Path-Integral Approach to Quantum Theory  

In General > s.a. formulations of quantum theory.
* History: Introduced in quantum mechanics by Feynman, they have since pervaded all areas of physics where fluctuation effects are important.
* Idea: One converts a problem formulated in terms of operators into one of sampling classical paths with a given weight; 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 transition amplitude that a system with an action S, in state \(|a\rangle\) at time t, will be in state |b\(\rangle\) at time t' is given by

\(\langle\)b, t' | a, t\(\rangle\) = tt' \(\cal D\)(all interpolating q) exp{iS[q]/\(\hbar\)},

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.
@ General references: Feynman PhD(42); DeWitt-Morette et al PRP(79); Marinov PRP(80); Khandekar & Lawande PRP(86); Cartier & DeWitt-Morette JMP(00); Ingold LNP(02)qp [and dissipative systems]; Dowker et al JPA(10)-a1002 [and Hilbert space]; Zinn-Justin TMP(11) [rev]; Moshayedi a1902-ln.
@ History: Derbes AJP(96)jul; Antoci & Liebscher AFLB(96)phy/97 [Wentzel as forerunner]; Inomata & Junker in(99)qp/98; Klauder qp/03; Albeverio & Mazzucchi JSP(04) [status]; Parrochia a1907; Hari Dass a2003 [Feynman and Dirac path integrals]; Robson et al a2105 [+ optics and photonics]; {> s.a. #Wentzel}.
@ Introductions / Texts: Brown ed-05; Feynman & Hibbs 65; Fried 72; in Felsager 81; Schulman 81; Scadron 91; Khandekar et al 93; Grosche ht/93-ln; DeWitt-Morette ed-JMP(95)#5; Grosche & Steiner 98 [handbook]; MacKenzie qp/00-ln; 't Hooft ht/02-conf; Zinn-Justin 04; Simon 05; Feynman & Brown ed-05 [PhD dissertation etc, r CQG(07)]; Cartier & DeWitt-Morette 07; Klauder 11; Rosenfelder a1209-ln; Fahssi a1303-ln; Nguyen JMP(16)-a1505 + YouTube [mathematical]; Gozzi et al 16.
@ Texts, heuristic: Ramond 81; Rivers 87; Kleinert 09 [with other applications]; Das 19.
@ Texts, constructive: Glimm & Jaffe 87; Rivasseau 91.
@ Texts, III: Swanson 92; Roepstorff 94; Chaichian & Demichev 01; Mazzucchi 09; Dittrich & Reuter 20.

The Measure > s.a. integration theory [functional integrals].
* 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); Botelho a0902.
@ Types of paths used: Bogojevic & Belic PLA(05) [jaggedness of paths]; Koch & Reyes IJGMP(15)-a1404 [with time-scale parameter, using differentiable paths].
@ Related topics: Swanson PRA(94) [and canonical transformations]; Dynin LMP(98)m.FA [time slicing construction].

Other Formulations > s.a. Polymer Representation; Schwinger's Action Principle.
* Closed-time version: The generating function is

Z[J+, J]:= J\(\langle\) 0|0+ \(\rangle\)J+ = \(\cal D\)φ+ \(\cal D\)φ exp{i (S[φ+] + J+[φ+] − S*[φ] − J[φ])} .

