Zoran Skoda diplomski Feynman integrals and semiclassical limit

Tema (Topic): Feynmanovi integrali i semiklasični limes

Smjer: fizika (ISTR)

Sažetak: Feynmanov pristup kvantnoj mehanici preko integrala po trajektorijama u terminima je klasičnih koordinata, pa je pogodan za proučavanje semiklasičnog limesa i semiklasičnog razvoja. Funkcionalni integral često nije egzaktno definiran, posebno u teoriji polja, nego je samo vrlo dobra i poučna heuristika.

Otvoreno je pitanje kako rigorozno kvantizirati kvantne teorije polja s netrivijalnim međudjelovanjem i simetrijama. Uspoređivanje klasične, simplektičke geometrije faznog prostora i formalizma funkcionalnih integrala vodi boljem razumijevanju samog koncepta kvantizacije. Cilj ovog diplomskog je upoznati se s osnovnim geometrijskim i fizikalnim aspektima koji su relevantni za pitanje kvantizacije u formalizmu Feynmanovih integrala te razumijevanje kratkovalne aproksimacije koja u tom slučaju vodi na semiklasični razvoj, a kod obične valne jednadžbe do limesa geometrijske optike. Student će upoznati i primjere kad naivni recepti za Feynmanov integral u konfiguracijskom prostoru trebaju korekcije koje su određene Maslovljevim indeksom iz geometrije faznog prostora. Sam semiklasični razvoj u više dimenzija, za razliku od 1-dimenzionalne WKB metode, također treba bolje razumijevanje Maslovljevog indeksa. Također je u praksi korisno znati kako kod nekih, mahom superintegrabilnih, sustava funkcionalni integral ima svojstvo lokalizacije, tj. da već vodeći član u semiklasičnom razvoju Feynmanovog integrala daje egzaktni rezultat.

Abstract: The Feynman’s approach to quantum mechanics via the path integral is in terms of classical coordinates, hence suitable for discussion of semiclassical expansion and limit. The functional integral is not rigorously defined in general, especially in field theory, but serves mainly as a very useful heuristics.

It is an open question how to quantize rigorously quantum field theories with nontrivial interaction and symmetries. A comparison of classical, symplectic geometry of the phase space and the formalism of functional integrals leads to better understanding of the very concept of quantization. This diploma work aims at acquaintance with basic geometric and physical aspects relevant for the quantization in the formalism of Feynman integrals and the understandinf of short-wave approximation which in this case leads to the semiclassical expansion and for the ordinary wave equation to the limit of geometrical optics. The student will learn some examples where the naive recipes for kthe Feynman integral in the configuration space need corrections in terms of Maslov index in geometry of phase space. The Maslov index plays role in the formulation of the semiclassical expansion in higher dimensions, which is more sophisticated than the 1-dimensional WKB method. It is also useful in praxis to understand how some, mainly superintegrable, systems, have the property of localization, where already the leading term in semiclassical expansion gives the exact result.

Basic literature

• R. P. Feynman, A. R. Hibbs, Quantum mechanics and path integrals, McGraw-Hill 1965.
• Leon A. Takhtajan, Quantum mechanics for mathematicians, Grad. Studies in Math. 95, AMS 2008: chapter 5, Path integral formulation of quantum mechanics; ch. 6, Integration in functional spaces
• Joel W. Robbin, Dietmar A. Salamon, Feynman path integrals and the metaplectic representation, Math. Z. 221 (1996), no. 2, 307–335, MR98f:58051, doi

Additional literature

• Joel W. Robbin, Dietmar A. Salamon, Phase functions and path integrals, Symplectic geometry (Proc., ed. D. Salamon), 203–-226, London Math. Soc. Lecture Note Ser. 192, Cambridge Univ. Press 1993, RobbinSalamonPhaseFunctionsPathIntegrals.djvu
• J.B. Keller, Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems, Annals of Physics 4 (1958), 180–188, MR99207 doi
• Richard J. Szabo, Equivariant cohomology and localization of path integrals, Lecture Notes in Physics, N.S. Monographs 63. Springer 2000. xii+315 pp. (preprint version: Equivariant localization of path integrals, hep-th/9608068)
• Pierre Cartier, Cécile DeWitt-Morette, Functional integration,

  J. Math. Phys. 41 (2000), no. 6, 4154–4187, doi MR2001m:58023

• nlab:semiclassical approximation
• P. Muratore-Ginanneschi, Path integration over closed loops and Gutzwiller’s trace formula, Phys. Rep. 383 (2003), no. 5-6, 299–-397. MR2004g:81060, doi
• A. S. Mishchenko, V. E. Shatalov, B. Yu. Sternin, Lagrangian manifolds and the Maslov operator, Springer Series in Soviet Mathematics, 1990. x+395 pp. MR91e:58191 (Russian orig. Мищенко А.С., Стернин Б.Ю., Шаталов В.Е. Лагранжевы многообразия и метод канонического оператора. М. Наука, 1978г. 352 с.)
• V. P. Maslov, Stationary-phase method for Feynman’s continual integral, Theoret. and Math. Phys., 2:1 (1970), 21–25; Russian original: ТМФ, 2:1 (1970), 30–35 pdf.
• P. Cartier, C. DeWitt-Morette, Functional Integration. Action and symmetries, Cambridge University Press, 2004.

Very specialized related references

• Lorenzo Zanellia, Paolo Guiottoa, Franco Cardin, Integral representations of the Schrödinger propagator, Rep. Math. Phys. 62 (2008), no. 1, 19–-56, MR2009m:81075, doi
• Johan Ulm, Quantization as a Kan extension (pdf), preprint 2009 (related to An Exercise in Kantization and ncafe quantum conjecture)
• А.С.Мищенко, Б.Ю.Стернин, В.Е.Шаталов, Метод канонического оператора Маслова, Комплексная теория, Московский институт электронного машиностроения, М., 1974.

Related entries: semiclassical approximation, deformation quantization, geometric quantization, Maslov index, microlocal analysis