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].

* __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 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]; Assar & Putz RPMP(06)mp/05 [massless particles,
momentum]; 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]; Yngvason LNP(15)-a1401 [relativistic, and entanglement]; Caban et al PRA(14) [and relativistic Einstein-Podolsky-Rosen correlations]; Balachandran a1609 [symplectic localization, rev].

@ __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].

@ __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].

**Other Contexts**

@ __References__: Pestun a1608 [in geometry].

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