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 \(\langle\)q', t' | q, t\(\rangle\) = G(q', t'; q, t):
- Spectrum: Use

\(\langle\)q', t' | q, t\(\rangle\) = n ψn(q') ψn*(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 = −iτ, and later extend to real t; Then exp(iS/\(\hbar\)) \(\mapsto\) exp(−SE/\(\hbar\)), which looks like a partition function in statistical mechanics, with \(\hbar\) replacing kT, and thermal fluctuations replaced by quantum ones; Often \(S_{\rm E}\) 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]; Stojiljković 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]; Greenberg & Mishra PRD(04)mp [parastatistics]; Field AP(06) [applications]; Grujić et al PLA(06) [energy expectation values]; Albeverio et al a1907 [with magnetic field, rigorous]; > s.a. parametrized theories; path integrals [including non-standard analysis].

Other Systems > s.a. black-hole radiation; constrained and dissipative systems; quantum theory and formulations; quantum oscillator.
@ General references: Edwards & Gulyaev PRS(64) [free particle, curved coordinates]; Fujikawa NPB(97)ht/96, ht/96-proc [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 CMP(08)-a0705 [on a Riemann manifold, rigorous]; Ludewig a1910.
@ 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]; Duru a1911 [pendulum potential].
@ Spinning: Lemmens PLA(96) [Ehrenfest model]; Cabra et al JPA(97); Ünal FP(98); Lopez & Stephany ht/00-proc; Grinberg PLA(03) [Ising and XY models]; Kowalski-Glikman & Rosati a1912 [arbitrary spin]; > s.a. quantum particles.
@ Non-commutative theories: Kempf ht/96; Dragovich & Rakić TMP(04)ht/03; Gitman & Kupriyanov EPJC(08)-a0707; Neves & Abreu a1206.
@ Brownian motion: Lavenda PLA(79); Stepanov & Sommer JPA(90); Watabe & Shibata JPSJ(90); Botelho NCB(02), NCB(03).
@ Integrable systems: Anderson & Anderson AP(90).
@ On discrete spaces: Dorlas & Thomas JMP(08); Ajaib a1403 [1D, and Euclidean propagator]; Penney et al NJP(17)-a1604 [quantum circuit]; Tyagi & Wharton a2103 [for multi-qubit entangled states]; > s.a. cellular automaton.
@ Other configuration spaces: Farhi & Gutmann IJMPA(90) [half-line]; Toms ht/04 [curved, Schwinger action principle]; Inomata & Junker PLA(12) [on a conical space]; Sakoda a1804, a1809 [particle in a finite interval and on the half-line].
@ With no action / Lagrangian: Kazinski et al JHEP(05)ht [Lagrange structure]; Sharapov IJMPA(14)-a1408 [Peierls bracket].
@ 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]; Andrzejewski et al PTP(11)-a0904 [higher-derivative theories]; Cahill a1501, Amdahl & Cahill a1611 [actions that are not quadratic in their time derivatives]; Vanchurin a1912 [strongly coupled, dual system path integrals]; > s.a. casimir effect; Darboux Space; knots in physics; quantum computing; quantum particle models; scattering.

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]; Chiou CQG(13) [timeless, as integral over paths in the constraint surface]; Padmanabhan a1901 [lattice regularization, conceptual and pedagogical issues]; Koch & Muñoz PRD-a2012.
@ Fermions / Dirac: Kull & Treumann IJTP(99)qp; Gaveau & Schulman AP(00); Ichinose CMP(14); see also Feynman's chessboard.
@ In 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-proc, IJMPA(01) [manifold with symmetries]; Krtouš CQG(04); Singh & Mobed MPLA(12)-a1008 [new approach].
@ Phase-space path integrals for systems on Riemannian manifolds: Kuchař JMP(83); Ferraro & Leston IJMPA(01).

Quantum Field Theory > see gauge theories, quantum gravity and quantum field theories; non-commutative fields and gauge theories.

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