Symmetries in Physical Theories |
In General
> s.a. Central Charge;
crystals [including generalized symmetries].
* Idea: A symmetry is a mapping
of a structured object onto itself which preserves the structure.
* History: Symmetries
entered physics as properties of structures (from an earlier notion that
applied to architecture), then motions, finally laws and actions; There,
they led to a fruitful connection with conserved quantities, through
Hamel's and Noether's work; Their importance was recognized especially
after Wigner's work in the 1930s.
* Applications: They are at
the basis of gauge theories and particle statistics; They help establish
physical laws and control their validity by imposing restrictions; They
are associated with conservation laws in dynamical theories; They lead
to the classification of crystals.
* Origin: They can be
postulated as fundamental (e.g., in the action), or they might emerge
dynamically (e.g., in solutions of field equations); Symmetries and
physical laws might also arise naturally from some essentially random
dynamics (e.g., as proposed in Froggatt & Nielsen 91).
Kinds of Symmetries
> s.a. gauge symmetries [Lie groupoids as generalized symmetries];
hamiltonian dynamics; Supersymmetry;
symplectic manifolds [reduction].
* According to what they act
on: Acting on the variables of the action, leading to conserved
quantities; On the equations of motion or field equations; On a solution
of the equations of motion or field equations; On the underlying
manifold (internal/external).
* Global vs local: In a modern
understanding, global symmetries are approximate and gauge symmetries may be emergent
[@ Witten a1710].
* According to what they
leave invariant: Symmetries realized by a unitary operator, under
which wave functions may pick up a phase; Gauge symmetries, which leave
all the physics invariant – all the observables in classical
mechanics, and all the amplitudes in quantum mechanics.
@ General references: Sudarshan FP(95) [classifying systems in terms of symmetry groups];
Hon & Goldstein PhSc(06)oct [types of symmetry arguments];
Giulini in(09)-a0802 [types, in Pauli's work];
Healey BJPS(09) [empirical vs theoretical];
Mouchet EPJH(13)-a1111 [different meanings];
Strocchi a1502
[symmetry, symmetry breaking and gauge symmetries].
@ Internal vs external:
Wisnivesky IJMPA(00) [unified];
Kantorovich SHPMP(03) [internal];
Aldaya & Sánchez-Sastre JPA(06)mp;
Kim & Noz a1007-proc [internal];
László JPA(17)-a1512 [unification];
> s.a. types of gauge theories.
@ Discrete:
Krauss GRG(90) [local];
Varlamov IJTP(01)mp/00;
Ishimori et al PTPS(10)-a1003,
12 [in particle physics];
Sozzi 12.
@ Special types: Wotzasek AP(95)ht [hidden];
Anco & Bluman JMP(96) [non-local, and conservation laws];
Mostafazadeh & Samani MPLA(00) [topological];
Cariglia RMP(15)-a1411 [hidden];
Czachor QSMF(14)-a1412 [relativity of arithmetics];
Gomes IJMPA(16)-a1510-ln [emergent];
Andersson et al JHEP(21)-a1909 [nilpotent symmetries and grand unification];
> s.a. wave equations.
Spacetime Symmetries
> s.a. geometry in quantum gravity; killing
fields; lorentzian geometry; Relativity.
* Types: A vector field
X may generate different kinds of symmetries, Isometry,
\(\cal L\)X
gab = 0;
Conformal isometry, \(\cal L\)X
gab = α
gab, with
α a function; Affine collineation,
\(\cal L\)X
Γabc
= 0; Projective collineation, \(\cal L\)X
Γabc
= 2 δa[b
f, c], with f
a function; Curvature collineation, \(\cal L\)X
Rabcd
= 0.
@ General references:
Katzin & Levine JMP(81);
Hall CQG(89),
GRG(98);
Duggal & Sharma 99;
Hall 04;
Harte CQG(08)-a0805 [approximate, and conservation laws];
Roberts BJPS(08)
[dynamical symmetries vs empirical symmetries];
Saifullah NCB(07)-a0902 [classification];
Houri & Yasui CQG(15)-a1410 [test];
Ayón-Beato & Velázquez-Rodríguez PRD(16)-a1511 [residual symmetries of a gravitational Ansatz].
@ In gravity theories: Bojowald IJMPD(16)-a1712 [Lagrangian and Hamiltonian gravity];
Tomitsuka et al a2012 [asymptotic symmetries];
> s.a. asymptotic flatness.
@ Other field theory: Halliwell PRD(91) [parametrization of hypersurface embedding];
Costa & Fogli 12;
Smolić CQG(15)-a1501 [symmetry inheritance, scalar fields];
Alexandre et al PRD(20)-a2006 [quantum field theories with PT symmetry].
> And gravity:
see bianchi models; general relativity;
initial-value formulation; minisuperspace;
solutions with symmetries; supergravity.
> Special types:
see axisymmetry; Collineations;
conformal invariance; Helical Symmetry;
Superrotations; Supertranslations;
Translations.
