Path-Integral Approach to Quantum Theory| 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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