In Classical Theories
@ Classical mechanics: Grigore JPA(90); Karpov et al PRA(00)qp [Hegerfeldt theorem for classical waves]; Campbell et al PT(04)jan [non-linearity and discreteness]; Gournay & Tiedra de Aldecoa a1101 [localisation observables and variation of energy along classical orbits]; > s.a. random walk.
@ Classical field theories: Calcagni et al PLB(08)-a0712; Stockman Phy(10) [of optical energy, in disordered materials].
In Quantum Mechanics > s.a. types of states
[semiclassical]; uncertainty relations.
* Non-relativistic: For a single particle, localization is described by the Born localization principle, which states that the particle is localized in a spatial region K if its wave function vanishes outside K; Two wave functions localizes in disjoint spatial regions K and K' are orthogonal.
* Relativistic: Localization in terms of single-time observables leads to conflict with the requirement of causality; It is not sharp for interacting point particles (Currie, Jordan and Sudarshan), for which it is replaced by symplectic localization of observables (as shown by Brunetti, Guido and Longo, Schroer and others), or in quantum field theory (Newton-Wigner), where it becomes modular localization; It is related to the concepts of uncertainty, entropy, phase space volume.
* Hegerfeldt theorem: A wave function for a particle whose time evolution is determined by a positive (or bounded from below) Hamiltonian, and initially confined to a finite volume, will instantaneously develop infinite tails and violate causality, even in relativistic quantum mechanics; This paradoxical result arises in relativistic theories from a failure of objectivity of localization – the so-called 'Jericho effect'.
* Solid-state physics: Localization of electrons is a dramatic effect of destructive interference; The phenomenon is usually perceived as arising from extrinsic disorder that breaks the discrete translational invariance of a perfect crystal lattice (e.g., defects or impurities for phonons, disorder for electrons); However, intrinsic localized modes, a.k.a. discrete breathers, are typical excitations in perfectly periodic but strongly non-linear systems.
* Atomic physics: Non-dispersing electron wave packets can be obtained by applying a circularly polarized microwave field.
@ General references: Wightman RMP(62); Szabo ht/96 [and path integrals]; Bracken & Melloy JPA(99)qp [of electrons, solution]; Melloy FP(02) [formalism, non-relativistic particles and Dirac particles]; Detournay et al PRD(02)ht [generalized commutation relations]; Dunstan qp/05 [interaction with the environment]; Mould qp/05 [and q-rules]; De Bièvre mp/06 [rev]; Morchio & Strocchi JPA(07)qp/06-fs [consequences]; Pinamonti RVMP(07) [intrinsic, from symmetry group]; Schroer a0711 [vs quantum field theory]; Sträng JPA(08)-a0708 [squeezed states]; Pennini et al PLA(08) [and Husimi distributions, Wehrl entropy]; Ziegler a1310 [localized states]; Caban et al PRA(14) [and relativistic Einstein-Podolsky-Rosen correlations]; Balachandran a1609 [symplectic localization, rev].
@ Massless particles: Assar & Putz RPMP(06)mp/05 [momentum]; Kosinski & Maslanka a1806 [gauge symmetry and Newton-Wigner operator].
@ Relativistic: Yngvason LNP(15)-a1401 [and entanglement]; Hoffmann a1806 [no minimum width]; Anastopoulos & Savvidou a1807 [in terms of time-of-arrival observables].
@ Spontaneous localization: Nicrosini & Rimini FP(03)qp/02 [relativistic]; Bedingham PRD(13)-a1306 [and single-particle energy diffusion]; Smirne & Bassi SRep(15)-a1408 [dissipative model].
@ Hegerfeldt theorem: Wickramasekara & Bohm JPA(02) [for t-semigroups]; Debs & Redhead SHPMP(03) [relativistic, Jericho effect].
@ Measurement-induced: Irby PRA(03)qp/02 [gamma rays]; Cable et al PRA(05)qp/04 [relational degrees of freedom].
@ In condensed matter, mesoscopic physics: Janssen 01 [fluctuations and localization]; Hamza et al CMP(12)-a1108 [in disordered quantum spin systems]; Serbyn et al PRX(15) [many-body localization and ergodicity breakdown]; > s.a. Anderson Localization [random media].
@ In atomic physics: Maeda et al PRL(09) and Stroud Phy(09) [non-dispersing electron wave packets]; Moreno et al PRL(10) [disordered boundaries].
In Quantum Field Theory > s.a. lattice field
theory [localization in random lattices]; quantum particles.
* Particles: A sharp localization as in non-relativistic quantum mechanics is impossible in local quantum field theory; One often sees the Compton wavelength used as limit for localizability, but its spatial probability density can be much narrower; > s.a. Compton Wavelength.
@ General references: John PT(91)may [light]; Schroer ht/98-ln; Zavialov TMP(04) [and deformed Heisenberg algebra]; Schroer a0711, SHPMP(10)-a0912, SHPMP(10)-a0912 [quantum mechanics, quantum field theory and quantum gravity]; Grundling & Neeb LMP(10)-a1001 [via automorphisms of the CARs]; Schroeren a1007 [criterion for a quantum field theory to admit particles]; Schroer SHPMP-a1107 [and causal quantum theory]; Schroer Sigma(14)-a1407 [modular localization]; Pestun et al a1608-in [foreword to book]; Pestun & Zabzine a1608-in [intro]; > s.a. fock space [for local quanta].
@ Newton-Wigner position operator: Newton & Wigner RMP(49); Halvorson PhSc(01)mar [vs Reeh-Schlieder]; Westra a1012 [alternative]; Choi a1412 [corresponding spin operator]; Hoffmann a1806 [no minimum width]; > s.a. Position.
@ Spacetime localization: Jaekel & Reynaud PLA(96)qp; > s.a. quantum-gravity phenomenology [events]
@ Gauge theories: Tetradis PLB(00)hp/99 [as dual Josephson junction]; Białynicki-Birula & Białynicki-Birula PRA(09)-a0903 [photons and coherent states]; Benini & Le Floch a1608-in [supersymmetric 2D theories]; > s.a. photons.
@ Other theories: Krekora et al PRL(04) [no limitations for electrons]; Kubicki et al a1606 [2D Dirac fermions in flat spacetime]; Willett a1608-in [3D supersymmetric theories on compact manifolds].
@ References: Pestun a1608 [in geometry].
– journals – comments
– other sites – acknowledgements
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