Abstract
We approach the relationship between classical and quantum theories in a new way, which allows both to be expressed in the same mathematical language, in terms of a matrix algebra in a phase space. This makes clear not only the similarities of the two theories, but also certain essential differences, and lays a foundation for understanding their relationship. We use the Wigner-Moyal transformation as a change of representation in phase space, and we avoid the problem of “negative probabilities” by regarding the solutions of our equations as constants of the motion, rather than as statistical weight factors. We show a close relationship of our work to that of Prigogine and his group. We bring in a new nonnegative probability function, and we propose extensions of the theory to cover thermodynamic processes involving entropy changes, as well as the usual reversible processes.
Similar content being viewed by others
References
E. Wigner,Phys. Rev. 40, 749 (1932).
J. E. Moyal,Proc. Camb. Phil. Soc. 45, 99 (1959).
H. Fröhlich,Riv. Nuovo Cimento 3, 490 (1973).
B. Misra, I. Prigogine, and M. Courbage,Proc. Nat. Acad. Sci. USA 76, 4768 (1979).
M. Schönberg,Riv. Nuovo Cimento IX, 1139 (1952).
F. A. M. Frescura and B. J. Hiley,Found. Phys. 10, 7 (1980).
J. von Neumann,Math. Ann. 104, 570 (1931).
T. Takabaysi,Prog. Theor. Phys. 11, 341 (1954).
D. Bohm, inThe Many Body Problem, C. DeWitt, ed. (Wiley, NY, 1959).
D. Bohm and G. Carmi,Phys. Rev. 133 A, 319 and 332 (1964).
T. Bastin, H. P. Noyes, J. Amson, and C. W. Kilmister,Int. J. Theor. Phys. 18, 445 (1979).
Y. Aharanov, M. Pendleton, and A. Peterson,Int. J. Theor. Phys. 2, 123 (1969).
J. von Neumann,Mathematical Foundations of Quantum Mechanics (Princeton, 1955).
E. Wigner, inPerspectives in Quantum Theory, W. Yourgrau and A. van der Merwe, eds. (MIT, Cambridge Mass., 1971).
L. Cohen, inContemporary Research in the Foundations and Philosophy of Quantum Theory, Hooker, ed. (Reidel, Dordrecht, Holland, 1973), p. 66.
E. B. Davies,Quantum Theory of Open Systems (Academic Press, London, 1976).
C. George, I. Prigogine, and L. Rosenfeld,K. Dan. Vidensk. Selsk. Mat.-fys. Meddel. 38, 1 (1972); C. George, F. Henin, F. Mayne, and I. Prigogine,Hadronic J. 1, 520 (1978); B. Misra,Proc. Nat. Acad. Sci. USA 75, 1627 (1978); B. Misra, I. Prigogine, and M. Courbage,Physica 98A, 1 (1979); C. George and I. Prigogine,Physica A99, 369 (1979).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bohm, D., Hiley, B.J. On a quantum algebraic approach to a generalized phase space. Found Phys 11, 179–203 (1981). https://doi.org/10.1007/BF00726266
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00726266