Relationship between Quantum and Classical Mechanics  

In General > s.a. origin of quantum mechanics; quantum probabilities; quantum statistical mechanics [relationship with classical]; semiclassical states.
* Dirac's view: Classical mechanics is formulated using commutative quantities (c-numbers) while quantum mechanics requires non-commutative ones (q-numbers).
* Formalism: Two formalisms that treat classical and quantum theory similarly are the phase-space formulation of quantum theory (possibly based on the Wigner function) and the Koopman-von Neumann operator approach to classical mechanics.
* Scales: Quantum effects are hard to see in the macroworld, but the reason is more related with the way quantum systems interact with one another than with size per se.
* Non-classical aspects: Negativity (the necessity of negative values in quasiprobability representations of quantum states such as the Wigner representation) and contextuality; In multipartite systems entanglement plays a central role, but other measures of non-classicality in single systems can be converted into entanglement; > s.a. classical limit [degree of classicality].
* Issues: The study of the relationship between classical and quantum regimes of a theory, how the correspondence principle really works; The main questions are, Which states of the quantum theory have a classical interpretation? What predictions does the quantum theory make for the classical observables on them, and their fluctuations?
* Remark: In an abstract mathematical sense, quantum mechanics adds a metric on phase space to the symplectic structure used in classical mechanics.
@ Reviews and books: Park 90; Gutzwiller AJP(98)apr [interplay, RL]; Landsman qp/05-ch; Arndt & Zeilinger pw(05)mar; Bokulich 08; Vedral SA(11)jun.
@ General references: Woo AJP(86)oct; Landsberg FP(88); 't Hooft JSP(88); Hemion IJTP(90); Sibelius FP(89); Floyd IJMPA(00)qp/99 [trajectory representation]; Bergeron JMP(01)qp; Ghose FP(02)qp/01, & Samal FP(02)qp/01; Page FP(09)qp/02; Bartlett & Rowe JPA(03)qp/02; Mittelstaedt IJTP(05)qp/02 [and quantum logic]; Neumaier IJMPB(03)qp [axiomatic]; Loris & Sasaki PLA(04)qp/03 [simple theorems]; Krüger qp/04 [quantum mechanics does not imply classical mechanics]; Panković et al qp/04 [as phase transition]; Curtis & Ellis EJP(06) [perturbations and probabilities]; Dreyer JPCS(07)qp/06 [classicality]; Khrennikov qp/06 [mathematical]; Nikolić AIP(07)-a0707; Spekkens PRL(08)-a0711 [negativity and contextuality]; de Gosson a0808, de Gosson & Hiley FP(11)-a1001 [common features]; Caruso et al AP(11) [formal equivalence]; Kisil a1204 [critique of Dirac's point of view]; Klauder JPA(12)-a1204 [coexistence, enhanced quantization]; 't Hooft a1308-conf; Stoica a1402 [principle of quantumness]; Hung a1407 [using Fisher information]; Wolfe a1409-PhD [using entanglement, non-locality and contextuality to distinguish quantum theory from classical mechanics and other probabilistic theories]; de Gosson RVMP(15)-a1501 [and the metaplectic representation]; Rosaler Topoi-a1511 ['formal' vs 'empirical' approaches]; Zinkernagel a1603-in [can all systems be treated quantum-mechanically?]; Zinkernagel SHPMP(16)-a1603 [the classical/quantum divide]; Renkel a1701 [building a bridge].
@ Classical mechanics from quantum mechanics: Bracken qp/02 [as deformation of quantum mechanics]; Isidro et al IJGMP(09)-a0808, IJMPA(09)-a0808 [Ricci flow]; Carcassi a0902 [as many-particle limit]; Hájíček FP(09), JPCS(12) [maximum-entropy packets]; Blood a1009; Terekhovich a1210 [from the path integral formulation]; Oliveira PhyA(14) [transition induced by continuous measurements]; Hájíček JPCS(15)-a1412; Kastner a1707-talk [the role of distinguishability].
@ Quantum mechanics from classical mechanics: Heslot PRD(85); Ghose qp/00; Blasone et al PRA(05)qp/04, AP(05) [path-integral approach for 't Hooft's derivation]; Bracken qp/06-conf [semiquantum mechanics]; Khrennikov TMP(07) [quantum mechanics as approximation to classical statistical mechanics]; Bender et al JPA(08) [quantum-like behavior of systems with complex energy]; Wetterich a0809 [four-state system]; Raftery et al PRX(14)-a1312 [dissipation-induced, experimental observation].
@ Unified descriptions: Koide et al JPCS(15)-a1412 [generalized variational principle].
@ Quantum theory not from quantization: Isidro qp/01; Galapon JMP(04)qp/02.

Specific Aspects and Interpretations > s.a. Koopman-von Neumann formalism; quantum formalism [ambiguities] and foundations.
@ Alternative / interpolating theories: Tammaro FP(12) [non-classical, non-quantum theory]; Massar & Patra PRA(14)-a1403 [polygon theories]; Spekkens a1409 [quasi-quantization and epistemic restrictions on statistical distributions]; Fabris et al IJMPA(15)-a1509-proc [introducing quantum effects in classical theories]; > s.a. atomic physics [classical atomic models]; classical mechanics [non-quantum systems]; quantum probability theory.
@ And decoherent histories: Halliwell PRL(99)qp, qp/99-proc; Gell-Mann & Hartle PRA(14)-a1312 [adaptive coarse grainings]
@ In Bohm / pilot-wave interpretation: Shifren et al PLA(00) [effective potential]; Allori et al JOB(02)qp/01; Allori & Zanghì FP(09)qp/01-in; Poirier JCP(04)-a0802; Bowman FP(05); Trahan & Poirier JCP(06)-a0802, JCP(06)-a0802; Poirier & Parlant JPC(07)-a0803; Matzkin & Nurock SHPMP(08) [mismatch]; Poirier JChemP(08)-a0803; Struyve a1507; Romano a1603-in.
@ Related topics: Greenberg et al PRL(95) [invariant tori and matrix mechanics]; Wilkie & Brumer PRA(97), PRA(97) [Liouville dynamics]; Muga et al PLA(98) [observables]; Carcassi a1203 [homogeneous bodies and reducibility].
> Related topics: see decoherence; classical limit [including correspondence principle]; Contextuality; Correspondence Principle; Ehrenfest Dynamics; Ehrenfest Time; locality [localization, localized states]; macroscopic systems [including coupled/hybrid classical and quantum systems]; quantum chaos; quantum gravity [neither classical nor quantized theory]; quantum statistical mechanics; Weyl Quantization.


main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 1 aug 2017