Path Integrals for Specific Theories| 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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