Abstract
A survey of recent developments concerning rigorously defined infinite dimensional integrals, mainly of the type of “Feynman path integrals,” is given. Both the theory and its applications, especially in quantum theory, are presented. As for the theory, general results are discussed including the case of polynomially growing phase functions, which are handled by exploiting the connection with probabilistic functional integrals. Also applications to continuous measurement theory and the stochastic Schrödinger equation are given. Other applications of probabilistic methods in non relativistic quantum theory and in quantum field theory, and their relations with statistical mechanics, are discussed.
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REFERENCES
S. Albeverio, Wiener and Feynman path integrals and their applications, in Proceedings of Symposia in Applied Mathematics, Vol. 52 (1997), pp. 163–194.
S. Albeverio, M. Schachermayer, and M. Talagand, Lectures on Probability Theory and Statistics, Proceedings of École d'Été de Probabilités de Saint-Flour 2000, P. Bernard, ed., Lecture Notes in Mathematics 1816 (Springer, Berlin, 2003).
S. Albeverio, A. M. Boutet de Monvel-Berthier, and Z. Brzeźniak, The trace formula for Schrödinger operators from infinite dimensional oscillatory integrals, Math. Nachr. 182: 21–65 (1996).
S. Albeverio and Z. Brzeźniak, Finite-dimensional approximation approach to oscillatory integrals and stationary phase in infinite dimensions, J. Funct. Anal. 113:177–244 (1993).
S. Albeverio and Z. Brzeźniak, Oscillatory integrals on Hilbert spaces and Schrödinger equation with magnetic fields, J. Math. Phys. 36:2135–2156 (1995).
S. Albeverio, Z. Brzeźniak, and Z. Haba, On the Schrödinger equation with potentials which are Laplace transforms of measures, Potential Anal. 9:65–82 (1998).
S. Albeverio, J. E. Fenstad, R. Høegh-Krohn, and T. Linstrøm, Pure and Applied Mathematics, Vol. 122 (Academic Press, Orlando, FL, 1986).
S. Albeverio and H. Gottschalk, Scattering theory for quantum fields with indefinite metric, Comm. Math. Phys. 216:491–513 (2001).
S. Albeverio, H. Gottschalk, and M. Yoshida, Representing Euclidean quantum fields as scaling limit of particle systems, J. Stat. Phys. 108:361–369 (2002).
S. Albeverio, H. Gottschalk, and J. L. Wu, Convoluted generalized white noise, Schwinger functions, and their analytic continuation, Rev. Math. Phys. 8:763–817 (1996).
S. Albeverio, H. Gottschalk, and J. L. Wu, Models of local relativistic quantum fields with indefinite metric (in all dimensions), Comm. Math. Phys. 184:509–531 (1997).
S. Albeverio, G. Guatteri, and S. Mazzucchi, Phase space Feynman path integrals, J. Math. Phys. 43:2847–2857 (2002).
S. Albeverio, G. Guatteri, and S. Mazzucchi, Representation of the Belavkin equation via Feynman path integrals, Probab. Theory Related Fields 125:365–380 (2003).
S. Albeverio, G. Guatteri, and S. Mazzucchi, Representation of the Belavkin equation via phase space Feynman path integrals, to appear in IDAQP.
S. Albeverio, A. Hahn, and A. Sengupta, Rigorous Feynman path integrals, with applications to quantum theory, gauge fields, and topological invariants, SFB 611, Preprint, Bonn, No. 58, 2003, to appear in Proc. Conf. Mathematical Legacy of Feynman's Path Integral, Lisbone 2003.
S. Albeverio, V. N. Kolokol'tsov, and O. G. Smolyanov, Représentation des solutions de l'équation de Belavkin pour la mesure quantique par une version rigoureuse de la formule d'intégration fonctionnelle de Menski, C. R. Acad. Sci. Paris Sér. I Math. 323:661–664 (1996).
S. Albeverio, V. N. Kolokol'tsov, and O. G. Smolyanov, Continuous quantum measurement: Local and global approaches, Rev. Math. Phys. 9:907–920 (1997).
S. Albeverio and R. Høegh-Krohn, Mathematical Theory of Feynman Path Integrals, Lecture Notes in Mathematics, Vol. 523 (Springer-Verlag, Berlin, 1976).
S. Albeverio, R. Høegh-Krohn, J. E. Fenstad, and T. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics (Academic Press, New York, 1986), [also translated into Russian by A. K. Svonskin and M. A. Shubin (Mir, Moscow, 1990)].
S. Albeverio and S. Mazzucchi, Generalized Fresnel Integrals, Preprint of the University of Bonn, No. 59 (2003).
S. Albeverio and S. Mazzucchi, Feynman Path Integrals for Polynomially Growing Potentials, Preprint of the University of Trento, UTM 638 (2003).
S. Albeverio and M. Röckner, Stochastic differential equations in infinite dimensions: Solutions via Dirichlet forms, Probab. Theory Related Fields 89:347–386 (1991).
