Summary
The possibility, first noticed by Wigner, that the form of the Hamiltonian together with the Heisenberg equations do not imply canonical commutation relations for the harmonic oscillator variables, is re-examined here. It is shown that this freedom of commutation relations just expresses the known possibility of quantizing the system by Bose or Fermi (and para-Fermi) statistics, and of altering the zero-point energy.
Riassunto
Si esamina nuovamente qui la possibilità, riscontrata per primo da Wignee, che la forma dell’hamiltoniano assieme alle equazioni di Heisenberg non implichino relazioni di commutazione canoniche per le variabili dell’oscillatore armonico. Si dimostra che questa libertà delle relazioni di commutazione esprime proprio la possibilità di quantizzare il sistema con la statistica di Bose o di Fermi (e para-Fermi), e di alterare l’energia di punto zero.
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References
E. P. Wigner:Phys. Rev.,77, 711 (1950).
L. O’Riffeartaigh andC. Ryan:Proc. Roy. Iri. Acad.,A 62, 93 (1963), to which the reader is referred for the detailed results.
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Supported by U.S. Army Trecom, U.S. Army Research Office Durham, A.F.O.S.R. and N.S.F.
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Boulware, D.G., Deser, S. « Ambiguity » of harmonic-oscillator commutation relations. Nuovo Cim 30, 230–234 (1963). https://doi.org/10.1007/BF02750763
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DOI: https://doi.org/10.1007/BF02750763