Symmetries
in Physical Theories |

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

* __History__: They
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; 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__:

* __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]; > 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].

@ __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; 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__:
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];
Henkel Symm(15)-a1509
[dynamical symmetries and causality]; Fang et al CTP(16)-a1601
[quantifying approximate symmetries of Hamiltonians or states, degree of
symmetry].

@ __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].

@ __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. 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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