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Zbl 1222.46001
Albeverio, Sergio A.; Høegh-Krohn, Raphael J.; Mazzucchi, Sonia
Mathematical theory of Feynman path integrals. An introduction. 2nd corrected and enlarged edition. (English)
[B] Lecture Notes in Mathematics 523. Berlin: Springer. x, 176~p. EUR~32.05 (2008). ISBN 978-3-540-76954-5/pbk

The first edition of this book [(1976; Zbl 0337.28009)] has become a classic in its field. It dates back to the 1970s when the names of the two authors stood for a prolific collaboration in mathematical physics. They made important contributions to the model building in constructive quantum field theory and championed Dirichlet form methods for quantum (field) theory. The time was then ripe for a central role of infinite dimensional and stochastic analysis in quantum physics, and thus also to take a fresh look at the Feynman integral. The Feynman integral is something like the Dr Jekyll and Mr Hyde of physics and mathematics. For the physicist it is the workhorse for the quantization of any classical theory. From a purely classical input -- the Lagrangian -- it produces the corresponding quantum model directly, and seemingly without ambiguity. For the mathematician almost any such achievement of the physicists -- in particular the less trivial ones -- is either ill-defined or outright meaningless. This begins with the fact that, contrary to the assertion of Gelfand and Yaglom, and as pointed out early on by Cameron, the Feynman integral is not an integral with respect to some suitable measure on the (infinite dimensional) space of paths over which it is supposed to ``average'' (one also speaks of a ``sum over histories''). This fact has generated a multitude of approximation schemes. Some used finite dimensional approximations, such as replacing the paths by polygons, others used analytic continuation schemes, notably in the time parameter, which informally links the ill-defined Feynman integral to the well defined Feynman-Kac formula for the heat equation. Another approach in the setting of infinite dimensional analysis tries to work with the Feynman ansatz directly, reinterpreting the integral as the action of a generalized function as in the theory of distributions or generalized functions. White noise analysis furnishes a suitable framework for this extension of distribution theory to infinite dimensional analysis. \par With any of these approaches the challenge does not stop there. The universality of the Feynman ansatz is deceptive. In operator based quantum physics, dynamics is typically given in terms of singular perturbations of self-adjoint operators which require case by case existence theorems. Likewise in the various Feynman path ``integral'' approaches, the decisive question from a mathematical point of view is which types of dynamics they can reliably handle. \par How does the ``Mathematical Theory of Feynman Path Integrals'' text of 1976 fit into this picture? The authors' approach falls into the framework of infinite dimensional analysis. It is based on a definition of oscillatory ``Fresnel integrals'' on a real separable Hilbert space of continuous ``paths''. They are given initially by their action on exponentials and from these they are extended linearly to Fourier integrals with respect to bounded complex measures on a Hilbert space $\mathcal{H}$, obtaining thus a Banach algebra of Fresnel integrable functions on $\mathcal{H}.$ Properties such as Euclidean invariance and a Fubini theorem are shown for the Fresnel integral, and in the central part of the treatise, solutions of the Schrödinger equation $$i\partial _{t}\psi (x,t)=-\tfrac{1}{2}\,\Delta _{x}\psi (x,t)+V(x)\psi (x,t)$$ are constructed as Fresnel integrals over paths. The method is shown to work for potentials $V$ and initial values $\psi (\,.\, ,0)$ which are Fourier integrals with regard to bounded complex measures on $\mathbb{R}^{n}.$ {\it For these the Feynman integral is thus given a direct meaning, without recourse to finite dimensional or any other limiting procedures!