Time in Quantum Theory| 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. particle effects in quantum gravity;
uncertainty relations [time-energy].
* Time evolution operator:
The operator T(t, t'):=
exp{−iH(t−t')/\(\hbar\)};
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 FP(17)-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];
Favalli & Smerzi a2003 [in the Page & Wootters proposal].
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];
Ron a1704.
@ Based on statistics:
Bonifacio NCB(99)qp,
qp/99-proc [statistical time and intrinsic decoherence];
Caticha AIP(11)-a1011,
Vaughan a2011 [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 PRD(17)-a1706 [internal clock and its effect on quantization].
Related Concepts and Effects
> s.a. arrow of time; causality in quantum theory;
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];
> s.a. clocks.
@ 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;
Ashmead a2101;
> 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];
Singha Deo & Satpathi a1806 [signal propagation in negative time].
@ 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;
Pashby SHPMP(15) [history and prospectus].
@ 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];
Smolin & Verde a2104 [time as transition between indefinite and definite].
@ 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 PRD(17)-a1610,
Bryan & Medved FP(18)-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 AP(18)-a1601 [path-integral approach];
Kitada et al PLA(16)-a1606 [local time].
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