Abstract
We present an operator quantization scheme on a continuous direct product of Hilbert spaces over a time interval as an extension of the quantization using Feynman path integrals. We define the continuous direct product as a Hilbert space with two principal bases: the Fock and the Feynman ones. The Fock basis, defined by a complete set of commuting operators at different times, serves for a definition of the operator calculus. The Feynman basis, simultaneously diagonalizing the complete set of commuting operators, leads to path integrals constructed without time slicing as a spectral representation of certain operator functions. The construction of quantum theory and the corresponding path integrals for the harmonic oscillator is demonstrated both in the configuration and phase spaces. The extension of the theory to coherent states and anticommuting variables is performed.
Similar content being viewed by others
References
Feynman R. P., Hibbs A. R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York, 1965.
Marinov M. S.: Phys. Reports60 (1980) 1.
Schulman L. S.: Techniques and Applications of Path Integration. J. Wiley and Sons, New York, 1981.
Janiš V., Souček J., Souček V.: J. Math. Phys.24 (1983) 834.
Friedrichs K. O.: Mathematical Aspects of the Quantum Theory of Fields. Interscience, New York, 1953.
Wiener N.: Nonlinear Problems in Random Theory. J. Wiley and Sons, New York, 1958.
Goto T., Naka S.: Progr. Theor. Phys.56 (1976) 1318.
Araki H., Woods E. J.: Publ. Res. Inst. Math. Sci. A2 (1967) 154.
Guichardet A.: Comm. Math. Phys.5 (1967) 262.
Khandekar D. E., Lawande S. V.: Phys. Reports137 (1986) 115. Czech. J. Phys. 40 (1990)
Kugo T., Ojima I.: Suppl. Progr. Theor. Phys.66 (1979) 1.
Klauder J. R., Sudarshan E. C. G.: Fundamentals of Quantum Optics. W. A. Benjamin, New York, 1968.
Casher A., Lurie D., Revzen M.: J. Math. Phys.9 (1968) 1312.
Langer J. S.: Phys. Rev.167 (1968) 183.
Chou K., Su Z., Hao B., Yu Lu: Phys. Reports118 (1985) 1.
Umezawa H., Aritmitsu T.:in Path Integrals from meV to MeV (Eds. M. C. Gutzwiller, A. Inomata, J. R. Klauder, L. Streit). World Scientific, Singapore, 1986.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Janiš, V. Feynman quantization: An operator approach to path integration over commuting and anticommuting variables. Czech J Phys 40, 836–856 (1990). https://doi.org/10.1007/BF01597956
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01597956