Conservation Laws, Conserved Quantities |
In General
> s.a. asymptotic flatness at spatial and
null infinity; lagrangian dynamics.
* Applications: They are
useful for studying the evolution of a system without knowing details about
the motion, to give a classification of all possible evolutions, etc, and
to reduce the order of the differential equations describing the motion.
From Symmetries > s.a. differential equations;
noether symmetries; killing fields
[Killing vectors and tensors]; symmetries of dynamical theories.
* Idea: Generators of symmetries
give rise to conserved currents, whose integrals are conserved quantities.
* Types: Conserved
quantities can correspond to Noether, Mei and Lie symmetries in a
Lagrangian dynamical system; Topological charges on the other hand
do not give rise to symmetry transformations on the fields.
* Superpotential:
An (n−2)-form U(ξ) in an n-dimensional
spacetime, associated with an infinitesimal Lagrangian symmetry ξ,
such that the boundary term in the conserved quantity Q(ξ)
= Qbulk(ξ)
+ Qbdry(ξ) is
Qbdry(ξ) = ∫∂(Σ) U(ξ) ;
The term also denotes a potential θabc
for a stress-energy pseudotensor, θmn
= ∂a
θm[na].
@ General references: Levi-Civita RAL(1899),
translation Saccomandi & Vitolo RCD(12)-a1201 [for Hamilton's equations];
Wigner PT(64)mar;
Katzin JMP(73) [quadratic constants of the motion];
Rosen & Freundlich AJP(78)oct [general framework];
Norbury EJP(88) [pedagogical, for momentum];
Schulte SHPMP(08) [and particle families];
Smith SHPMP(08) [in Lagrangian mechanics];
Ivanova et al PhyA(09)-a0806;
Fang et al PLA(10) [types, and proposal of new types];
Ngome Abiaga PhD(11)-a1103 [Van Holten's covariant algorithm];
Nucci PLA(11) [Lie symmetries of Lagrangian systems];
Anco IJMPB(16)-a1512.
@ Non-Noether symmetries:
Hojman JPA(92)
+ González-Gascón JPA(94)
+ Lutzky JPA(95);
Anco & Bluman PRL(97) [for field theories];
Kara & Mahomed IJTP(00);
Chavchanidze mp/02,
JGP(03)mp/02;
Birtea & Tudoran IJGMP(12);
but see Deser a1905;
> s.a. lagrangian dynamics.
@ Superpotential: in Francaviglia & Raiteri CQG(02)gq/01;
Katz & Livshits CQG(08)-a0807 [from variational derivatives, gravity theories];
Sardanashvily IJGMP(09)-a0906 [and gauge conservation laws];
Adamek & Novotny a1606 [new superpotential].
@ Related topics: Kara et al IJTP(99) [approximate symmetries];
Cicogna MMAS(13)-a1307 [and generalizations].
Related Concepts
> s.a. Chevreton Tensor; geodesics;
energy; observables;
stress-energy pseudotensor;
superselection rules.
* Continuity equation:
A local, differential form of conservation equation; If u is
a velocity/flow 4-vector and f a source strength, then
∇a ua = f , or ρ,t + ∇ · (ρv) = f in 3+1 form .
@ General references: Horwood JMP(07) [higher-order first integrals].
@ Continuity equation:
Belevich JPA(09);
Schild PRA(18)-a1808 [with quantum clock];
Laba & Tkachuk a2004 [in a space with minimal length];
Herrmann a2005.
@ Energy-momentum conservation:
Giachetta et al G&C(99) [gauge approach];
Weatherall SHPMP-a1702 [as a consequence of the gravitational field equations];
Deser & Pang PLB(19)-a1811
[are all identically conserved local symmetric tensors variations of some coordinate invariant action?];
Pitts a1909
[in general relativity, and Cartesian mental causation; ?];
> s.a. energy-momentum tensors;
gravitational energy-momentum.
@ Time-dependent invariants of motion:
Sarris & Proto PhyA(05) [complete sets of non-commuting observables].
@ Related topics:
Boyer AJP(05)oct [center of energy, illustrations];
> s.a. wigner function.
Specific Types of Theories > s.a. graph theory in physics;
noether theorem; non-local theories.
* In Newtonian mechanics:
Only 4 were known at the end of the XIX century, mass, linear momentum,
angular momentum and energy.
* In relativistic mechanics:
For geodesic motion in curved spacetime, each Killing vector field
ξa gives rise to a
constant of the motion, pa
ξ a, and so does
each Killing tensor.
* In relativistic field theory: They are expressed by
T ab;b
= 0; If the spacetime has a Killing vector field ξa,
then T ab
ξb is a conserved current.