} \par The remainder of the monograph is devoted to extensions of this principal result. These touch upon the extension from free actions to more general quadratic ones (oscillators), including Gibbs and ground state expectations, and from there to quasifree states and relativistic free fields. \par While the scope of manageable interactions was still limited, the 1976 book set a new standard with regard to those two opposing challenges: following as closely as possible Feynman's intuitive and direct construction and maintaining proper mathematical rigor. \par The new edition goes way beyond the habitual corrections and additions; roughly half of its volume is taken up by a chapter called Some Recent Developments. A closer look reveals five subchapters. \par The first one takes a fresh look at infinite dimensional oscillatory integrals on the basis of Hörmander's definition in the finite dimensional case. Infinite dimensional ones are then obtained by a limiting procedure which extends the authors' earlier definition. In the next section this is put to use to go beyond bounded perturbations: the important example of fourth order polynomials $V$ can now be treated, a case that, contrary to the models treated in the first edition, calls for a non-perturbative approach. There follows a section on the so-called semiclassical approximation where stationary phase methods are used for an approximate solution of the Feynman integrals. Alternate approaches are treated next: analytic continuation from the heat equation with its Feynman-Kac path integrals, sequential approximations based on the Trotter product formula, and white noise analysis where the Feynman ``integral'' is understood and constructed as a generalized function, i.e., a continuous linear mapping on a suitable test function space (the larger spaces of generalized functions introduced in [{\it Yu. G. Kondratiev} and {\it L. Streit}, ``Spaces of white noise distributions: Constructions, descriptions, applications. I'', Rep. Math. Phys. 33, No. 3, 341--366 (1993; Zbl 0814.60034)] are more useful than the one presented in the present monograph). To develop all of these methods in full detail would have been impossible in the present setting; for the reader who wants to know more, those approaches are very well documented, and further commented in ``Notes''. (The number of references has jumped from 56 to an extremely useful 475 in the present edition, for the next edition one might suggest inclusion also of [{\it Z.-Y. Huang} and {\it J.-A. Yan}, Introduction to infinite dimensional stochastic analysis. Dordrecht: Kluwer Academic Publishers (2000; Zbl 0972.58013)].) The final section is dedicated to a number of recent applications: magnetic fields, time dependent perturbations, phase space integrals as championed by J.~Klauder, models for quantum mechanical measurement, and finally a mathematically solid reformulation of Witten's path integral for the Chern-Simons gauge field theory model. \par It is good to have this new book. Not only for the more recent results it contains, but also as a point of departure for so many questions that are still open in the realm of infinite dimensional oscillatory integrals. Here are a few interesting topics: -- A systematic approach to changes of variables in infinite dimensional oscillatory integrals. -- Is it a mere coincidence that admissible polynomial potentials in the sense of {\it H. Doss} [``Sur une résolution stochastique de l'équation de Schrödinger à coefficients analytiques'', Commun. Math. Phys. 73, 247--264 (1980; Zbl 0427.60099)] are quasi exactly solvable in the sense of {\it A. G. Ushveridze} [Quasi-exactly solvable models in quantum mechanics. Bristol: Institute of Physics Publishing (1994; Zbl 0834.58042)]? -- Can we better understand why for vast classes of singular potentials $V$ the Feynman integral for the fundamental solution of the Schrödinger equation nevertheless admits a convergent perturbation expansion, while the Feynman-Kac integral for the corresponding heat equation ``of course" does not; cf.\ [{\it T. Kuna, L. Streit} and {\it W. Westerkamp}, ``Feynman integrals for a class of exponentially growing potentials'', J. Math. Phys. 39, No. 9, 4476--4491 (1998; Zbl 0934.58014)]? \par To discuss these and many other topics -- and as a wonderful friend -- we doubly miss Raphael Høegh-Krohn who passed away so terribly early.
[Ludwig Streit (Bielefeld)]

MSC 2000:: *46-02 Research monographs (functional analysis)
81-02 Research monographs (quantum theory)
81S40 Path integrals in quantum mechanics
46G12 Measures and integration on abstract linear spaces
58D30 Spaces and manifolds of mappings in appl. to physics
28C20 Set functions and measures and integrals in infinite-dim. spaces
47N50 Appl. of operator theory in quantum physics

Keywords: constructive quantum field theory; Feynman integral; white noise analysis; Fresnel integrals; Schrödinger equation; Feynman-Kac path integrals

Citations: Zbl 0337.28009; Zbl 0814.60034; Zbl 0972.58013; Zbl 0427.60099; Zbl 0834.58042; Zbl 0934.58014

Cited in: Zbl 1153.81325

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