Mathematical Problems in Theoretical Physics pp 265-275 | Cite as

# Some applications of functional integration

Functional Integration General Plenary Session

First Online:

## Abstract

We discuss some new developments in the theory of functional integration and some applications in nonrelativistic quantum theory, quantum field theory, statistical mechanics, solid state physics and hydrodynamics.

## Keywords

Gauge Group Functional Integration Trace Formula Feynman Path Feynman Path Integral
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

## Ch. I.

- 1).The approach mentioned here originates in work by K. Ito. Extensive references are in [1], [2]. For recent work in connection with other definitions see [5],[6] (who also discusses the relation with work by Cameron and Storvick) and [7]. There are other methods of defining Feynman path integrals e.g. A) Time division combined with path approximation (e.g. [5] resp. [22]); B) Reduction to probabilistic methods (e.g. [23]); C) Use of Poisson integrals (see e.g. [24]); D) Various combinations and extensions. Full references will be given in [25].Google Scholar
- 2).See [3], [4], [8]. The derivation of asymptotic expansions for the solution of Schrödinger equation has been discussed also by methods of p.d.e. by Maslov, Duistermaat, Fujiwara (e.g. [22]) and others. There is a rich literature concerning compact manifolds (see e.g. [8]). The basic ideas for writing in the WKBJ approach the leading term in the trace formula for Schrödinger operators goes back to M. Gutzwiller (for discussions:and references, also in connection with quantum fields see e.g. [9]), The approach to the leading term as h → O is also discussed from a mathematical definition of Feynman path integrals in [10], [11]. The trace formula has been proven recently by methods of p.d.e. by Chazarain [12]. Recent very interesting work on the structure of classical orbits with prescribed period has been given by C. Conley and E. Zehnder [13].Google Scholar
- 3).A note on probabilitstic methods related to the Feynman path integral methods. We shall mention some recent related work which uses probabilistic path integrals rather than Feynman path integrals. 1. Asymptotics-of-function-space-integrals-with-respect-to-Gaussian measures There is a tradition (Donsker, Schilder, Pinkus, Varadhan) in such studies, see e.g. [14]. Recent results related to the one in 12) have been obtained by R. Ellis, J. Rosen concerning the case where the formal complex Gaussian measure of (2), is replaced by a Gaussian probability measure. Here instead of the stationary points of the phase one has to look at minimum points. The expansion corresponds to the one given in our case. The results have applications to the convergence of associated random variables and by this to some problems in classical statistical mechanics (for a very recent result see [15]). Related methods with other applications have been used recently by I. Davies and A. Truman [16] (see also [11] for manifolds and e.g. [17], for quantum fields). 2. Methods of stochastic-equations. The methods have been used in connection with solutions of the Schrödinger equations [18] and their eigenvalues and eigenfunctions [12], [19]. Recently an interesting relation between heuristic Feynman path integrals for the Coulomb problem on ℝ
^{3}and Feynman path integrals for the harmonic oscillator on ℝ^{4}has been discovered by I.H. Duru and H. Kleinert [20]. Ph. Blanchard and M. Sirugue [21] have proven the validity of this relation replacing Feynman path integrals by Wiener ones.Google Scholar - [1]S. Albeverio, R. Høegh-Krohn, Mathematical Theory of Feynman Path Integrals, Lecture Notes in Maths. 523, Springer (1976).Google Scholar
- [2]S. Albeverio, R. Høegh-Krohn, pp. 3-57 in “Feynman Path Integrals”, Ed. S. Albeverio et al., Lect. Notes in Phys. 106, Springer (1979).Google Scholar
- [3]S. Albeverio, R. Høegh-Krohn, Inv. Math. 40, 59–106 (1977).Google Scholar
- [4]S. Albeverio, Ph. Blanchard, R. Høegh-Krohn, Preprint (1981).Google Scholar
- [5]A. Truman, pp. 73–102 in book [2]; and to appear in Phys. Repts.Google Scholar
- [6]G.W. Johnson, Univ. of Nebraska, Lincoln Preprint (1981)Google Scholar
- [7]R.H. Burkhart, University of North Carolina, Preprint (1978).Google Scholar
- [8]S. Albeverio, Ph. Blanchard, R. Høegh-Krohn, Bielefeld-Bochum Preprint 1980 (to appear Comm. Math. Phys.).Google Scholar
- [9]C. De Witt-Morette, A. Maheshwari, B. Nelson, Phys. Repts. 50, 255–372 (1979).Google Scholar
- [10]A. Truman, pp. 73–102 in Ref. [2] and these Proceedings.Google Scholar
- [11]D. Elworthy, to appear Proc. Marseille Conf. June 1981; and these Proc.Google Scholar
- [12]J. Chazarain, Comm. P.D.E. 5, 595–644 (1980); B. Helffer, D. Robert, Nantes Preprint(1981).Google Scholar
- [13]C. Conley, E. Zehnder, Bochum Preprint 1981Google Scholar
- [14]B. Simon, Functional Integration and Quantum Pysics, Academic Press 1979.Google Scholar
- [15]Eisele, J. Ellis, Preprint in preparation.Google Scholar
- [16]I. Davies,A. Truman, Edinburgh Preprint 1981.Google Scholar
- [17]D. Williams, these Proceedings H. Hogreve, R. Schrader, R. Seiler, pp. 282–289 in Ref. [2].Google Scholar
- [18]S. Albeverio, R. Høegh-Krohn, L. Streit, J. Math. Phys. 18, 907–917 (1977)Google Scholar
- 22A).L. Streit, Phys. Repts. 1981; G. Jona-Lasinio, F. Martinelli, E. Scoppola, these Proc.Google Scholar
- [18a]H. Doss, Comm. Math. Phys. 73, 247–264 (1980); W. Kirsch., P. Martinelli, these Proc.Google Scholar
- [19]R. Carmona, B. Simon, Princecon Preprint 1981; F.Guerra, Phys.Reports 1981Google Scholar
- [20]I.H. Duru, H. Kleinert, Phys. Letts. B 84, 185–188 (1979), and Freie Univ. Preprint 1981.Google Scholar
- [21]Ph.Blanchard, M. Sirugue, J. Math. Phys. 22, 1372–1376 (1981)Google Scholar
- [22]D. Fujiwara, to appear J. Analys. Math.Google Scholar
- [23]P. Jørgensen, Aarhus Preprint 1981Google Scholar
- 29A).T. Hida, L. Streit, these Proc.Google Scholar
- 30A).J. Klauder, I. Daubechies, Bell Lab. Preprint.Google Scholar
- [24]M. Sirugue, these Proc.Google Scholar
- [25]S. Albeverio, Ph. Blanchard, R. Høegh-Krohn, book in prep. for Encycl. of Maths. (Ed. G.C. Rota).Google Scholar
- [26]J. Glimm, A. Jaffe, Quantum Physics., Springer (1981).Google Scholar

