Time in Quantum Theory  

In General > s.a. time / hilbert space [rigged].
* Formal role: Time does not correspond to a dynamical variable/operator (by examining a clock, we do not measure t, just the variable "position of the handles''), but it is necessary in formulating all approaches and interpretations, and it drives the evolution; Measurements are made "at an instant of time'', histories are not measurable.
* Philosophy: The separation between being (kinematics) and becoming (dynamics) generates inextricable difficulties.

Time as an Observable / Operator > s.a. uncertainty relations [time-energy].
* Time evolution operator: The operator T(t, t'):= exp{–iH(tt')/\(\hbar\)}; If H is self-adjoint, T is unitary.
* Result (Pauli): The existence of a self-adjoint time operator canonically conjugate to a Hamiltonian implies that both operators possess completely continuous spectra spanning the entire real line; Pauli concluded that constructing a time operator is impossible in general, although it can be done in specific cases.
@ General references: Giannitrapani IJTP(97)qp/96; Oppenheim et al LNP(99)qp/98; Belavkin & Perkins IJTP(98)qp/05 [unsharp measurement]; Galapon O&S(01)qp/00, PRS(02)qp/01 [including discrete semibounded H], remarks Hall JPA(09)-a0811; Kitada qp/00; Hahne JPA(03)qp/04; Bostroem qp/03; Olkhovsky & Recami IJMPB(08)qp/06; Wang & Xiong AP(07)qp/06; Strauss a0706 [forward and backward time observables]; Arai LMP(07) [spectrum]; Wang & Xiong AP(07); Brunetti et al FP(10)-a0909; Prvanović PTP(11)-a1005; Zagury et al PRA(10)-a1008 [unitary expansion of the time evolution operator]; Greenberger a1011-conf [and mass]; Strauss et al CRM(11)-a1101 [self-adjoint operator indicating the direction of time]; Burić & Prvanović a1102 [in extended phase space]; Yearsley PhD(11)-a1110 [approaches]; Mielnik & Torres-Vega CiP-a1112; Bender & Gianfreda AIP(12)-a1201 [matrix representation]; Fujimoto RJHS-a1403; Roberts a1404 [general framework, and time observables in extensions of standard quantum theory]; Dias & Parisio a1507; Kinyanjui & Wamalwa a1507.
@ Re Pauli's argument: Galapon PRS(02)qp/99, qp/03-proc, Wang et al JPA(03)qp/02, qp/03-wd [critique]; Isidro PLA(05)qp/04 [bypassing Pauli's theorem]; Gessner a1305 [symmetric but not self-adjoint]; Leon & Maccone a1705 [overcoming the objection].
@ Related topics: Kundrát & Lokajíček PRA(03) [3D oscillator]; Durt a1003 [predictions on decay-time distributions]; de la Madrid NPA-a1306 [and quantum decay].

Other Views and Proposals > s.a. observables [multi-time]; time [two-time physics].
* Covariant view: According to Reisenberger & Rovelli, time has a special role in quantum mechanics because of idealized instantaneous measurements, but the idealization can be dropped, and spacetime states defined.
* Relational time: One promotes all variables in a system to quantum operators, and later chooses one of the variables to operate like a "clock"; Conditional probabilities are computed for variables of the system to take certain values when the "clock'' specifies a certain time; The framework is attractive in contexts where assuming the existence of an external, perfectly classical clock, appears unnatural, as in quantum cosmology.
@ General references: Aniello et al AP(16)-a1605 [time as a function on the space of physical states].
@ Based on statistics: Bonifacio NCB(99)qp, qp/99-proc [statistical time and intrinsic decoherence]; Caticha AIP(11)-a1011 [entropic time].
@ Covariant: Reisenberger & Rovelli PRD(02)gq/01 [spacetime states].
@ Relational time: Gambini et al NJP(04)gq, PRD(04)gq [and decoherence], IJMPD(04)gq [black-hole information]; Milburn & Poulin IJQI(06)qp/05 [two oscillators]; Moreva et al PRA(14)-a1310 [time emergent from quantum entanglement]; Małkiewicz & Miroszewski a1706 [internal clock and its effect on quantization].

