take furthermore into account the $G$-Theory point of view (cf. [HPRS,S]) where the role of group action is particularly emphasized. As an example illustrating the method, the case of two particles is worked out in detail. Previous works (cf. [BR1,BR2]) aiming at a proof of a spin-statistics theorem for non-relativistic quantum mechanics are re-considered from the point of view of our approach, enabling us to clarify several points.

@article{AMBP_2004__11_2_205_0,
author = {Papadopoulos, Nikolaos A. and Paschke, Mario and Reyes, Andr\'es and Scheck, Florian},
title = {The spin-statistics relation in nonrelativistic quantum mechanics and projective modules},
journal = {Annales Math\'ematiques Blaise Pascal},
pages = {205--220},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {11},
number = {2},
year = {2004},
doi = {10.5802/ambp.193},
mrnumber = {2109608},
zbl = {1086.81056},
language = {en},
url = {www.numdam.org/item/AMBP_2004__11_2_205_0/}
}
Papadopoulos, Nikolaos A.; Paschke, Mario; Reyes, Andrés; Scheck, Florian. The spin-statistics relation in nonrelativistic quantum mechanics and projective modules. Annales Mathématiques Blaise Pascal, Tome 11 (2004) no. 2, pp. 205-220. doi : 10.5802/ambp.193. http://www.numdam.org/item/AMBP_2004__11_2_205_0/

[1] Atiyah, M. F. K-theory, Benjamin, New York, 1967 | MR 224083

[2] Berry, M.V.; Robbins, J.M. Indistinguishability for quantum particles: spin, statistics and the geometric phase, Proc. R. Soc. Lond. A, Volume 453 (1997), pp. 1771-1790 | Article | MR 1469170 | Zbl 0892.46084

[3] Berry, M.V.; Robbins, J.M. Quantum indistinguishability: alternative constructions of the transported basis, J. Phys. A: Math. Gen., Volume 33 (2000), p. L207-L214 | Article | MR 1768751 | Zbl 1010.81040

[4] Heil, A.; Papadopoulos, N.A.; Reifenhauser, B.; Scheck, F. SCALAR MATTER FIELD IN A FIXED POINT COMPACTIFIED FIVE-DIMENSIONAL KALUZA-KLEIN THEORY, Nuclear Physics B, Volume 281 (1987), pp. 426-444 | Article | MR 869560

[5] Laidlaw, M.G.G.; DeWitt, C.M. Feynman Functional Integrals for Systems of Indistinguishable Particles, Phys. Rev. D, Volume 3 (1971), pp. 1375-1378 | Article

[6] Leinaas, J.M.; Myrheim, J. On the Theory of Identical Particles, Nuovo Cim. B, Volume 37 (1977), pp. 1-23 | Article

[7] Papadopoulos, N.; Paschke, M.; Reyes, A.; Scheck, F. (In preparation)

[8] Paschke, M. Von Nichtkommutativen Geometrien, ihren Symmetrien und etwas Hochenergiephysik (2001) (Ph.D. thesis, Mainz University)

[9] Reyes, A. (Ph.D. thesis (in preparation), Mainz University)

[10] Sladkowski, J. Generalized G-Theory, Int. J. Theor. Phys., Volume 30 (1991), pp. 517-520 | Article