Uncertainty Relations in Quantum Theory  

In General > s.a. equivalence principle; fluctuations [including classical uncertainty relations]; locality in quantum theory; physics teaching.
* Idea: Some observables can have no uncertainty, but not all of them, and any change in the expectation value of an observable quantity must be associated with some degree of uncertainty.
* Heisenberg's version: A lower bound for the product of the measurement error and the disturbance, stating that in observing the world we inevitably disturb it by introducing an unavoidable recoil; In the Heisenberg form the principle is valid only under certain circumstances, but a new universal error-disturbance relation has been proposed.
* Later version: The uncertainty relation was later reformulated in terms of standard deviations, where the focus was exclusively on the indeterminacy of predictions.
* Beating the quantum uncertainties: Some proposals have been made of types of measurements that have an effect smaller than the standard uncertainty, for example weak measurements.
@ General references: DeWitt JMP(62) [and commutation relations]; Hilgevoord & Uffink FP(91) [in prediction and inference]; Franson PRA(96) [and changes of expectation values]; Hilgevoord AJP(02)oct [and standard deviation]; Busch et al PRP(07)qp/06 [rev, conceptual]; Marburger AJP(08)jun [re early derivation]; Dumitru PiP(10)-a1005; Fujikawa & Umetsu PTP(11)-a1012 [and probability]; Partovi PRA(11) [and majorization theory]; Matía-Hernando & Luis PRA(12) [different measures of quantum uncertainty]; Dumitru a1205; Li et al JPA(14)-a1302 [as inequality for bipartite correlation functions, and experiment]; de Gosson JPA(13)-a1303 [in the Born-Jordan quantization scheme]; news pw(13)may [tradeoff between measurement uncertainties]; Dumitru a1501; Narasimhachar et al NJP(16)-a1505 [general framework]; Ozawa a1507-conf; Busch & Reardon-Smith a1901 [uncertainty region]; Werner & Farrelly FP(19)-a1904 [overview, meanings]; Wigderson & Wigderson a2006 [variations]; Urbanowski Symm(20)-a2010; Cazacu et al a2012 [mathematical]; Duan a2103 [interpretation].
@ For mixed states: Andersson & Heydari JMP(14)-a1302, PS(15)-a1412; Belfield & Brody a2012 [higher-order uncertainty bounds].
@ Approaches: Ivan et al a1205 [invariant theoretic approach]; Renes et al Quant(17)-a1612 [operational approach]; Mann et al a2005 [successive measurements].
@ Universal form / general theory: Deutsch PRL(83) [non-canonically-conjugate variables]; Friedland et al PRL(13)-a1304; Kechrimparis & Weigert JPA(16)-a1509; Bagchi & Pati PRA(16)-a1511 [arbitrary unitary operators]; Li et al SRep(16)-a1610; Huang & Huang a1807 [in different quantum theories].
@ Quantum vs classical uncertainties: Cini & Serva PLA(92); Luo TMP(05); Beretta PhD(81)qp/05 [quantum thermodynamics]; Busch PS(10)-a1004-conf; Huang & Huang PLA(10) [classical statistical and quantum uncertainty relations]; Usha Devi & Karthik AJP(12)aug-a1108 [in the classical limit]; Berta PhD(13)-a1310 [quantum side information]; Gattus & Karamitsos EJP(20)-a2102 [dimensional analysis]; > s.a. diffusion; fluctuations.
@ From lack of information: Rozpędek et al NJP(17)-a1606; Zhao et al a2105 [tests].
@ Origin of uncertainties: Anderson & Halliwell PRD(93)gq [quantum + thermal fluctuations]; Wesson GRG(04)gq/03 [from higher dimensions]; Arbatsky qp/06 [derivation from "certainty principle"]; Downes et al a1108 [uncertainty relation for the spacetime metric]; Fujikawa PRA(12)-a1205 [incorporating both intrinsic quantum fluctuations and measurement effects]; Girolami et al PRL(13)-a1212 [intrinsic quantum uncertainty on a single observable]; Thuan a1507 [spacetime curvature]; Adami a2005-FQXi [from the indeterminism inherent in mathematical logic].
@ Beating the quantum uncertainties: Polzik & Hammerer AdP(15)-a1405 [trajectories without quantum uncertainties]; > s.a. types of measurements.
@ Uncertainty conservation relations: Wang et al a1711 [theory and experiment].
> Related topics: see histories formulations; mixed states; optics; Reference Frame; semiclassical states [minimal-uncertainty].