S. Albeverio and M. Röckner, Classical Dirichlet forms on topological vector spaces— The construction of the associated diffusion process, Probab. Theory Related Fields 83:405–434 (1989).
S. Albeverio and M. Röckner, Dirichlet forms, quantum fields, and stochastic quantization, in Stochastic Analysis, Path Integration, and Dynamics, Pitman Res. Notes Math. Ser., Vol. 200 (Longman Sci. Tech., Harlow, 1989), pp. 1–21.
S. Albeverio and B. Rüdiger, Infinite dimensional Stochastic differential equations obtained by subordination and related Dirichlet forms, J. Funct. Anal. 204:122–156 (2003).
S. Albeverio, B. Rüdiger, and J. L. Wu, Analytic and Probabilistic Aspects of Lévy Processes and Fields in Quantum Theory, O. E. Bandorff-Nielsen, T. Mikosch, and S. L. Resnick, eds. (Birkhäuser Boston, Boston, MA, 2001), pp. 187–224.
S. Albeverio and J. Schäfer, Abelian Chern-Simons theory and linking numbers via oscillatory integrals, J. Math. Phys. 36:2157–2169 (1995).
S. Albeverio and A. Sengupta, A mathematical construction of the non-Abelian Chern- Simons functional integral, Comm. Math. Phys. 186:563–579 (1997).
R. Azencott and H. Doss, L'équation de Schrödinger quand h tend vers zéro: Une approche probabiliste (French) [The Schrödinger equation as h tends to zero: A probabilistic approach], in Stochastic Aspects of Classical and Quantum Systems (Marseille, 1983), Lecture Notes in Math., Vol. 1109 (Springer, Berlin, 1985), pp. 1–17.
V. P. Belavkin, A new wave equation for a continuous nondemolition measurement, Phys. Lett. A 140:355–358 (1989).
G. Ben Arous and F. Castell, A probabilistic approach to semi-classical approximations, J. Funct. Anal. 137:243–280 (1996).
L. Bertini, G. Jona-Lasinio, and C. Parrinello, Stochastic quantization, stochastic calculus, and path integrals: Selected topics, Progr. Theoret. Phys. Suppl. 111:83–113 (1993).
R. H. Cameron, A family of integrals serving to connect the Wiener and Feynman integrals, J. Math. Phys. 39:126–140 (1960).
P. Cartier and C. DeWitt-Morette, Functional integration, J. Math. Phys. 41:4154–4187 (2000).
D. M. Chung, Conditional analytic Feynman integrals on Wiener spaces, Proc. AMS 112:479–488 (1991).
K. L. Chung and J. C. Zambrini, Introduction to Random Time and Quantum Randomness, Monographs of the Portuguese Mathematical Society (McGraw-Hill, Lisbon, 2001).
A. B. Cruzeiro, L. Wu, and J. C. Zambrini, Bernstein processes associated with a Markov process, in Stochastic Analysis and Mathematical Physics (Santiago, 1988), Trends. Math. (Birkhäuser Boston, Boston, MA, 2000), pp. 41–72.
I. Daubechies and J. R. Klauder, Quantum-mechanical path integrals with Wiener measure for all polynomial Hamiltonians II, J. Math. Phys. 26:2239–2256 (1985).
G. F. De Angelis, G. Jona-Lasinio, and V. Sidoravicius, Berezin integrals and Poisson processes, J. Phys. A 31:289–308 (1998).
M. De Faria, H. H. Kuo, and L. Streit, The Feynman integrand as a Hida distribution, J. Math. Phys. 32:2123–2127 (1991).
H. Doss, Sur une résolution stochastique de l'équation de Schrödinger à coefficients analytiques, Comm. Math. Phys. 73:247–264 (1980).
D. Elworthy and A. Truman, Feynman maps, Cameron-Martin formulae, and anharmonic oscillators, Ann. Inst. H. Poincaré Phys. Théor. 41:115–142 (1984).
M. V. Fedoriuk and V. P. Maslov, Semi-Classical Approximation in Quantum Mechanics (D. Reidel, Dordrecht, 1981).
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View (Springer-Verlag, New York, 1987).
H. Gottschalk, Particle Systems with Weakly Attractive Interactions, Preprint of the University of Bonn (Sept. 2002), (www.lqp.uni-goettingen.de/papers/02/08/).
H. Gottschalk and H. Thaler, Interacting Quantum Fields with Indefinite Metric on Globally Hyperbolic Space-Times, Preprint of the University of Bonn, SFB 611 n.10.
L. Gross, Abstract Wiener spaces, in Proc. 5th Berkeley Symp. Math. Stat. Prob., Vol. 2 (1965), pp. 31–42.
T. Hida, H. H. Kuo, J. Potthoff, and L. Streit, White Noise (Kluwer, Dordrecht, 1995).
K. Ito, Wiener integral and Feynman integral, in Proc. Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2 (California University Press, Berkeley, 1961), pp. 227–238.
