zbMATH — the first resource for mathematics

The spin-statistics relation in nonrelativistic quantum mechanics and projective modules. (English) Zbl 1086.81056
Summary: In this work we consider non-relativistic quantum mechanics, obtained from a classical configuration space \({\mathcal Q}\) of indistinguishable particles. Following an approach proposed in M. Paschke Von Nichtkommutativen Geometrien, ihren Symmetrien und etwas Hochenergiephysik. Ph.D. thesis, Mainz University, 2001, wave functions are regarded as elements of suitable projective modules over \(C({\mathcal Q})\). We take furthermore into account the \(G\)-Theory point of view 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 [M. V. Berry and J. M. Robbins, Proc. R. Soc. Lond. A 453, 1771–1790 (1997; Zbl 0892.46084); J. Phys. A, Math. Gen., 33, L207–L2l4 (2001; Zbl 1010.81040)] 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.

81S05 Commutation relations and statistics as related to quantum mechanics (general)
Full Text: DOI Numdam EuDML
[1] Atiyah, M. F., K-theory, (1967), Benjamin, New York
[2] Berry, M. V.; Robbins, J. M., Indistinguishability for quantum particles: spin, statistics and the geometric phase, Proc. R. Soc. Lond. A, 453, 1771-1790, (1997) · Zbl 0892.46084
[3] Berry, M. V.; Robbins, J. M., Quantum indistinguishability: alternative constructions of the transported basis, J. Phys. A: Math. Gen., 33, L207-L214, (2000) · 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, 281, 426-444, (1987)
[5] Laidlaw, M. G.G.; DeWitt, C. M., Feynman functional integrals for systems of indistinguishable particles, Phys. Rev. D, 3, 1375-1378, (1971)
[6] Leinaas, J. M.; Myrheim, J., On the theory of identical particles, Nuovo Cim. B, 37, 1-23, (1977)
[7] Papadopoulos, N.; Paschke, M.; Reyes, A.; Scheck, F.
[8] Paschke, M., Von Nichtkommutativen Geometrien, ihren Symmetrien und etwas Hochenergiephysik, (2001)
[9] Reyes, A.
[10] Sladkowski, J., Generalized G-theory, Int. J. Theor. Phys., 30, 517-520, (1991)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.