Momentum| Momentum |
In General, and Particle Mechanics > s.a. hamiltonian dynamics
and systems; phase space.
* Idea: The conserved quantity related
to spatial translation invariance of a theory (from movimentum).
* In classical non-relativistic mechanics:
A particle with velocity v has momentum p = \(m\,{\bf v}\).
* In special relativistic mechanics:
A particle with 4-velocity ua
has 4-momentum pa
= mua.
@ General references: Sibelius FP(90) [mechanical and wave-theoretical aspects];
Crenshaw PLA(05) [electromagnetic, and Fresnel relations];
Roche EJP(06) [general definition];
Lee SHPMP(11)
[examples of momentum non-conservation in classical mechanics].
@ History: Gillespie AJP(95)apr [why "p"? :-)];
Graney TPT(13)-a1309 [John Buridan's 14th century concept].
@ In special relativity:
Sonego & Pin EJP(05),
EJP(05);
Adkins AJP(08)nov;
Riggs TPT(16) [vs Newtonian dynamics].
@ Other systems: Liu et al PRA(11) [free particle on a 2-sphere, geometric momentum];
Exner a1205-fs [on graphs];
Liu JMP(13)
[particle on a curved hypersurface].
> Related topics:
see conservation laws.
Momentum-Space Geometry
> s.a. finsler geometry.
@ General references:
Freidel & Smolin a1103 [and photon propagation];
Amelino-Camelia et al CQG(12)-a1107 [distant observers and phenomenology];
Kowalski-Glikman IJMPA(13)-a1303 [curved, rev];
Freidel et al IJMPD(14)-a1405-GRF [dynamical momentum space and string theory];
Lobo & Palmisano IJMPcs(16)-a1612 [isometry group and Planck-scale-deformed co-products].
@ Curved momentum space and spacetime: Freidel & Rempel a1312 [scalar quantum field theory in curved momentum space];
Gutierrez-Sagredo et al a1907-conf [non-commutative spacetimes];
Lizzi et al NPB(20)-a2001 [for κ-Minkowski spacetime];
Relancio & Liberati a2002 [cotangent bundle geometry],
a2008 [constraints].
@ Relative locality:
Kowalski-Glikman IJGMP(12)-a1205-proc [and curved momentum space];
Amelino-Camelia et al a1307;
Banburski & Freidel PRD(14)-a1308 [non-commutativity related to Snyder spacetime];
Amelino-Camelia a1408
[non-linear composition law and the soccer-ball problem];
> s.a. Fermi Surface.
@ Phenomenology of curved momentum space:
Amelino-Camelia et al PLB(16)-a1605,
a1609 [dual redshift and dual lensing];
> s.a. modified thermodynamics [photon gas in curved momentum space];
Carmona et al PRD(19)-a1907,
a1912 [deformed kinematics].
In Field Theory
* In general: The momentum
density of matter Tab as
seen by an observer ta is
− tb
T ab.
@ For an electromagnetic field:
Babson et al AJP(09)sep;
Spavieri & Gillies G&C(10)-a1005 [speed of light in moving media, and photon mass];
Brevik & Ellingsen AP(11)-a1008,
comment Griffiths AP(12) [in media];
Griffiths AJP(12)jan [RL];
Crenshaw AP(13) [field and matter momentum in a linear dielectric];
Franklin AJP(14)sep [static electromagnetic fields];
Corrêa & Saldanha PRA(16)-a1601 [and reflection by a quantum mirror];
Singal AJP(16)oct;
Brevik AP(17)-a1610 [Minkowski momentum]; Johns a2105 [and flow of field energy].
@ Internal electromagnetic momentum and "hidden" momentum:
Boyer AJP(15)may-a1408,
PRE(15)-a1408;
> s.a. magnetism.
@ For a fluid: Vishwakarma ASS(09)-a0705,
Jagannathan AJP(09)may [pressure contribution to fluid momentum density].
> For an electromagnetic field:
see energy-momentum tensor [including the Abraham-Minkowski dilemma, for light];
fields in media; maxwell theory.
> For gravity: see canonical general
relativity [various formulations]; gravitational energy-momentum.
In Quantum Theory
> s.a. quantum field theory in generalized
backgrounds; wigner function.
* Quantum mechanics:
A momentum operator conjugate to a configuration variable is one with the
right commutation relations; If classically the momentum is associated
to a vector field ua
on configuration space C, a quantum momentum operator
can be defined by û ψ(x):=
i (£u
+ \(1\over2\)div u) ψ(x), where
the divergence is calculated using the volume element on C
with respect to which the operator must be self-adjoint.
* For a wave: A 1-particle
wave with wave vector k has momentum p
= ħk, or p = h/λ.
@ General references: Jordan AJP(75)dec;
Roy et al RMF-a0706 [in general coordinates];
Gaveau & Schulman JPA(12)-a1206 [relative momentum of identical particles];
Berry EJP(13) [five different, equivalent definitions];
Xiao & Liu a1605 [canonical momentum vs geometric momentum].
@ Radial momentum: Paz EJP(01)qp/00;
Mosley mp/03;
Liu & Xiao a1411.
@ Momentum representation:
Lombardi a1906 [for the hydrogen atom].
@ Non-trivial configuration spaces:
Shikano & Hosoya JMP(08) [on a half-line];
Liu et al IJGMP(13)-a1212 [on a 2-sphere, and coherent states].
@ Maximum momentum: Ching & Ng MPLA(14)-a1311 [effect on wave equations];
> s.a. deformation quantization;
deformed uncertainty relations.
@ Other systems, quantum field theory: de Haan ht/06 [electron mechanical momentum in QED];
Arzano CQG(14)-a1305 [3D semiclassical gravity with point particles, deformed Fock space];
Oliveira & Saldanha PRA(15)-a1507 [hidden momentum in a hydrogen atom in an external electric field].
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