Full text loading...
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
A new perspective on functional integration
1.F. Rabelais, Literally “bone marrow as a producer of substance” in Gargantua, Prologue de l’auteur.
2.C. Morette, “On the definition and approximation of Feynman’s path integral,” Phys. Rev. 81, 848–852 (1951).
3.C. DeWitt-Morette, B. Nelson, and T.R. Zhang, “Caustic problems in quantum mechanics with applications to scattering theory,” Phys. Rev. D 28, 2526–2546 (1983).
4.M.G.G. Laidlaw and C. Morette-DeWitt, “Feynman functional integrals for systems of indistinguishable particles,” Phys. Rev. D 3, 1375–1378 (1971).
5.R.H. Cameron and W.T. Martin, “Transformation of Wiener integrals under a general class of linear transformations,” Trans. Am. Math. Soc. 58, 184–219 (1945).
6.C. DeWitt-Morette, A. Maheshwari, and B. Nelson, “Path integration in non-relativistic quantum mechanics,” Phys. Rep. 50, 266–372 (1979). This article includes a summary of earlier articles.
7.B. Nelson and B. Sheeks, “Fredholm determinants associated with Wiener integrals,” J. Math. Phys. 22, 2132–2136 (1981).
8.K.D. Elworthy, Stochastic Differential Equations on Manifolds (Cambridge University, Cambridge U.K., 1982).
9.A path is said to be if where
10.Here and in the rest of this paper we use the Einstein summation convention over repeated indices.
11.In standard applications, D is a differential operator and G the corresponding Green operator, taking into account the boundary conditions of the domain of D.
12.It is not true, as is still often stated, that the case has no mathematical foundation. See for example references 13–16.
13.C. Morette-DeWitt, “Feynman’s Path Integral, definition without limiting procedure,” Commun. Math. Phys. 28, 47–67 (1972);
13.C. Morette-DeWitt, “Feynman Path Integrals, I. Linear and affine techniques, II. The Feynman-Green function,” Commun. Math. Phys. 37, 63–81 (1974).
14.S.A. Albeverio and R.J. Ho/egh-Krohn, Mathematical Theory of Feynman Path Integrals, Springer-Verlag Lecture Notes in Mathematics, Vol. 523 (Springer-Verlag, Berlin, 1976).
15.P. Krée, “Introduction aux théories des distributions en dimension infinie” Bull. Soc. Math. France 46, 143–162 (1976),
15.and references therein, in particular, Seminar P. Lelong, Springer Verlag Lecture Notes in Mathematics Vols. 410 and 474 (1972)–(1974).
16.P. Cartier and C. DeWitt-Morette, “Intégration fonctionnelle; éléments d’axiomatique,” C. R. Acad. Sci. Paris 316, Série II, 733–738 (1993).
17.Y. Choquet-Bruhat and C. DeWitt-Morette, “Supplement to Analysis, Manifolds and Physics,” Armadillo preprint, Center for Relativity, University of Texas, Austin, TX 78712.
18.Linear changes of variable of integration in an infinite-dimensional space are sufficiently powerful and varied for the purposes of this paper. In a later publication we shall present nonlinear changes, simplifying and generalizing earlier works such as Ref. 19.
19.A. Young and C. DeWitt-Morette, “Time substitutions in stochastic processes as a tool in path integration,” Ann. Phys. 69, 140–166 (1986).
20.More precisely, for every smooth function f in we assume that the continuous function on T is the primitive of a function in
21.Any function in with compact support will do.
22.That is, continuous with continuous first-order derivatives.
23.C. DeWitt-Morette, K.D. Elworthy, B.L. Nelson, and G.S. Sammelman, “A stochastic scheme for constructing solutions of the Schrödinger equation,” Ann. Inst. H. Poincaré A 32, 327–341 (1980).
24.C. DeWitt-Morette and T.-R. Zhang, “Path integrals and conservation laws,” Phys. Rev. D 28, 2503–2516 (1983).
25.The credit for the technique used in deriving (III.38 is due to B. Nelson and B. Sheeks, Ref. 7. It is made simpler and more general here by not integrating by parts. The first calculation giving the ratio of functional determinants in terms of finite determinants can be found in Refs. 6 and 26.