And Dynamical Theories
> s.a. conservation laws; formulations of classical
mechanics; interaction; Lambda Symmetries;
Mechanical Similarity.
* Idea: We say that a physical
theory has a certain symmetry if, given a solution q(t) for
the equations of motion, the transformation q \(\mapsto\) q':=
f(q) yields another solution; These transformations usually
have a group structure.
* Conservation laws: They
can be obtained from Noether's theorem, Lutzky's theorem, bi-Hamiltonian
formalism, or bidifferential calculi.
* Canonical framework:
A transformation q \(\mapsto\) q' can be extended
to a canonical one.
@ General references: Caratù et al draft-77;
Barnich & Brandt NPB(02)ht/01 [field theory, covariant theory];
Gitman & Tyutin BJP(06)ht/05 [equivalent Lagrangian and Hamiltonian systems];
de León et al IJGMP(04)mp [field theory, multisymplectic];
Bogoyavlenskij CMP(05) [hidden structures];
Ferrario & Passerini EJP(07) [Lagrangian vs dynamical];
Wilczek MPLA(10) [symmetry transmutation];
Boozer EJP(12) [in classical mechanics];
Andersson et al CQG(14)
+ Bäckdahl CQG+ [conditions for existence of symmetry operators for field equations];
Fang et al CTP(16)-a1601
[quantifying approximate symmetries of Hamiltonians or states, degree of symmetry];
Ali et al a2012
[global symmetries in quantum gravity].
@ Dynamical symmetries:
Henkel Symm(15)-a1509 [and causality];
Leviatan a1901-proc;
Gryb & Sloan a2103 [dynamical similarities].
@ And conservation laws:
Lange SHPMP(07);
Cicogna MMAS(13)-a1307 [and generalizations];
Peng JDEA(14)-a1403 [for difference systems];
Sharapov Sigma(16)-a1607 [based on variational tricomplex with a presymplectic structure];
Strocchi a1711.
@ And constraints: Lee & Wald JMP(90);
Giulini MPLA(95)gq/94;
Chitaia et al PRD(97),
PRD(97);
> s.a. constrained systems;
symmetries in quantum physics.
@ Noether symmetries:
Rosenhaus & Katzin JMP(94) [for differential equations];
Pons & García IJMPA(00)ht/99 [constrained systems];
Brading & Brown ht/00 [gauge symmetries];
García & Pons IJMPA(01)ht/00 [canonically realized in enlarged phase space].
@ Non-Noether: Chavchanidze GMJ(01)mp,
mp/01,
JGP(03)mp/02 [and bi-Hamiltonian systems],
mp/02 [and conservation laws].
@ Non-linear realizations: Love MPLA(05)ht [supersymmetry and others, rev];
> s.a. sigma-models.
> Types of theories:
see fields with higher spin; gauge theory;
integrable system; lagrangian dynamics;
noether theorem [non-local theories]; special relativity.
References > s.a. dualities;
lie algebra; physics teaching.
@ Books, I: Lederman & Hill 04 [and physics];
Zee 07 [and physics].
@ Books: Weyl 52;
Takens 77;
Elliott & Dawber 79;
Rosen 83;
García Doncel et al 87;
Yaglom 88;
Bunch 89;
Froggatt & Nielsen 91;
Rosen 95;
Stewart & Golubitsky 93 [I];
Singer 01 [II/III];
Brading & Castellani ed-03 [philosophical];
Prakash 03 [including super];
Mainzer 05 [and complexity];
Debs & Redhead 07 [objectivity, invariance, and convention];
Haywood 10
[group theory, r CP(120#2];
Goldberg 13;
Sundermeyer 14 [in fundamental physics].
@ General articles:
Feynman TPT(66);
Rosen FP(90);
Gières ht/97-proc;
Kosso BJPS(00) [observation];
Suppes FP(00) [invariance and covariance];
Chester ISPS(02)-a1202;
Brading & Castellani qp/03-ch;
Esposito & Marmo in(04)mp/05 [rev];
Zuber a1307-conf [and Klein's Erlangen program];
Alamino a1305 [generalization to symmetry on average];
Lederer a1401 [philosophical approach];
Mouchet a1503 [discovered or invented?];
Das & Kunstatter a1609-JAFS [and unification].
@ And group theory: Rosen AJP(81)apr [RL];
Guay & Hepburn PhSc(09)apr [groups vs groupoids];
> s.a. group theory; Semigroups.
@ Other mathematics: Yanofsky & Zelcer FoS(16)-a1502;
> s.a. differential equations (ordinary
and partial); integral equations;
killing fields.
@ History:
Brading SHPMP(02) [Noether & Weyl];
Katzir HSPBS(04) [origin];
Hon & Goldstein SHPSA(05) [evolution];
Wilczek MPLA(10)-a1008-in [and BCS theory];
Maldacena a1410
[in particle physics, and the Higgs boson].
@ And teaching:
Mangelsdorf & Heald AJP(90)feb;
Bloembergen AJP(90)feb;
Rosen AJP(90)aug;
Hill & Lederman TPT(00)phy.
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