Abstract
An extension of the formalism of quantum mechanics to the case where the canonical variables are valued in a field ofp-adic numbers is considered. In particular the free particle and the harmonic oscillator are considered. In classicalp-adic mechanics we consider time as ap-adic variable and coordinates and momentum orp-adic or real. For the case ofp-adic coordinates and momentum quantum mechanics with complex amplitudes is constructed. It is shown that the Weyl representation is an adequate formulation in this case. For harmonic oscillator the evolution operator is constructed in an explicit form. For primesp of the form 4l+1 generalized vacuum states are constructed. The spectra of the evolution operator have been investigated. Thep-adic quantum mechanics is also formulated by means of probability measures over the space of generalized functions. This theory obeys an unusual property: the propagator of a massive particle has power decay at infinity, but no exponential one.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Vladimirov, V.S., Volovich, I.V.: Superanalysis, Differential calculus. Theor. Math. Phys.59, 3–27 (1984)
Volovich, I.V.:p-adic space-time and string theory. Theor. Math. Phys.71, 337–340 (1987)
Volovich, I.V.:p-adic string. Class. Quant. Grav.4, L83-L87 (1987)
Volovich, I.V.: Number theory as the ultimate physical theory. Preprint CERN-TH. 4981/87
Grossman, B.:p-adic strings, the Weyl conjectures and anomalies. Phys. Lett. B197, 101–106 (1987)
Freund, P.G.O., Olson, M.: Non-archimedean strings. Phys. Lett. B199, 186–190 (1987)
Freund, P.G.O., Witten, E.: Adelic string amplitudes. Phys. Lett. B199, 191–195 (1987)
Volovich, I.V.: Harmonic analysis andp-adic strings. Lett. Math. Phys.16, 61–67 (1988)
Aref'eva, I. Ya., Dragovič, B., Volovich, I.V.: On thep-adic summability of the anharmonic oscillator. Phys. Lett. B200, 512–514 (1988)
Gervais, J.-L.:p-adic analyticity and Virasoro algebras for conformal theories in more than two dimensions. Phys. Lett. B201, 306–310 (1988)
Frampton, P.H., Okada, Y.:p-adic stringN-point function. Phys. Lett.60, 484–486 (1988)
Marinari, E., Parisi, G.: On thep-adic five point function. Preprint ROM2F-87/38
Freund, P.G.O., Olson, M.:p-adic dynamical systems. Nucl. Phys. B297, 86–97 (1988)
Aref'eva, I. Ya., Dragovič, B., Volovich, I.V.: On the adelic string amplitudes. Phys. Lett. B209, 445–450 (1988)
Frampton, P.H., Okada, Y.: Effective scalar field theory ofp-adic string. Phys. Rev. D (to appear)
Grossman, B.: Adelic conformal field theory. Preprint Rockefeller University DOE/ER/40325-16
Meurice, Y.: The classical harmonic oscillator on Galois andp-adic fields, preprint ANL-HEP-PR-87-114
Borevich, Z.I., Shafarevich, I.R.: Number theory. New York: Academic Press 1966
Gelfand, I.M., Graev, M.I., Pjatetski-Shapiro, I.I.: Theory of representations and automorphic functions. Moscow: Nauka 1966
Vladimirov, V.S.: Distributions over the field ofp-adic numbers. Usp. Math. Nauk43, 17–53 (1988)
Koblitz, N.:p-adic numbers,p-adic analysis, and zeta-functions. Berlin, Heidelberg, New York: Springer 1984
Dwork, B.: Lectures onp-adic differential equations. Berlin, Heidelberg, New York: Springer 1982
Vinogradov, I.M.: Elements of Number theory. Moscow: Nauka 1965
Weyl, H.: The theory of groups and quantum mechanics. New York: Dover 1931
Karasev, M.V., Maslov, V.P.: Asymptotic and geometric quantization. Usp. Math. Nauk.39, 115–173 (1984)
Weil, A.: Sur certains groupes d'operateurs unitaries. Acta Math.111, 143–211 (1964)
Manin, Yu. I.: Non-Archimedean integration andp-adicL-functions of Jacquet and Lenglands. Usp. Math. Nauk.31, 5–54 (1976)
Feynman, R.P., Hibbs, A.R.: Quantum mechanics and path integrals. New York: McGraw-Hill 1965
Maslov, V.P.: Operator methods. Moscow: Nauka 1973
Glimm, J., Jaffe, A.: Quantum physics. A functional integral point of view. Berlin, Heidelberg, New York: Springer 1987
Vladimirov, V.S., Volovich, I.V.: Vacuum state inp-adic quantum mechanics. Phys. Lett. B217, 411–415 (1989)
Vladimirov, V.S., Volovich, I.V.:p-adic Schrödinger type equation, Preprint NBI-HE-87-77 (1988)
Zelenov, E.I.:p-adic quantum mechanics forp=2. Theor. Math. Phys. (to appear)
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Rights and permissions
About this article
Cite this article
Vladimirov, V.S., Volovich, I.V. p-adic quantum mechanics. Commun.Math. Phys. 123, 659–676 (1989). https://doi.org/10.1007/BF01218590
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01218590