Path Integral Approach to Quantum Theory  

In General [> s.a. formulations of quantum theory.]
* Idea: One gives a set of histories, the amplitude for each history, a rule for summing over histories (measure), and a complete and exclusive set of observables; Then we can find (relative) probabilities; For example, the probability amplitude that a system with an action S, in state |a at time t, will be in state |b at time t' is given by

b, t' | a, t = tt' (all interpolating q) exp{iS[q]/},

summing over all paths q(t) that move forward in time; Shows that quantum mechanics is a generalization of classical stochastic theory in which the probability measure is replaced by a quantum measure.
@ History: Derbes AJP(96); Antoci & Liebscher AFLB(96)phy/97 [Wentzel as forerunner]; Inomata & Junker in(99)qp/98; Klauder qp/03; Albeverio & Mazzucchi JSP(04) [status]; {> s.a. #Wentzel}.
@ Texts, reviews: Feynman & Hibbs 65; Fried 72; DeWitt-Morette et al PRP(79); Marinov PRP(80); in Felsager 81; Schulman 81; Khandekar & Lawande PRP(86); Khandekar et al 91; Scadron 91; Grosche ht/93; DeWitt-Morette ed-JMP(95)#5; Grosche & Steiner 98 [handbook]; Cartier & DeWitt-Morette JMP(00); MacKenzie qp/00-ln; Ingold qp/02-ln [and dissipative systems]; 't Hooft ht/02-in; Zinn-Justin 04; Simon 05; Feynman & Brown ed-05 [PhD thesis etc, r CQG(07)]; Cartier & DeWitt-Morette 07.
@ Texts, heuristic: Ramond 81; Rivers 87; Kleinert 06 [with other applications]; Das 06.
@ Texts, constructive: Glimm & Jaffe 87; Rivasseau 91.
@ Texts, III: Swanson 92; Roepstorff 94; Chaichian & Demichev 01; Dittrich & Reuter 01.

The Measure
* Choice: When the space of histories is a linear space, use a Gaussian measure.
@ Mathematical: Albeverio & Hoegh-Krohn 76; Cameron & Storvik 83; Yamasaki 85; Klauder in(86).
@ Related topics: Swanson PRA(94) [and canonical transformations]; Dynin LMP(98)m.FA [time slicing construction].

Closed-Time Version
* Generating function:

Z[J+, J–]:= J –0–|0+J + = + – exp{i (S[+] + J+[+] – S*[–] – J–[–])} .

@ References: Schwinger JMP(61); Keldysh ZETF(64); Korenman AP(66); Chou et al PRP(85); Manoukian NCB(87), NCA(88); Jordan PRD(86) [in curved spacetime]; Calzetta & Hu PRD(87) [in cosmology]; Cooper ht/95.

Concepts and Techniques > s.a. coherent states; lattice field theory; partial differential equations.
* Regularization: Can be done by using a lattice (the most common), or Klauder's continuous time regularization.
@ Hamiltonian and Lagrangian: Grosse-Knetter PRD(94)hp/93, hp/93-PhD; Lopuszanski mp/00 [classically equivalent L's].
@ Change of variables: Smolyanov & Smolyanova TMP(94); Kleinert & Chervyakov PLA(00)qp.
@ In phase space: Mizrahi JMP(75) [and Weyl transforms]; Takatsuka PRL(88); Sonego PRA(90) [Wigner functions, etc]; Marinov PLA(91); Farhi & Gutmann AP(92); Niemi & Tirkkonen AP(94)ht/93; Whelan gq/97-in [skeletonization]; Klauder qp/97; Shabanov & Klauder PLB(98)qp [symplectic manifolds]; Ferraro & Leston IJMPA(01)gq/00 [in curved spacetime]; Albeverio et al JMP(02); Ichinose CMP(06) [mathematical theory].
@ Other formulations: Hegseth qp/04 [in momentum space]; Stannett a0805-in [finitary reformulation].
@ World-line formalism: Schmidt & Schubert ht/98-in; Bastianelli & Zirotti NPB(02); Schubert ht/07-in [for QED].
@ Non-standard analysis: Nakamura JMP(91); Loo JMP(99)mp/00, JPA(00)mp [general], JMP(99)mp/00 [sho].
@ And renormalization: Henderson & Rajeev JMP(97)ht/96.
@ Approximation methods: Blau et al PLB(90) [geometrical, WKB]; Kleinert PLB(92); Wasilkowski & Wozniakowski JMP(96); Horvathy qp/07 [semiclassical, Maslov correction]; Paulin et al JSP(07) [low-temperature behavior]; Smirnov JPA(08) [limiting procedures]; Thrapsaniotis JPA(08) [based on central limit theorem].
@ Numerical: Wandzura PRL(86) [Monte Carlo]; Gerry & Kiefer AJP(88); Onofri & Tecchiolli PS(88); Sauer phy/01-in [rev]; Bogojevic et al PRL(05) [acceleration]; Moch & Schneider a0709-in [using difference equations].
@ And boundary conditions: Jaroszewicz PRL(88); Asorey et al qp/06-in [cannot describe highly non-local ones].

Related Topics > s.a. canonical quantum mechanics [factor ordering]; quantum systems; regge calculus; spacetime foam.
@ And configuration space topology: Laidlaw & Morette DeWitt PRD(71); Tanimura & Tsutsui AP(97) [on G/H].
@ And stochastic mechanics: Wang PLA(89); Boos JMP(07).
@ And classical mechanics: Ajanapon AJP(87); Gozzi PLB(88); Rivero qp/98; > s.a. classical mechanics, field theory.
@ Particle vs field: van Holten NPB(95)ht; Fujita a0801 [critical review].
@ Generalizations: Kauffmann ht/95 [arbitrary canonical transformations]; Djordjevic & Dragovich mp/00-in, MPLA(97)mp/00 [p-adic]; Smailagic & Spallucci JPA(03) [non-commutative plane]; Acatrinei JPA(07) [higher-order Lagrangians].
@ Other topics: Menskii TMP(83) [and group theory], TMP(92) [and continuous measurement]; Popov 88 [and collective excitations]; Sorkin in(90) [and causality]; Brun gq/94 [and decoherence]; Halliwell PLA(95)qp [path decomposition expansion]; Marchewka & Schuss PRA(00)qp/99 [and currents]; Samson JPA(00)qp [t discretization]; Shankaranarayanan & Padmanabhan IJMPD(01) [duality, and electromagnetism]; Dreisigmeyer & Young mp/01 [as semigroups]; Ashmead qp/03 [and fluctuations in time?]; Ord et al qp/04, FPL(06) [phase, physical basis]; Tumulka EJP(05)qp [and Bohmian formulation]; Bogojevic & Belic PLA(05) [jaggedness of paths]; Putrov ht/06 [energy representation]; Furuya JMP(06) [Riemann-type integral]; Jackiw a0711 [and charge fractionalization]; > s.a. Peierls Brackets; Stationary Phase; Steepest Descent; Trace Formulas.

Systems > see particle statistics; quantum field theories; quantum gauge theories; quantum gravity; other theories [including spectrum estimation].

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