@ General references: Hegseth qp/04 [in momentum space]; Stannett a0805-conf [computable formulation]; Stoyanovsky a0808 [Green-function-like distributions]; Rubin TMP(08) [calculation method, differential equation]; Kochan APoly-a0812 [using only classical equations of motion], IJMPA(09), JGP(10), PRA(10)-a1001 [non-Lagrangian systems]; Ootsuka & Tanaka PLA(10)-a0904 [Lagrangian, in terms of Finsler geometry]; Nagao & Nielsen PTP(13)-a1205 [complex action theory, with future included]; Mou et al a1902 [real time, complex field variables]; Buchholz & Fredenhagen a1905 [and dynamical algebras].
@ 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-proc [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]; Yamashita JMP(11) [in terms of Brownian motions and stochastic integrals].
@ World-line formalism: Schmidt & Schubert ht/98-conf; Bastianelli & Zirotti NPB(02); Schubert AIP(07)ht [for QED].
@ Closed-time version: 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: It 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, PhD(93)hp; Łopuszański mp/00 [classically equivalent Lagrangians].
@ Change of variables: Smolyanov & Smolyanova TMP(94); Kleinert & Chervyakov PLA(00)qp; Johnson-Freyd a1003 [for fields taking values on a general fiber bundle]; Cugliandolo et al a1806 [with one degree of freedom].
@ Non-standard analysis: Nakamura JMP(91); Loo JMP(99)mp/00, JPA(00)mp [general], JMP(99)mp/00 [sho].
@ Approximation methods: Blau et al PLB(90) [geometrical, WKB]; Kleinert PLB(92); Wasilkowski & Wozniakowski JMP(96); Horváthy CEJP(11)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].
@ Diagrammatic expansions: Halliwell PLA(95)qp [path decomposition expansion]; Johnson-Freyd LMP(10)-a1003, JMP(10)-a1004.
@ Numerical: Wandzura PRL(86) [Monte Carlo]; Gerry & Kiefer AJP(88)nov; Onofri & Tecchiolli PS(88); Samson JPA(00)qp [time discretization]; Sauer phy/01-in [rev]; Bogojevic et al PRL(05) [acceleration]; Moch & Schneider PoS-a0709 [using difference equations]; Grimsmo et al PLB(13) [consequences of modified discrete-time lattice actions]; Alexandru et al a2007 [Monte Carlo method, approaches to the sign problem].
@ And boundary conditions: Jaroszewicz PRL(88); Asorey et al qp/06-proc [cannot describe highly non-local ones].
@ Related topics: Henderson & Rajeev JMP(97)ht/96 [and renormalization]; Jizba & Kleinert PRD(10)-a1007 [superstatistics approach]; Sekihara a1201 [Metropolis algorithm]; Halliwell & Yearsley PRD(12)-a1205, JPCS(13)-a1301 [amplitudes for spacetime regions and the quantum Zeno effect]; Sokolovski PRD(13)-a1301 [probabilities for classes of paths in spacetime]; LaChapelle a1505 [functional integral representations of C*-algebras]; Jizba & Zatloukal PRE(15)-a1506 [local-time representation]; Cahill a1501 [without using the Hamiltonian, for theories that are not quadratic in time derivatives]; Amaral & Bojowald AP(18)-a1601 [and the problem of time]; Malgieri et al AJP(16)sep [using energy-dependent propagators]; Kochetov a1811 [continuous-time formulation]; Morales-Ruiz a1910 [differential Galois approach]; Trapasso a2004 [time-frequency analysis].
> Related topics: see Peierls Brackets; Stationary-Phase Approximation; Steepest-Descent Approximation; Trace Formulas.

Related Topics > s.a. canonical quantum mechanics [canonical transformations, factor ordering]; quantum systems; regge calculus; spacetime foam.
@ And configuration-space topology: Laidlaw & Morette DeWitt PRD(71); Tanimura & Tsutsui AP(97) [on G/H]; Mouchet a2010; > s.a. topology in physics.
@ And stochastic mechanics: Wang PLA(89); Boos JMP(07).
@ Particle vs field: van Holten NPB(95)ht; Fujita a0801 [critical review].
@ In non-commutative spaces: Smailagic & Spallucci JPA(03) [non-commutative plane]; Mignemi & Štrajn PLA(16)-a1509 [1D and 2D Snyder space].
@ Other generalizations: Kauffmann ht/95 [arbitrary canonical transformations]; Djordjević & Dragovich mp/00-proc, MPLA(97)mp/00 [p-adic]; Acatrinei JPA(07) [higher-order Lagrangians]; Lloyd & Dreyer a1302 [universal path integral, as sum over all computable structures]; Savvidy MPLB(15)-a1501 [integral over random surfaces, gonihedric action]; > s.a. generalized uncertainty principle.
@ 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]; Marchewka & Schuss PRA(00)qp/99 [and currents]; 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]; Putrov TMP(08)ht/06 [energy representation]; Furuya JMP(06) [Riemann-type integral]; Jackiw in(08)-a0711 [and charge fractionalization]; Witten a1009 [and branes in a two-dimensional A-model]; Green et al a1607 [and entangled states]; Gozzi PLA(18)-a1702 [quantum identities for the action]; Terekhovich a1909-in [ontology]; > s.a. pilot-wave interpretation; quantum measurements [stochastic path integral formalism]; representations [tomography].

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

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