Path Integrals for Specific Theories  

Non-Relativistic Mechanics > s.a. formulations of classical mechanics; histories formulations; statistical mechanics.
* Results: All quantities of interest can be obtained from the Green's function q', t'| q, t = G(q', t'; q, t):
- Spectrum: Use

q', t'| q, t = n n(q') n*(q) exp{–i En(t'–t)} ,

and perform Fourier transforms;
- Ground state: Euclideanize and take the limit t → .
* Smoothness: One has to include all continuous paths; The use of the Wiener measure then is ok, and includes the contribution from the action, even if the latter is not well-defined for all paths.
* Euclideanized version: It is often convenient to evaluate the integral (extending analytically the integrand to complex t) in imaginary time, with t = –i, and later extend to real t; Then exp(iS/) exp(–SE/), which looks like a partition function in statistical mechanics, with replacing kT, and thermal fluctuations replaced by quantum ones; Often SE is positive definite, and the integral thus exponentially damped.
* Relationships: Choosing a space of paths is equivalent to choosing a polarization in geometric quantization.
@ General references: Dirac PZS(33), RMP(45); Feynman RMP(48); DeWitt-Morette CMP(72), CMP(74), et al PRP(79); Klauder PRD(79); Hartle PRD(91); Anderson PRD(94)gq/93; Gudder JMP(98); Ansoldi et al EJP(00)qp/99 [propagator, simple].
@ Spectrum calculations: Feynman PR(55) [electron in polarizable lattice, lowest energy]; Stojiljkovic et al PLA(06) [efficient calculation].
@ Related topics: DeWitt-Morette & Zhang PRD(83) [conservation laws]; Kleinert PLB(89) [approximate formula]; Kleinert & Chervyakov PLB(99)ht, PLB(00)ht/99 [reparametrization invariance]; Grujic et al PLA(06) [E expectation values]; > s.a. parametrized theories, path integrals [including non-standard analysis].

Other Systems > s.a. black-hole radiation; constrained systems; quantum mechanics and formulations; quantum oscillator.
@ General references: Edwards & Gulyaev PRS(64) [free particle, curved coordinates]; Fujikawa NPB(97)ht/96, ht/96-in [H atom]; Strunz PRA(96) [open]; Grosche PS(98) [radial Coulomb]; Asorey et al JPCS(07)-a0712 [with boundaries].
@ Supersymmetric quantum mechanics: Catterall & Gregory PLB(00); Fine & Sawin a0705 [on a Riemann manifold, rigorous].
@ Solvable ones: Dykstra et al PLB(93); Grosche & Steiner ht/93; Grosche ht/93, JPA(95), JPA(96).
@ With H unbounded below: Carreau et al AP(90).
@ Simple potentials: Goodman AJP(81)sep [infinite well]; Nevels et al PRA(93) [infinite barrier].
@ Spinning: Lemmens PLA(96) [Ehrenfest model]; Cabra et al JPA(97); Ünal FP(98); Lopez & Stephany ht/00-in; Grinberg PLA(03) [Ising and XY models].
@ Non-commutative theories: Kempf ht/96; Dragovich & Rakic TMP(04)ht/03; Gitman & Kupriyanov EPJC(08)-a0707; > s.a. non-commutative fields.
@ Brownian motion: Lavenda PLA(79); Stepanov & Sommer JPA(90); Watabe & Shibata JPSJ(90); Botelho NCB(02), NCB(03).
@ Integrable systems: Anderson & Anderson AP(90).
@ Special configuration spaces: Farhi & Gutmann IJMPA(90) [half-line]; Toms ht/04 [curved, Schwinger action principle]; Dorlas & Thomas JMP(08) [discrete].
@ Related topics: Balaban CMP(85) [3D]; Caves PRD(86), PRD(87); Gangopadhyay & Home PLA(88); Gavazzi JMP(89) [fermions]; Zhao & Pan PLA(89) [Zassenhaus formula]; Junker JPA(90); Tolpin AP(90); Bitar et al PRL(91); Loo JPA(00)mp [with vector potential]; Muslih mp/00, mp/00 [singular system]; Greenberg & Mishra PRD(04)mp [parastatistics]; Kazinski et al JHEP(05)ht [systems with Lagrange structure, no action principle]; Field AP(06) [applications]; Andrzejewski et al a0904 [higher-derivative theories]; > s.a. casimir effect; Darboux Space; generalized quantum mechanics [supersymmetric]; knots in physics; quantum computing; quantum particle models.

Relativistic Particle Mechanics > s.a. quantum particle models.
* Remark: Here the path can move forward and backward in time; Interpreted as pair creation.
@ General references: Redmount & Suen IJMPA(93); Halliwell & Ortiz IJMPD(94)gq/93 [composition law for propagator]; Kleinert PLA(96) [spinless, Coulomb potential].
@ Fermions / Dirac: Kull & Treumann IJTP(99)qp; Gaveau & Schulman AP(00); also Feynman's chessboard.
@ Curved spacetime: Hawking CMP(77); Toms PRD(87); Grosche PLA(88); Kleinert AP(97)ht/96; Bastianelli et al PLB(00) [dimensional regularization]; Tanimura ht/01-in, IJMPA(01) [manifold with symmetries]; Krtous CQG(04).
@ Phase-space path integrals for systems on Riemannian manifolds: Kuchar JMP(83); Ferraro & Leston IJMPA(01).

Quantum Field Theory > see gauge theories, quantum gravity and quantum field theories.

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