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) [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 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.

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