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Transactions of the Moscow Mathematical Society

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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
Published electronically: November 16, 2005
MathSciNet review: 2193434
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Abstract | References | Similar Articles | Additional Information

Abstract: A family $ \{M_{p}\}_{p\in\mathbb{R}^d}$ of cylindrical measures is constructed on the space of functions ($ =$trajectories) $ f: [0,+\infty)\to\mathbb{R}^d$ such that for every $ t\geq0$ the formula

$\displaystyle \psi(t,p)= \int M_{p}(df)\,\psi_0(f( t))$ (1)

represents a solution of the Cauchy problem

$\displaystyle \frac\partial{\partial t} \psi = i \widehat H \psi \quad (t\geq0),\quad \psi(0,\cdot)=\psi_0$ (2)

(with respect to the required function $ \psi: [0,\infty)\times\mathbb{R}^d\to\mathbb{C}^s$, $ {d,s\in\mathbb{N}}$), 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 $ t$, and are Fourier transforms of countably additive (and in general matrix-valued) measures in the space variables.

The images of the measures $ M_p$ obtained by restricting trajectories to finite intervals $ [0,T]$ 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 $ M_p$ similarly to the way Markov transition probabilities give the distribution of a Markov process.

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  • 1. V. P. \cyr{M}aslov, Kompleksnye markovskie tsepi i kontinualnyi integral Feinmana dlya nelineinykh uravnenii, 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, 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, Kontinualnye 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, 10.1007/BF02557405
  • 9. I. I . Gikhman and A. V. Skorokhod, Teoriya sluchainykh 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, 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, 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.), Kvantovye sluchainye 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

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Additional Information

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

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