@ Gauge theories: Chodos CMP(79) [Yang-Mills theory];
Przeszowski JPA(89) [non-abelian currents];
Deser & Henneaux MPLA(95)ht [abelian and non-abelian currents];
Barnich et al LMP(04)gq [n−2 forms in curved spacetime, classification];
Manno et al JGP(08) [second-order field equations and variational principles];
Berche AJP(16)aug [classical and quantum mechanics];
Burns AACA(19)-a1906 [local charge conservation implies Maxwell's equations].
@ Spinor fields: Anco & Pohjanpelto PRS(03)mp/02 [any spin].
@ Gravity: Wald & Zoupas PRD(00)gq/99;
Papadopoulos JMP(06)gq/05 [essential constants];
Obukhov & Rubilar PRD(06)gq,
PRD(07)-a0712,
PLB(08)-a0712;
Fabris a1208-proc [and cosmology];
García-Parrado a1606 [4D spacetime with a Killing vector];
Hopfmüller & Freidel PRD(18)-a1802 [along null surfaces];
Andersson et al a1812
[linearized gravity, spin and orbital angular momentum];
Aoki et al a2005;
Lopes de Lima a2103-in;
> s.a. gravitational action and energy;
higher-order theories; quasilocal energy.
@ Gravity, quasilocal charges:
Kim et al PRL(13);
Chen et al a1312;
Hyun et al PRD(14)-a1406 [and holography];
Bart a1908 [general prescription].
@ Generalized gravity:
Alves et al GRG(08)-a0710 [in massive graviton theory];
Lompay & Petrov JMP(13)-a1309 [metric-torsion theories];
Obukhov & Puetzfeld PRD(14)-a1405 [metric-affine gravity];
Setare & Adami NPB(16)-a1511 [Chern-Simons-like theories];
Obukhov et al PRD(15)-a1511 [generally covariant theories];
Ghodrati et al EPJC(16)-a1606 [in f(R) gravity].
@ Hamiltonian systems:
Cariglia et al JMP(14)-a1404 [covariant algorithm using conformal Killing tensors];
Torres del Castillo a1408.
@ Quantum mechanics: Sharp & D'Amico PRB(14)-a1403 [metric-space formulation];
Ohsawa JMP(15)-a1410 [and symmetries, in semiclassical Gaussian wave packet dynamics];
Torres del Castillo & Herrera a1510 [and unitary symmetries of the Hamiltonian];
Aharonov et al a1609 [questions about the meaning];
> s.a. quantum spin models; relativistic
quantum mechanics; schrödinger equation.
@ Particle dynamics:
Katzin & Levine JMP(74);
Cetto & de la Peña AJP(84)jun
[relationship between energy and adiabatic invariant J];
Hojman et al JMP(86),
Del Castillo & Hojman JMP(90) [geodesic motion];
Igata et al PRD(11)-a1005 [constrained, particle around a black hole];
Feroze MPLA(10)-a0905 [for curved spacetimes];
Cariglia et al CQG(14)-a1401 [Killing tensors and canonical geometry];
Maughan & Torre GRG(18) [and affine symmetries];
> s.a. diffusion.
@ Approaches, special situations: Popovych & Sergyeyev PLA(10) [two independent variables, normal form of evolution equations];
Peng a1607 [for differential-difference equations];
Ruggieri & Speciale JMP(17)-a1612 [pdes, algorithmic computation];
Dray in(17)-a1701 [piecewise conserved quantities].
> Related topics: see angular
momentum; charge [including non-conservation]; electromagnetism;
energy; energy-momentum tensors;
interaction; momentum.
References > s.a. Superposition Principle.
@ General: Takens 77;
Uhlenbeck in(83);
Amigo & Reeh FdP(88)
[additive constants of motion]; Serre 00;
Jiang & Li IJTP(07) [classical vs quantum];
Popovych & Samoilenko JPA(08) [complete description for second-order 2-dimensional equations];
Haywood 10 [group theory];
Lange PhSc(11) [conceptual];
Herrmann a1901 [unified formulation];
Maudlin et al SHPMP-a1910
[history, role, and semiclassical gravity].
@ Other types: Gorni & Zampieri a1412 [non-local constants of motion in Lagrangian dynamics];
Post & Winternitz a1501
[N-th order integrals of motion].
@ Related topics: Olszewski AJP(83)oct [non-conservation of E, p, L in non-static spacetime];
Sidharth CSF(00)qp/98 [statistical view];
Fagotti a1408 [quasi-conserved operators];
Reintjes a1510 [constrained systems of conservation laws];
Palese & Winterroth a2012-proc [Noether
theorems, 'proper' and 'improper' conservation laws, and conservation of energy in general relativity].
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