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.

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