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)
[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 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.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
25 jun 2008