Symmetries in Physical Theories |

**In General**
> s.a. Central Charge;
crystals [including generalized symmetries].

* __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 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];
Laszlo 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];
László & Andersson a1909 [local, without gauge bosons];
> s.a. wave equations.

**Spacetime Symmetries**
> s.a. killing fields; lorentzian geometry;
Relativity.

* __Types__: A vector field
*X* may generate different kinds of symmetries, __Isometry__,
\(\cal L\)_{X}
*g*_{ab} = 0;
__Conformal isometry__, \(\cal L\)_{X}
*g*_{ab}
= *α* *g*_{ab},
with *α* a function; __Affine collineation__,
\(\cal L\)_{X}
Γ^{a}_{bc}
= 0; __Projective collineation__, \(\cal L\)_{X}
Γ^{a}_{bc}
= 2 δ^{a}_{[b}
*f*_{, c]}, with *f*
a function; __Curvature collineation__, \(\cal L\)_{X}
*R*^{a}_{bcd}
= 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 a1511 [residual symmetries of a gravitational Ansatz];
Bojowald IJMPD-a1712 [in Lagrangian and Hamiltonian gravity].

@ __And field theory__: Halliwell PRD(91) [parametrization of hypersurface embedding];
Costa & Fogli 12;
Smolić CQG(15)-a1501 [symmetry inheritance, scalar fields].

> __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].

@ __Dynamical symmetries__:
Henkel Symm(15)-a1509 [and causality];
Leviatan a1901-proc.

@ __And conservation laws__: Lange SHPMP(07);
Cicogna MMAS(13)-a1307 [and generalizations];
Peng 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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