Configuration-Momentum Uncertainty Relations > s.a. modified and deformed relations [including entropic]; phenomenology [systems, tests, violations].
* Idea: In any preparation of a system, uncertainties are constrained to satisfy Δq Δp ≥ \(\hbar\)/2; In the usual approach to quantum theory, the bound can be traced back to the [q, p] commutation relations.
* In terms of creation-annihilation: The uncertainty relation is expressed by \(\langle a^\dagger a\rangle \ge \langle a^\dagger \rangle \langle a\rangle\).
@ General references: Heisenberg ZP(27); Schrödinger SPAW(30), translation BulgJP-qp/99; Gamow SA(58)jan; Lévy-Leblond AJP(72)jun [non-simultaneous measurements]; Fefferman BAMS(83); Landsberg FP(88) [and classical and quantum mechanics]; Chisolm AJP(01)mar-qp/00 [restrictions]; D'Ariano FdP(03)qp/02; Rigolin EJP(15)-phy/05 [derivation]; Schürmann & Hoffmann FP(09)-a0811; Mandilara & Cerf PRA(12)-a1201; Hedenmalm JAM(12)-a1203 [in the sense of Beurling]; Rudnicki PRA(12) [sharper bounds for σrσp]; Busch et al PRA(14)-a1311 [for qubit measurements], JMP(14)-a1312 [measurement uncertainty relations]; Boughn & Reginatto EJP(18)-a1712 [revisiting Heisenberg's microscope]; > s.a. localization.
@ For angular momentum: Franke-Arnold et al NJP(04); Dürr pw(04)oct; Dammeier et al NJP(15)-a1505; Lake et al Univ-a1912 [in quantum geometry].
@ And phase space geometry: Curtright & Zachos MPLA(01)ht; Anastopoulos & Savvidou AP(03)qp; de Gosson PLA(03) [phase-space quantization], qp/04 ["quantum blobs"], mp/06 [and symplectic non-squeezing]; de Gosson & Luef PRP(09) [symplectic capacity]; de Gosson FP(12)-a1106; Werner a1601-proc [for general phase spaces]; > s.a. phase space.
@ And complementarity: Uffink & Hilgevoord PhyB(88)qp/99; Björk et al PRA(99)qp; Basso & Maziero a2007.
@ In terms of information: Gibilisco & Isola mp/05, JMP(07)mp, Chakrabarty APS(04)qp/05, Sánchez-Moreno et al JPA(11) [and Fisher information].
@ Error-disturbance relation: Brown & Redhead FP(81); Hofmann PRA(03)qp/02; Wulleman PE(03)qp/06, CoP(06)#3; Rozema et al PRL(12) + news UT(12)sep [violation in weak measurements of photons]; Busch et al PRL(13), a1402 [proof of Heisenberg's relation, as characterizing measuring devices]; Ipsen a1311 [for finite-dimensional systems]; news pw(13)nov [uncertainty vs disturbance debate]; Ozawa a1403-proc [reformulation]; Bastos et al PRD(14)-a1406 [phase-space non-commutative formulation]; Dressel & Nori PRA(14) [definitions of error and disturbance]; Fujikawa et al PRA(15)-a1412 [and Hardy's paradox]; Ozawa a1505 [interpretation]; Nishizawa & Chen a1506 [universal form]; Zhao et al PRA(17)-a1512 [quantum-walk-based experimental test]; Inoue & Ozawa a2009 [violation by Stern-Gerlach measurements].
@ And entanglement: Rigolin FPL(02)qp/00, qp/01; Hari Dass et al IJMPB(13)-a1107 [for entangled states]; Berta et al PRA(14)-a1302 [relation between entanglement and uncertainty].
@ Related topics: Landsberg Mind(47) [philosophy]; Yu PLA(96); Hewitt-Horsman qp/03 [and many worlds]; Sarris et al PLA(04) [as invariant of motion]; Busch & Pearson JMP(07) [for error-bar widths]; Kryukov PLA(07)-a0710 [geometric]; Zozor et al PRA(11)-a1112 [uncertainty relations based on moments of arbitrary order]; Rudnicki et al PRA(12)-a1204 [for coarse-grained measurements]; Malbouisson PRA(13)-a1307 [in a cavity at finite temperature]; Tomassini & Viaggiu CQG(14)-a1308 [spacetime uncertainty relations]; Bosyk et al PRA(14) [geometric]; Majumdar & Pramanik a1410-proc [applications in quantum information]; Li & Qiao SRep-a1502 [new form]; Kechrimparis & Weigert JPA(18)-a1703 [linear combinations of position and momentum observables]; Pollack & Miret-Artés PRA(19)-a1808 [for time averaged weak values]; Huang et al ChPB(18)-a2003 [new product forms]; > s.a. Gerbe [reformulation].

Time-Energy Uncertainty Relation > s.a. mixed states; time in quantum mechanics.
@ General references: Aharonov & Bohm PR(61); Kijowski RPMP(76); Sorkin FP(79); Busch FP(90), FP(90); Kobe & Aguilera-Navarro PRA(94); Pfeifer & Fröhlich RMP(95); Hilgevoord AJP(96)dec, AJP(98)may; Busch in(02)qp/01 [types, history]; Brunetti & Fredenhagen RVMP(02)qp [rigorous derivation]; Miyadera FP(16)-a1505, comment Yasuda a1809 [in quantum measurements]; Kieu PRS(19)-a1702 [time-dependent Hamiltonians]; Urbanowski MPLA-a1810, a1908-conf [not universally valid]; Bertoni et al NJP(20)-a2001 [entropic, algebraic approach]; Campaioli a2004-PhD [improved bounds on the speed limit of quantum evolution]; Roberts & Butterfield JPCS(20)-a2007 [it does not allow particle creation].
@ Time-mass: Kudaka & Matsumoto JMP(99)qp [τ and m as operators]; Ram mp/02; Dodonov & Dodonov PS-a1504 [exact inequalities].
@ Related topics: Pegg PRA(98) [operator conjugate to H]; Aharonov et al PRA(02)qp/01 [and estimating the Hamiltonian]; Gillies & Allison FPL(05) [time-temperature]; Karkuszewski qp/05 [upper bound on uncertainties]; Denur AJP(10)nov [and quantum phenomena].

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 20 may 2021