 Probability in Quantum Physics

In General > s.a. foundations; interpretations; hidden variables; many-worlds interpretation; pilot-wave theory; quantum mechanics.
* Role: Probabilities are an essential part of the interpretation, obtained from inner products (0 ≤ cos2θ ≤ 1).
* Idea: Probabilities do not behave like in classical physics; The basic objects for questions Q are probability amplitudes A(Q), from which the probabilities are calculated as P(Q) = |A(Q)|2; Quantum probability is a variant of contextual probability.
* Calculation: In general, when an event can occur in several different ways, its probability amplitude is the sum of those for the individual ways (interference),

A(Q) = ∑i A(Q, i) ;

However, if an experiment is capable of determining which alternative is followed, then interference is lost; For example, if Q is a question whose answer depends on what the system does up to a time t, then

P(Q) = ∑x |A(x at tf, a)|2 = |A(Q)|2 ,    where    tf > t .

* Remark: Independence on t in the above calculation is equivalent to unitarity; One point of view is that probability doesn't change in time; We often ignore the fact that when we talk about time dependence we're talking about different experiments.
* QBism: Quantum Bayesianism or the quantum-Bayesian approach to quantum theory, the personalist Bayesian view of probability in quantum theory; This view is widely held in general but not by many physicists; Some physicists who have argued for it are Caves, Fuchs, Schack, and Mermin (Don Page thinks that there are both frequentist and personalist probabilities); It has profound implications for the meaning of quantum mechanics.
> Related topics: see Probability Current; Wigner's Friend.

References > s.a. measurement in quantum theory; probabilities in physics [negative probabilities, general probabilistic theories]; QBism.
@ Intros, reviews: Cufaro Petroni FP(92); Meyer 95; Sudarshan qp/01; Rédei & Summers SHPMP(07)qp/06 [and von Neumann algebras]; Sontz a0902 [simple introduction]; Janotta & Hinrichsen JPA(14)-a1402; Khrennikov 16 [and classical]; Svozil a1707 [and correlations]; Schleißinger a2001 [simple].
@ General references: Accardi PRP(81); Accardi & von Waldenfels ed-85; Van den Berg et al PhSc(90)mar; Halpin PhSc(91)mar; Youssef MPLA(91); Farina AJP(93)may; Gudder IJTP(93); Ismael BJPS(96) [conceptual]; Velleman AJP(98)nov; Noyes & Etter PE(99)qp/98; Adler qp/00-proc [postulated vs emergent]; Barnum et al PRS(00)qp/99; Khrennikov qp/01 [context-dependent]; Rylov qp/01 [dynamically based]; Belavkin IDAQP(00)m.PR/05 [history]; Dreyer qp/06 [emergent probabilities]; Lehrer & Shmaya PRS(06) [qualitative approach]; Tipler qp/06; de la Torre EJP(08); Rave a0806 [interpretation with closed loops and phases]; Page PLB(09) [insufficiency of quantum state]; Bub a1005-in [and information theory]; Janssens PhD(10)-a1011; Leifer & Spekkens PRA(13)-a1107 [quantum theory as a causally neutral theory of Bayesian inference]; Blackman & Hsiang PE-a1110 [from large number of degrees of freedom]; Pfister MS-a1203; Aerts & Sassoli de Bianchi a1401, a1401; Hiley LNCS(14)-a1408-conf [structure processes, and non-commutative probability theory]; Yukalov & Sornette PTRS-a1601 [general definition]; Porta Mana a2007 [conditional probabilities]; Niestegge Ent(20)-a2009 [algebraic origin].
@ And quantum foundations: Khrennikov qp/01-conf; Wilce FP(10); Holik et al AP(14)-a1211 [origin]; Fröhlich & Schubnel a1310; de Ronde in-a1506 [probabilities as objective knowledge]; Garola a1806 [epistemic interpretation]; de Ronde et al a1903 [interpretation]; > s.a. origin of quantum theory.
@ From classical probability: Slavnov TMP(06)qp/07; Grigorescu PhyA(08)-a0711 [classical Fokker-Planck equation and quantum Brownian motion]; Groessing et al a1403.
@ And classical probabilities: Khrennikov & Loubenets FP(04)qp/02; Khrennikov JMP(02), AIP(05)qp/03; Nyman IJTP(10)-a0906 ["quantum-like representation algorithm"]; Farenick et al JMP(11)-a1110 [classical and non-classical randomness, in terms of operator-valued measures]; Hardy a1303-ch; Dzhafarov & Kujala FP(14)-a1305, a1312-PLoS; Khrennikov a1406-conf; Hofmann a1606-proc; Garner & Müller a2004.
@ And decoherence: Bacciagaluppi SHPMP(07)qp [time-directed probability]; Jordan & Chisolm PLA(09)-a0801.
@ Objective vs subjective probabilities: Mohrhoff AJP(01)aug-qp/00; Saunders ch(05)qp/04; Srednicki PRA(05)qp; Huber BJPS(05) [as basis for scientific reasoning]; Ballentine AIP(07)-a0710; Maudlin SHPMP(07); Szabó SHPMP(07); Glynn BJPS(10)#1 [deterministic chances]; > s.a. ψ-Epistemic Theories; ψ-Ontic Theories.
@ Non-Kolmogorov: Szabó IJTP(94), IJTP(95)gq/94, FPL(95)gq/94, qp/96; Khrennikov JMP(00)qp, a0709 [and Bell inequalities].
@ Bayesian approach: Caves et al PRA(02)qp/01; Pitowsky SHPMP(03)qp/02; Schack qp/02 [and Hardy's axioms for quantum mechanics]; Appleby FP(05)qp/04, O&S(05)qp/04; Marlow AP(06)qp/05 [histories], gq/06, JMP(06)qp; Caves et al SHPMP(07)qp/06 [concept of certainty]; Bub SHPMP(07); Rau AP(09)-a0710 [quantum vs classical]; Fuchs & Schack RMP(13).
@ Quasi-probability representation: Ferrie RPP(11)-a1001; Ferrie et al PRA(10) [necessity of negative probabilities]; Ryu et al PRA(13)-a1206 [operationally defined, for qudits].
@ Related topics: Page qp/95, IJMPD(96)gq/95 ["sensible quantum mechanics", without probabilities]; Shirokov qp/06 [on set of states]; Kupczynski a0810-conf [statistical predictions]; Döring & Isham JMP(12)-a1102 [as truth values in suitable sheaf topoi]; Loeliger & Vontobel a1201-conf [factor-graph representation]; Stairs SHPMP(11); Henson a1210 [Consistent Exclusivity]; Sokolovski PRD(13)-a1301 [for classes of Feynman paths in spacetime]; Nenashev PS(14)-a1406 [from Zurek's envariance and Gleason's theorem]; > s.a. bell inequalities; Born Rule; experiments on quantum mechanics; Gleason's Theorem; histories formulations [including extended probabilities]; measure theory; mixed states; quantum collapse [GRW]; representations [tomographic].