Dispersion

In General > s.a. causality.
* Idea: The dependence of the index of refraction of a wave in a medium on frequency; A non-trivial dependence n(), or a non-constant n() = v₀ / v_p with v_p = d/dk, is equivalent to a non-linear relation = 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(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: 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.

In Classical Theories > s.a. FRW spacetimes; phenomenology of gravity; 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.
@ Electromagnetic waves: Lucarini et al RNC(03) [optics]; Marino et al AP(07) [in a lattice of oscillators].
@ General references: Hagedorn 64; Harko & Cheng ApJ(04)ap [multidimensional cosmology]; Amore & Raya Chaos(06)mp/05 [non-linear Klein-Gordon]; Gratton & Perazzo AJP(07)feb [from dimensional analysis].

In Quantum Mechanics
* Idea: Free particle propagation is dispersive, since p = k and E = , related by E = p²/2m, so = k²/2m and v_p = p/m.

In Quantum 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 = f₁ T_ab k^a k^b + f₂ C_acbd k^a k^{b c d} ,

where ^a is the polarization vector.

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,

E² = p² + _photon (E³/m_P) .

or DSR models;
@ References: Shore CP(03)gq [QED in curved spacetime]; Buhmann & Welsch PQE(07)qp/06 [in macroscopic QED].
> Related topics: see Chandrasekhar Limit; graviton; matter phenomenology and photon phenomenology in quantum gravity.

Kramers-Kronig Relations
* Idea: Integral relations between the real and imaginary parts of the index of refraction n() 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.
@ References: Wigner ed; in Arfken; in Weinberg 95; Kowar a0901 [and causality].

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