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 / vp
with vp = 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 ka 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 vp = 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 vp = p/m.
In Quantum Theory > s.a. photon.
* In QED: Quantum field
theory effects can change the relation gab ka kb
= 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
gab ka kb = f1 Tab ka kb
+ f2 Cacbd
ka kb
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/mP)
.
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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sep 2009