Localization |

**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 IJGMP(17)-a1609 [symplectic localization, rev];
Taillebois & Avelar a2009 [proper-time approach];
Villaseñor et al a2103 [in phase space].

@ __Massless particles__: Assar & Putz RPMP(06)mp/05 [momentum];
Kosiński & Maślanka AP(18)-a1806 [gauge symmetry and Newton-Wigner operator].

@ __Relativistic__: Yngvason LNP(15)-a1401 [and entanglement];
Hoffmann JPB(18)-a1806 [no minimum width];
Anastopoulos & Savvidou JMP-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,
Stolz a1810-proc
[in disordered quantum spin systems/chains];
Serbyn et al PRX(15) [many-body localization and ergodicity breakdown];
Pretko & Nandkishore PRB(18)-a1712 [extended objects];
> 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 JPA(17)-a1608 [preface];
Pestun & Zabzine JPA(17)-a1608-in [intro];
Castrigiano a1711
[Dirac and weyl fermions are the only causal systems];
Papageorgiou & Pye JPA(19)-a1902 [in the relativistic vs non-relativistic regimes];
Feintzeig et al FP(21)-a2104 [in the classical limit];
> 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 JPB(18)-a1806 [no minimum width];
Hoffmann JPA(19)-a1811 [not possible for spinning particles];
Schwartz & Giulini IJGMP(20)-a2004 [classical perspective];
> s.a. Position.

@ __Spacetime localization__: Jaekel & Reynaud PLA(96)qp;
Kiosses a2010 [scalar fields in Minkowski space];
> 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];
Hoffmann JPA(20)-a1909 [photons];
> 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];
Lizzi et al a1912-proc [in κ-Minkowski spacetime].

**Other Contexts**

@ __References__: Pestun a1608-ch [in geometry].

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