The spin-statistics relation in nonrelativistic quantum mechanics and projective modules

Nikolaos A. Papadopoulos[1]; Mario Paschke[2]; Andrés Reyes[1]; Florian Scheck[1]

  • [1] Johannes-Gutenberg-Universität Institut für Physik - ThEP. Staudinger Weg 7 Mainz, D-55128 Germany
  • [2] Max Planck Institute for Mathematics in the Sciences Inselstrasse 22-26 Leipzig, D-04103 Germany

Annales mathématiques Blaise Pascal (2004)

  • Volume: 11, Issue: 2, page 205-220
  • ISSN: 1259-1734

Abstract

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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 ( 𝒬 ) . 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.

How to cite

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Papadopoulos, Nikolaos A., et al. "The spin-statistics relation in nonrelativistic quantum mechanics and projective modules." Annales mathématiques Blaise Pascal 11.2 (2004): 205-220. <http://eudml.org/doc/10506>.

@article{Papadopoulos2004,
abstract = {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 [8], 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 (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.},
affiliation = {Johannes-Gutenberg-Universität Institut für Physik - ThEP. Staudinger Weg 7 Mainz, D-55128 Germany; Max Planck Institute for Mathematics in the Sciences Inselstrasse 22-26 Leipzig, D-04103 Germany; Johannes-Gutenberg-Universität Institut für Physik - ThEP. Staudinger Weg 7 Mainz, D-55128 Germany; Johannes-Gutenberg-Universität Institut für Physik - ThEP. Staudinger Weg 7 Mainz, D-55128 Germany},
author = {Papadopoulos, Nikolaos A., Paschke, Mario, Reyes, Andrés, Scheck, Florian},
journal = {Annales mathématiques Blaise Pascal},
language = {eng},
month = {7},
number = {2},
pages = {205-220},
publisher = {Annales mathématiques Blaise Pascal},
title = {The spin-statistics relation in nonrelativistic quantum mechanics and projective modules},
url = {http://eudml.org/doc/10506},
volume = {11},
year = {2004},
}

TY - JOUR
AU - Papadopoulos, Nikolaos A.
AU - Paschke, Mario
AU - Reyes, Andrés
AU - Scheck, Florian
TI - The spin-statistics relation in nonrelativistic quantum mechanics and projective modules
JO - Annales mathématiques Blaise Pascal
DA - 2004/7//
PB - Annales mathématiques Blaise Pascal
VL - 11
IS - 2
SP - 205
EP - 220
AB - 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 [8], 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 (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.
LA - eng
UR - http://eudml.org/doc/10506
ER -

References

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  1. M. F. Atiyah, K-theory, (1967), Benjamin, New York MR224083
  2. M.V. Berry, J.M. Robbins, Indistinguishability for quantum particles: spin, statistics and the geometric phase, Proc. R. Soc. Lond. A 453 (1997), 1771-1790 Zbl0892.46084MR1469170
  3. M.V. Berry, J.M. Robbins, Quantum indistinguishability: alternative constructions of the transported basis, J. Phys. A: Math. Gen. 33 (2000), L207-L214 Zbl1010.81040MR1768751
  4. A. Heil, N.A. Papadopoulos, B. Reifenhauser, F. Scheck, SCALAR MATTER FIELD IN A FIXED POINT COMPACTIFIED FIVE-DIMENSIONAL KALUZA-KLEIN THEORY, Nuclear Physics B 281 (1987), 426-444 MR869560
  5. M.G.G. Laidlaw, C.M. DeWitt, Feynman Functional Integrals for Systems of Indistinguishable Particles, Phys. Rev. D 3 (1971), 1375-1378 
  6. J.M. Leinaas, J. Myrheim, On the Theory of Identical Particles, Nuovo Cim. B 37 (1977), 1-23 
  7. N. Papadopoulos, M. Paschke, A. Reyes, F. Scheck 
  8. M. Paschke, Von Nichtkommutativen Geometrien, ihren Symmetrien und etwas Hochenergiephysik, (2001) 
  9. A. Reyes 
  10. J. Sladkowski, Generalized G-Theory, Int. J. Theor. Phys. 30 (1991), 517-520 

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