Abstract
We review the standard notion of Weyl system, which stems from the Weyl formulation of the canonical commutation relations of quantum mechanics, and propose an alternative definition based on the theory of projective representations. Next, we discuss some conceptual advantages of this alternative definition. Finally, we introduce a notion of physical equivalence of group representations and propose a further ‘purely conceptual’ definition of Weyl system based on this notion.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Schrödinger, “Quantisierung als Eigenwertproblem I” Ann. Phys., 79, 361 (1926).
E. Schrödinger, “Quantisierung als Eigenwertproblem II” Ann. Phys., 79, 489 (1926).
E. Schrödinger, “Quantisierung als Eigenwertproblem III” Ann. Phys., 80, 437 (1926).
E. Schrödinger, “Quantisierung als Eigenwertproblem IV” Ann. Phys., 81, 109 (1926).
W. Heisenberg, “Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen” Z. Phys., 33, 879 (1925).
M. Born and P. Jordan, “Zur Quantenmechanik” Z. Phys., 34, 858 (1925).
M. Born, W. Heisenberg, and P. Jordan, “Zur Quantenmechanik II” Z. Phys., 35, 557 (1926).
H. Weyl, Gruppentheorie und Quantenmechanik, Hirzel (1928).
F. Strocchi, An Introduction to the Mathematical Structure of Quantum Mechanics, World Scientific, Singapore (2005).
A. Wintner, Phys. Rev., 71, 738 (1947).
P. Aniello (in preparation).
B. Fuglede, Math. Scand., 20, 79 (1967).
M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press (1972).
C. R. Putnam, Commutation Properties of Hilbert Space Operators and Related Topics, Springer-Verlag (1967).
B. Fuglede, Math. Scand., 51,163 (1982).
M. H. Stone, “Linear transformations in Hilbert space, III: operational methods and group theory,” in: Proceedings of the National Academy of Sciences of the U.S.A. (1930), Vol. 16, p. 172.
J. von Neumann, “Die Eindeutigkeit der Schrödingerschen Operatoren” Math. Ann., 104, 570 (1931).
P. Aniello, J. Phys. A: Math. Theor., 42, 475210 (2009).
V. S. Varadarajan, Geometry of Quantum Theory, 2nd ed., Springer (1985).
J. Clemente-Gallardo and G. Marmo, Int. J. Geom. Meth. Mod. Phys., 6, 129 (2009).
I. E. Segal, Mathematical Problems of Relativistic Physics, American Mathematical Society (1963).
F. Gallone and A. Sparzani, J. Math. Phys., 20, 1375 (1979).
G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press (1995).
G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press (1989).
A. Ibort, V. I. Man’ko, G. Marmo, et al., J. Phys. A: Math. Theor., 42, 155302 (2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Margarita A. Man’ko, a shining example of a scientist, on the occasion of her 70th birthday.
Rights and permissions
About this article
Cite this article
Aniello, P. On the notion of Weyl system. J Russ Laser Res 31, 102–116 (2010). https://doi.org/10.1007/s10946-010-9130-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10946-010-9130-x