On Nambu's generalized hamiltonian mechanics
Recently, Y. Nambu stated the principles of a new analytical mechanics which allows an odd, as well as an even, number of phase space variables. In this paper we investigate if this mechanics is independent for the odd case of Dirac's classical mechanics because there are reasons which allow one to suspect that independence fails. We prove that Nambu's mechanics is independent of Dirac's mechanics, at least for the odd cases, so that the Nambu mechanics can, in principle, describe physical systems for which the Dirac mechanics is not suitable.
As regards the even case, it is natural to confront the Nambu mechanics with the Hamiltonian one, because both have the same dimension for their phase spaces. In this paper we restrict our study to the comparison of the canonical groups of both formalisms and prove that both groups are different.
García Sucre and Kálnay have suggested that perhaps the Nambu mechanics is the natural one for quarks. We end our study with a brief discussion on this subject whose conclusion seems to support their conjecture.
KeywordsField Theory Phase Space Elementary Particle Quantum Field Theory Physical System
Unable to display preview. Download preview PDF.
- Bergmann, P. G. and Goldberg, I. (1955).Physical Review 98, 531.Google Scholar
- Cohen, I. (1972).Formalismo de Hamilton Dirac. Universidad de Los Andes Report. Mérida, Venezuela.Google Scholar
- Cohen, I. (to be submitted for publication).Generalization of Nambu's Mechanics.Google Scholar
- Dirac, P. A. M. (1950).Canadian Journal of Mathematics 2, 129.Google Scholar
- Dirac, P. A. M. (1958).Proceedings of the Royal Society (London), Series A,246, 326.Google Scholar
- Dirac, P. A. M. (1964).Lectures on Quantum Mechanics. Belfer Graduate School of Sciences Monograph Series No. 2. Yeshiva University, New York.Google Scholar
- Garcia Sucre, M. and Kalnay, A. J. (1974). On the statistics consistent with Nambu's new quantum rules. Quarks? To be published in theInternational Journal of Theoretical Physics.Google Scholar
- Kilmister, C. W. (1964).Hamiltonian Dynamics. Longmans, Green and Co. Ltd., London.Google Scholar
- Marx, E. (1972).International Journal of Theoretical Physics Vol. 6, No. 4, p. 307.Google Scholar
- Nambu, Y. (1973).Physical Review D 7, 2405.Google Scholar
- Ruggeri, G. (to be submitted for publication).Nambu's Mechanics as a Class of Singular Generalized Dynamical Formalisms.Google Scholar