26.C. DeWitt-Morette, “The semiclassical expansion,” Ann. Phys. 97, 367–399 (1976). (Correct a misprint p. 385,1.5, the reference is Ref. 5.)
27.The scaling factor has been chosen in such a way that no physical constant enters in this normalization. With the notations of section A 3.6, we have the dimensional equations
28.Since the semiclassical expansion will proceed according to the powers of
29.S.F. Edwards and Y.V. Gulyaev, “Path integrals in polar coordinates,” Proc. R. Soc. London Ser. A 279, 224 (1964).
30.For a discussion of functional integrals when paths take their values in a Riemannian space, see Refs. 6, 23 and 31.
31.C. DeWitt-Morette, “Quantum mechanics in curved spacetimes; stochastic processes on frame bundles,” in Quantum Mechanics in Curved Space-Time, edited by J. Audretsch and V. de Sabbata (Plenum, New York, 1990), pp. 49–87.
32.More explicitly, for a given frame ρ at a point x of M, is a linear map from to mapping u into for u in
33.For instance, two geodesics on intersect at two antipodal points; they are the developments of two halflines with one common origin.
34.We refer the reader to section IV.7 for an explicit example where we use this strategy.
35.W. Greub and H.-R. Petry, “Minimal coupling and complex line bundles,” J. Math. Phys. 16, 1347–1351 (1975).
36.Y. Choquet-Bruhat and C. DeWitt-Morette, Analysis, Manifolds and Physics Part I: Basics, Part II: 92 Applications, (North Holland, Amsterdam, 1989).
37.We give the formula in the oscillatory case The reader is invited to work out the formulas for the case The definition of is given in equation (IV.22).
38.Roughly speaking, a polarization is a foliation of M whose leaves are Lagrangian submanifolds of dimension n.
39.F.A. Berezin, “Quantization,” Math. USSR, Izvestija 8, 1109–1165 (1974).
40.D. Bar-Moshe and M.S. Marinov, “Berezin quantization and unitary representations of Lie groups,” to appear in Berezin Memorial (1994), edited by R. Dobrushin et al. (Am. Math. Soc., Providence, 1995);
40.also, “Realization of compact Lie algebras in Kähler manifolds,” J. Phys. A. 27, 6287–6298 (1994).
41.We express our formulas directly in terms of the path x, and have no need for the scaling introduced earlier.
42.For they reduce to the vector fields given in equations (IV.36) and (IV.37).
43.Notice that the transition probabilities are functions of F which admit a period a well-known physical effect.
44.That is: a σ-additive functional from the σ-algebra of Borel subsets of into the complex numbers (see Ref. 45 for instance). We use a simplified terminology by refering to μ as a “measure” on
45.K.R. Parthasarathy, Probability Measures on Metric Spaces (Academic, New York, 1967).
46.The initial J stands for Fresnel or Feynman according to the worshipping habits of the reader.
47.According to the standard definition, this is the l.u.b. of the set of numbers where runs over the set of all partitions of into a finite number of Borel subsets.
48.In the space X, the integrator depends on and s, and should be written more explicitly as Similarly for in space Y.
49.N. Bourbaki, Intégration, Chapitre 9 (Masson, Paris, 1982).
50.N. Bourbaki, Espaces Vectoriels Topologiques (Masson, Paris, 1981).
51.P. Cartier, “A course on determinants,” in Conformal Invariance and String Theory, edited by P. Dita and V. Georgescu (Academic Press, New York, 1989).
52.A positive-definite continuous quadratic form is not necessarily invertible. For instance, let X be the space of sequences of real numbers with and define the norm by We can identify X with its dual the scalar product being given by The quadratic form corresponds to the map taking into The inverse of D does not exist as a map from into since the sequence 1,2,3,… is unbounded.
53.A better notation should perhaps be
54.The classical momentum is obtained by evaluating at the point
55.N. Bourbaki, Fonctions d’une Variable Réelle (Masson, Paris, 1982).
56.The letter H stands for “horizontal.”
Article metrics loading...