K. Ito, Generalized uniform complex measures in the Hilbertian metric space with their applications to the Feynman path integral, Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, Part 1 (California University Press, Berkeley, 1967), pp. 145–161.
G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman's Operational Calculus (Oxford University Press, New York, 2000).
G. Jona-Lasinio, Invariant measures under Schrödinger evolution and quantum statistical mechanics, in Stochastic Processes, Physics, and Geometry: New Interplays, I (Leipzig, 1999), CMS Conf. Proc., Vol. 28 (Amer. Math. Soc., Providence, RI, 2000), pp. 239–242.
G. Jona-Lasinio, Stochastic processes and quantum mechanics, Colloquium in Honor of Laurent Schwartz, Vol. 2 (Palaiseau, 1983), Astérisques, No. 132 (1985), pp. 203–216.
G. Jona-Lasinio, G. Martinelli, and E. Scoppola, Tunneling in One Dimension: General Theory, Instabilities, Rules of Calculation, Applications, Mathematics+physics, Vol. 2 (World Scientific, Singapore, 1986), pp. 227–260.
G. Jona-Lasinio and P. K. Mitter, On the stochastic quantization of field theory, Comm. Math. Phys. 101:409–436 (1985).
G. Jona-Lasinio and P. K. Mitter, Large deviation estimates in the stochastic quantization of Φ 4 2, Comm. Math. Phys. 130:111–121 (1990).
G. Jona-Lasinio and R. Sénéor, On a class of stochastic reaction-diffusion equations in two space dimensions, J. Phys. A 24:4123–4128 (1991).
G. Jona-Lasinio and R. Sénéor, Study of stochastic differential equations by constructive methods I, J. Stat. Phys. 83:1109–1148 (1996).
G. Kallianpur, D. Kannan, and R. L. Karandikar, Analytic and sequential Feynman integrals on abstract Wiener and Hilbert spaces, and a Cameron-Martin Formula, Ann. Inst. H. Poincaré, Prob. Th. 21:323–361 (1985).
T. Kuna, L. Streit, and W. Westerkamp, Feynman integrals for a class of exponentially growing potentials, J. Math. Phys. 39:4476–4491 (1998).
H. H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math. (Springer-Verlag, Berlin/Heidelberg/New York, 1975).
S. Leukert and J. Schäfer, A rigorous construction of Abelian Chern-Simons path integrals using white noise analysis, Rev. Math. Phys. 8:445–456 (1996).
V. Mandrekar, Some remarks on various definitions of Feynman integrals, in Lectures Notes in Math., K. Jacob Beck, ed. (1983), pp. 170–177.
V. P. Maslov, Méthodes opérationelles (Mir, Moscou, 1987).
P. Mullowney, A General Theory of Integration in Function Spaces, Pitman Research Notes in Mathematics, Vol. 153 (1987).
E. Nelson, Feynman integrals and the Schrödinger equation, J. Math. Phys. 5:332–343 (1964).
J. Rezende, The method of stationary phase for oscillatory integrals on Hilbert spaces, Comm. Math. Phys. 101:187–206 (1985).
M. Reed and B. Simon, Methods of Modern Mathematical Physics. Fourier Analysis, Self-Adjointness (Academic Press, New York, 1975).
B. Simon, Functional Integration and Quantum Physics (Academic Press, New York/ London, 1979).
B. Simon, The P(Φ) 2 Euclidean (Quantum) Field Theory, Princeton Series in Physics (Princeton University Press, Princeton, NJ).
O. G. Smolyanov and E. T. Shavgulidze, Path Integrals (Moskov. Gos. Univ., Moscow, 1990).
H. Thaler, Solution of Schrödinger equations on compact Lie groups via probabilistic methods, to appear in Potential Analysis.
A. Truman, The Feynman maps and the Wiener integral, J. Math. Phys. 19:1742–1750 (1978).
K. Yajima, Smoothness and non-smoothness of the fundamental solution of time dependent Schrödiger equations, Comm. Math. Phys. 181:605–629 (1996).
J. C. Zambrini, Feynman integrals, diffusion processes and quantum symplectic two-forms, J. Korean Math. Soc. 38:385–408 (2001).
T. J. Zastawniak, Equivalence of Albeverio-Høegh-Krohn-Feynman Integral for Anharmonic Oscillators and the Analytic Feynman Integral, Univ. Iagel. Acta Math., Vol. 28 (1991), pp. 187–199.
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BiBoS; IZKS; SFB611; CERFIM (Locarno); Acc. Arch. (Mendrisio)
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Albeverio, S., Mazzucchi, S. Some New Developments in the Theory of Path Integrals, with Applications to Quantum Theory. Journal of Statistical Physics 115, 191–215 (2004). https://doi.org/10.1023/B:JOSS.0000019836.37663.d9
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DOI: https://doi.org/10.1023/B:JOSS.0000019836.37663.d9