Related Concepts and Effects > s.a. arrow of time; quantum measurement; symmetries [time translation].
* Passage time: The shortest time after which, a quantum state is transformed into a state orthogonal to itself.
@ Clocks: Karpman IJTP(78) [time without clocks]; Anderson gq/95 [vs time]; Ashworth PRD(98)qp/97 [oscillator coherent states].
@ Real clocks: Hartle PRD(88); Egusquiza et al PRA(99); Egusquiza & Garay PRA(03)qp [Zeno effect]; Stodolsky in(06)qp/03 [coherence]; Corbin & Cornish FP(09)-a0811 [conditional probability interpretation and decoherence].
@ Time of measurement: Oppenheim et al PRA(99)qp/98; Rovelli FP(98)qp, comment Oppenheim et al FPL(00)qp/98; Home & Chattopadhayaya qp/99.
@ Time of occurrence: Brunetti & Fredenhagen PRA(02) [observable]; > s.a. quantum effects [time of arrival, decay, and other effects].
@ Transitions / jumps: Mensky PLA(96) [reversibility]; Schulman qp/01-in [time].
@ Uncertainties and time measurement: Braunstein et al AP(96)qp/95; Kitada gq/99; > s.a. uncertainty principle.
@ Time reversal: Costa De Beauregard PLA(78) [S-matrix, Feynman zigzag and Einstein correlation]; Pastawski et al qp/04 [time-reversal mirror]; Dowling et al PRL(05) [stochastic quantum dynamics]; Aharonov & Tollaksen a0706-in [rev].
@ Time ordering: Oppenheim et al JPA(02); McGuire et al qp/05.
@ Other effects: Ruseckas & Kaulakys PLA(01) [time spent in a state]; Brody JPA(03) [passage time]; Bernstein a0906 [the past]; Ludwin a1101 ["quantum twin paradox" and global proper time]; Vona & Dürr a1309-ch [measurement, role of the probability current]; > s.a. atomic physics [complex time]; causality violations [time travel]; Detector; Ehrenfest Time; modified quantum theories [time symmetric]; Retrodiction.

References > s.a. composite systems; Non-Linear Quantum Mechanics; quantum spacetime.
@ Reviews: Hilgevoord SHPMP(05) [history]; Fleming SHPMP(05); Zeh a0705-en.
@ General: Droz-Vincent PLA(88); Collins & Squires FP(93); Busch et al PLA(94); Kapitanski & Rodnianski qp/97; Briggs & Rost EPJD(00)qp/99; Muga et al ed-01; Hilgevoord AJP(02)mar [conceptual]; Elitzur & Dolev qp/02-conf [and histories]; Lokajicek qp/02-conf [and irreversibility]; Helfer a0812-FQXi [and measurement]; Billionnet IJMPA(10)-a0908 [and width of atomic levels]; Singh a1001-FQXi, IJMPD(10)-a1005-GRF [and operator time in quantum gravity]; Elçi JPA(10) [and classical mechanics]; Sato & Bohm a1011 [dynamical evolution time vs observer time]; Arce PRA(12)-a1202 [unification of the conditional probability and semiclassical interpretations]; Moyer a1305; Vona PhD(14)-a1403; York a1405; Butterfield a1406-in; Giovannetti et al PRD(15)-a1504 [consistent quantum description].
@ Time non-locality: Kim & Mahler ZN(01)qp; Suarez qp/01 [no time ordering]; Kracklauer qp/02 [+ s.a. qp/02]; Gainutdinov et al PLA(03)qp/02 [atom and surroundings].
@ Discrete: Balachandran & Chandar NPB(94) [from quantization]; Misra FP(95), Farias & Recami qp/97 ["chronon"]; Jaroszkiewicz & Norton JPA(97), JPA(97); Norton & Jaroszkiewicz JPA(98), JPA(98); Bruce PRA(01)qp; Valsakumar Pra(05)qp [and arrow of time]; Skulimowski FPL(06) [quantum time]; Isidro et al MPLA(08)-a0804 [and evolution of observables]; Isidro et al MPLA(08)-a0804 [and Heisenberg's equation as delay-differential equation]; > s.a. discrete spacetime.
@ And relativity, covariance: Mehlberg 80 [and causality]; Nikolić IJQI(09)-a0811 [time operator, pilot-wave theory]; Vaccaro PRS(15)-a1502 [quantum asymmetry between time and space]; Bauer EJP(17)-a1606; > s.a. relativistic quantum mechanics.
@ Quantum mechanics without time: Singh qp/01; Singh GRG(03)gq/02 [and non-commutative geometry]; Majid JMP(05)ht [spontaneous time generation]; Singh AiA(09)-a0901-in [and gravity and the cosmological constant]; Oreshkov & Cerf NJP(16)-a1406; Foreman a1609 [from subsystems]; Marletto & Vedral a1610, Bryan & Medved a1706 [evolution without evolution, from entanglement, à la Page and Wootters].
@ Other references: Saunders Syn(95) [wave-function collapse, actuality and passage]; Hitchcock gq/99 [complex systems], qp/01-talk [and information]; Foschini qp/98 [and logic]; Cramer in(01)qp/05 [and transactional interpretation]; Hiller et al PRL(04)qp/03 [undoing t evolution]; Mizel PRA(04)qp/03 [mimicking t evolution in the ground state]; Werbos IJTP(08) [backwards-time physics]; Ashmead a1005-conf [time variations in path integral]; Rotter FdP(13)-a1206 [in an open quantum system, and dynamical stabilization]; Kalinowski a1411 [time as a stochastic process]; Amaral & Bojowald a1601 [path-integral approach]; Kitada et al PLA(16)-a1606 [local time].

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