## Ch. II. The results have connections with many other domains, e.g.: — Affine Lie algebras and number theory (see e.g. [11]) — Current groups (see e.g. [11], [12]) — The energy representation can be looked upon as a representation of gauge groups. There are also some connections with recent work on constructing approximate Euclidean measures for Yang-Mills fields [13]. — More generally new developments in stochastic geometry (see e.g. contributions by Bismut, Pinsky, Stroock in the volume [4]).

- [1]See e.g. [14], [26] in Sect. I and [2], [3].Google Scholar
- [2]S. Albeverio, R. Høegh-Krohn, pp. 331–353 and 303–329, Quantum Fields-Algebras, Processes, Ed. L. Streit, Springer Wien (1980).Google Scholar
- [3]S. Albeverio, R. Høegh-Krohn, to appear in Phys. Repts.Google Scholar
- [4]See e.g. S. Albeverio, R. Høegh-Krohn, pp. 497–540 in Stochastic Integrals, Ed. D. Williams, Lecture Notes in Mathematics, Springer (1981), and references therein.Google Scholar
- [5]S. Albeverio, R. Høegh-Krohn, Compos. Math. 36, 37–52 (1978).Google Scholar
- [6]This was proven for d ≥ 5 by Ismagilov, for d ≥ 4 by Vershik, Gelfand, Graev [12] and Albeverio, Høegh-Krohn [2], for d ≥ 3 by Albeverio, Høegh-Krohn, TestardGoogle Scholar
- [7]S. Albeverio, R. Høegh-Krohn, D. Testard, J. Funct. Anal. 41, 378–396 (1981).Google Scholar
- [8]S. Albeverio, G. Gallavotti, R. Høegh-Krohn, Commun. Math. Phys. 70, 187–192 (1979).Google Scholar
- [9]S. Albeverio, R. Høegh-Krohn, J. Funct. Anal. 16, 39–82 (1974)Google Scholar
- [10]S. Albeverio, R. Høegh-Krohn D. Testard, A. Vershik, Bochum Preprint (1981)Google Scholar
- [11]B. Frenkel, Yale Preprint (1980).Google Scholar
- [12]A. Vershik, I. Gelfand, M. Graev, Compos. Math. 35, 299–334 (1977); Compos. Math. 42, 217–243 (1981).Google Scholar
- [13]B. Gaveau, Ph. Trauber, J. Funct. An. 42, 356–367 (1981); M. Asorey, P. Mitter, Comm. Math. Phys. 80, 43–48 (1981).Google Scholar

## Ch. III

- [1]J. Westwater, Comm. Math. Phys. 72, 131–174 (1980) and these Proc.Google Scholar
- [2]K. Symanzik, pp. 152–226 in Local Quantum Theory, Ed. R. Jost, Academic Press, N.Y. (1969)Google Scholar
- [3]S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lindstrøm, book in preparation.Google Scholar
- [4]S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, Trans. Am. Math. Soc. 252, 275–295 (1979).Google Scholar
- [5]S. Albeverio, F. Gesztesy, R. Høegh-Krohn, Bochum Prep. 1981 (to appear Ann. Inst. H. Poincaré B).Google Scholar

## Ch. IV

- [1]A basic reference for early work in the statistical approach is A.S. Manin, A.M. Yaglom, Statistical Fluid Mechanics, M.I.T. Press: (1975).Google Scholar
- [2]S. Albeverio, M. De Faria, R. Høegh-Krohn, J. Stat. Phys. 20, 585–595 (1979).Google Scholar
- [3]S. Albeverio, R. Høegh-Krohn, Bochum-Preprint (1979).Google Scholar
- [4]C.Boldrighini, S. Frigio, Commun. Math-. Phys. 72, 55–76 (1980).Google Scholar
- [5]R. Calinon, D. Merlini, Bochum Preprint (1981).Google Scholar
- [6]There is work in preparation by S. Albeverio, M. De Faria, D. Dürr, R. Uegh-Krohn and D. MerliniGoogle Scholar

## Ch. V

- [1]G.G. Emch, S. Albeverio, J.P. Eckmann, Rept. Math. Phys. 13, 73–85 (1978).Google Scholar
- [1a]E.B. Davies, Quantum theory of open systems, Acad. Press, N.Y. (1976).Google Scholar
- [2]S. Albeverio, R. Høegh-Krohn, G. Olsen, J. Reine u. angew. Math. 319, 25–37 (1980).Google Scholar

## Copyright information

© Springer-Verlag 1982