Dispersion |

**In General**
> s.a. causality; Refraction.

* __Idea__: Dispersion in general
is the phenomenon by which something is distributed over a wide area in space;
In optics (chromatic) dispersion is the separation of white light into colors
when going through certain devices, due to the fact that the phase velocity
(and therefore the index of refraction) of a wave in a medium depends on its
frequency.

* __Dispersion relation__: The
dependence of the index of refraction of a wave in a medium on frequency;
A non-trivial dependence *n*(*ω*), or a non-constant
*n*(*ω*) = *v*_{0}
/ *v*_{p} with *v*_{p}
= d*ω*/d*k*, is equivalent to a non-linear relation *ω*
= *f*(**k**); Therefore, in general a *dispersion relation*
is a relationship between the components of a wave vector \(k^a\) that in particle-like
terms is described by the shape of the "mass-shell" \(E = E({\bf p})\).

* __Remark__: The expression "dispersion
relation" is often used to denote integral relations of the Kramers-Kronig type
– see below.

* __Ordinary dispersion__: The refractive
index of a material changes with increasing wavelength; This stretches out the pulse
and reduces the group velocity–the speed at which the peak of the pulse travels.

* __Anomalous dispersion__: It occurs in
materials that absorb radiation in a certain range of wavelengths; On either side
of this absorption band, *n* changes sharply with wavelength; In these regions,
the components of radiation at the tail of the pulse interfere destructively, and
the peak of the wave is effectively pushed forward.

@ __References__:
Labuda & Labuda EPJH(14) [history of mathematical methods, Titchmarsh's theorem];
Zwicky a1610-ln [and analyticity].

**In Classical Theories** > s.a. phenomenology of gravity;
types of wave equations; wave phenomena.

* __Electromagnetic waves__: A flat vacuum
is non-dispersive, since *v*_{p} = *c* if
we neglect quantum field theory effects; In a medium, several effects can lead to dispersion.

@ __General references__: Hagedorn 64;
Gratton & Perazzo AJP(07)feb [from dimensional analysis].

@ __Electromagnetic waves__:
Lucarini et al RNC(03) [optics];
Marino et al AP(07)phy [in a regular lattice of oscillators];
Itin PLA(10) [anisotropic media].

@ __Other types of waves__:
Harko & Cheng ApJ(04)ap [multidimensional cosmology];
Amore & Raya Chaos(06)mp/05 [non-linear Klein-Gordon equation];
Likar & Razpet AJP(13)apr [surface waves on water];
Baggioli et al a1904 [gapped momentum dispersion relations].

> __Related topics__: see dirac equation;
FLRW spacetimes.

**In Quantum Mechanics**
> s.a. schrödinger equation [solutions].

* __Idea__: Free particle propagation is
dispersive, since **p** = \(\hbar\)**k** and *E*
= \(\hbar\)*ω*, related by *E* = *p*^{2}/2*m*,
so *ω* = \(\hbar\)*k*^{2}/2*m*
and *v*_{p} = *p*/*m*.

**In Quantum Field Theory** > s.a. photon.

* __In QED__: Quantum field theory effects
can change the relation *g*_{ab}
*k*^{a} *k*^{b}
= 0 into a dispersive one; An effective coupling to the background matter stress-energy
and the Weyl tensor, or a temperature effect, replaces it by one of the form

*g*_{ab} *k*^{a}
*k*^{b} = *f*_{1}
*T*_{ab} *k*^{a}
*k*^{b}
+ *f*_{2} *C*_{acbd}
*k*^{a} *k*^{b}
*ε*^{c}
*ε*^{d} ,

where *ε*^{a} is the polarization vector.

@ __References__: Gharibyan PLB(05)he/03 [possible observation of photon dispersion in vacuum];
Pomeau a1409 [in QED, estimate and suggested test];
Hojman & Asenjo a2003
[classical vs quantum dispersion relations, scalar and vector fields].

**Quantum-Gravity-Motivated Modifications**
> s.a. modified lorentz symmetry.

* __Idea__: Several models have been proposed
in which quantum-gravity effects modify the usual dispersion relations, such as DSR models,
and ones in which Lorentz invariance is broken and one has, for example for photons,

*E*^{2}
= *p*^{2} + *ξ*_{photon}
(*E*^{3}/*m*_{P}) ,

or DSR models.

@ __General references__: Rätzel et al PRD(11)-a1010 [restrictions on possible forms].

@ __Phenomenology__: Shore CP(03)gq [QED in curved spacetime];
Buhmann & Welsch PQE(07)qp/06 [macroscopic QED];
Biswas & Faruk a1706 [thermodynamics of a photon gas].

> __Related topics__:
see Chandrasekhar Limit; graviton;
matter phenomenology and photon phenomenology
in quantum gravity; phenomenology of uncertainty relations;
Spectral Dimension.

**Kramers-Kronig Relations**

* __Idea__: Integral relations between the real and
imaginary parts of the index of refraction *n*(*ω*) of waves in a medium, related
to the condition that the propagation of the waves be causal – They relate a dispersive process
to an absorption one; > s.a. equivalence principle.

@ __General references__: Wigner ed-64;
in Arfken 85;
in Weinberg 95;
in Alastuey et al 16.

@ __Related topics__: Kowar a0901 [and causality];
Bohren EJP(10) [history and meaning];
Akyurtlu & Kussow PRA(10) [and negative index of refraction];
Kinsler EJP(11) [and spacetime causality];
Llosa & Salvat a1905
[beyond the optical approximation].

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