A functional integral with respect to a countably additive measure representing a solution of the Dirac equation

Author:
N. N. Shamarov

Translated by:
J. Wiegold

Original publication:
Trudy Moskovskogo Matematicheskogo Obshchestva, tom **66** (2005).

Journal:
Trans. Moscow Math. Soc. **2005**, 243-255

MSC (2000):
Primary 28B99; Secondary 46G10, 81Q05

DOI:
https://doi.org/10.1090/S0077-1554-05-00150-0

Published electronically:
November 16, 2005

MathSciNet review:
2193434

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A family of cylindrical measures is constructed on the space of functions (trajectories) such that for every the formula

(1) |

represents a solution of the Cauchy problem

(2) |

(with respect to the required function , ), for the general evolution equation from some class containing the classical Dirac equation and the Schrödinger equation in its impulse representation, with ``model'' potentials that are independent of , and are Fourier transforms of countably additive (and in general matrix-valued) measures in the space variables.

The images of the measures obtained by restricting trajectories to finite intervals have bounded variation and are countably additive.

The integral kernels (``Green's functions'') of the corresponding solution operators, which can be approximated (using Trotter's formula) by integrals of finite multiplicity of the expressions explicitly defined by the ingredients of the original equation, are (matrix-valued) transition measures that give cylindrical measures similarly to the way Markov transition probabilities give the distribution of a Markov process.

**1.**V. P. Maslov,*\cyr Kompleksnye markovskie tsepi i kontinual′nyĭ integral Feĭnmana dlya nelineĭnykh uravneniĭ.*, Izdat. “Nauka”, Moscow, 1976 (Russian). MR**0479126****2.**J. Diestel and J. J. Uhl Jr.,*Vector measures*, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR**0453964****3.**O. G. Smoljanov and S. V. Fomīn,*Measures on topological linear spaces*, Usephi Mat. Nauk**31**(1976), no. 4, (190), 3–56 (Russian). MR**0420764****4.**O. G. Smolyanov and E. T. Shavgulidze,*A simple proof of a theorem of Tarieladze on the sufficiency of positively sufficient topologies*, Teor. Veroyatnost. i Primenen.**37**(1992), no. 2, 421–424 (Russian); English transl., Theory Probab. Appl.**37**(1992), no. 2, 402–404. MR**1211174**, https://doi.org/10.1137/1137090**5.**O.G. Smolyanov and E.T. Shavgulidze,*Pseudodifferential operators in superanalysis*, Uspekhi Mat. Nauk**41**(1986), 164-165.**6.**O. G. Smolyanov and E. T. Shavgulidze,*\cyr Kontinual′nye integraly*, Moskov. Gos. Univ., Moscow, 1990 (Russian). MR**1145004****7.**A. P. Robertson and Wendy Robertson,*Topological vector spaces*, 2nd ed., Cambridge University Press, London-New York, 1973. Cambridge Tracts in Mathematics and Mathematical Physics, No. 53. MR**0350361****8.**O. G. Smolyanov and A. Trumen,*Schrödinger-Belavkin equations and associated Kolmogorov and Lindblad equations*, Teoret. Mat. Fiz.**120**(1999), no. 2, 193–207 (Russian, with Russian summary); English transl., Theoret. and Math. Phys.**120**(1999), no. 2, 973–984. MR**1737286**, https://doi.org/10.1007/BF02557405**9.**I. I. Gikhman and A. V. Skorokhod,*\cyr Teoriya sluchaĭnykh protsessov. Tom II*, Izdat. “Nauka”, Moscow, 1973 (Russian). MR**0341540****10.**Edward Brian Davies,*One-parameter semigroups*, London Mathematical Society Monographs, vol. 15, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR**591851****11.**Hiroshi Watanabe and Yûsuke Itô,*A construction of the fundamental solution for the relativistic wave equation. I*, Tokyo J. Math.**7**(1984), no. 1, 99–117. MR**752112**, https://doi.org/10.3836/tjm/1270152993**12.**Hiroshi Watanabe,*Path integral for some systems of partial differential equations*, Proc. Japan Acad. Ser. A Math. Sci.**60**(1984), no. 3, 86–89. MR**763290****13.**R. S. Egikyan and D. V. Ktitarev,*The Feynman formula in phase space for systems of pseudodifferential equations with analytic symbols*, Mat. Zametki**51**(1992), no. 5, 44–50, 157 (Russian, with Russian summary); English transl., Math. Notes**51**(1992), no. 5-6, 453–457. MR**1186530**, https://doi.org/10.1007/BF01262176**14.**N. N. Shamarov,*Matrix-valued cylindrical measures of Markov type and their Fourier transforms*, Russ. J. Math. Phys.**10**(2003), no. 3, 319–333. MR**2012903****15.**A. S. Kholevo (ed.),*\cyr Kvantovye sluchaĭnye protsessy i otkrytye sistemy*, \cyr Matematika: Novoe v Zarubezhnoĭ Nauke [Mathematics: Recent Publications in Foreign Science], vol. 42, “Mir”, Moscow, 1988 (Russian). Translated from the English and edited by A. S. Kholevo. MR**963453**

Retrieve articles in *Transactions of the Moscow Mathematical Society*
with MSC (2000):
28B99,
46G10,
81Q05

Retrieve articles in all journals with MSC (2000): 28B99, 46G10, 81Q05

Additional Information

**N. N. Shamarov**

Affiliation:
Mechanics and Mathematics Department, Moscow State University, Vorob’evy Gory, Moscow 119234, Russia

DOI:
https://doi.org/10.1090/S0077-1554-05-00150-0

Published electronically:
November 16, 2005

Additional Notes:
This work was supported by the Russian Foundation for Fundamental Research, Grant No. 02-01-01074.

Article copyright:
© Copyright 2005
American Mathematical Society