Symmetries
in Physical Theories| Symmetries
in Physical Theories |
In General > s.a. Central
Charge; crystals.
* 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 1930's.
* Applications: They
are at the basis of gauge theories and particle statistics; They help establish
physical laws and control their validity by imposing restrictions;
classification of crystals.
> In mathematics: see differential
equations (ordinary and partial); group
theory; integral equations; killing
fields.
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.
@ Internal vs external: Wisnivesky IJMPA(00)
[unified];
Kantorovich SHPMP(03)
[internal]; Aldaya & Sánchez-Sastre JPA(06)mp; > s.a. types
of gauge theories.
@ Special types: Krauss GRG(90)
[discrete local]; Wotzasek AP(95)ht [hidden];
Anco & Bluman JMP(96)
[non-local, and conservation laws]; Mostafazadeh & Samani
MPLA(00)
[topological]; Varlamov
IJTP(01)mp/00 [discrete];
> s.a. wave equations.
Spacetime Symmetries > s.a. axisymmetry; conformal
invariance; Helical; killing
fields; Translations.
* Types: A vector field X may
generate different kinds of symmetries, Isometry, 
X gab =
0; Conformal
isometry, X gab = 
gab,
with
a function; Affine
collineation,
X
abc
= 0; Projective collineation, X
abc =
2 
a[b f, c],
with f a function; Curvature collineation, X Rabcd
= 0.
@ General references: Katzin & Levine JMP(81);
Hall CQG(89), GRG(98);
Duggal & Sharma
99; Hall 02; Harte a0805 [approximate,
and conservation laws]; Roberts BJPS(08)
[dynamical symmetries vs empirical
symmetries].
@ And field theory: Halliwell PRD(91)
[parametrization of hypersurface embedding].
> And gravity: see bianchi
models; general
relativity; initial value formulation; minisuperspace; solutions
with
symmetries; supergravity.
And Dynamical Theories > s.a. conservation
laws; formulations
of classical mechanics; lagrangian dynamics; 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 
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 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]; Lange SHPMP(07)
[and conservation laws]; Ferrario & Passerini EJP(07)
[Lagrangian vs dynamical].
@ And constraints: Lee & Wald JMP(90);
Giulini MPLA(95)gq/94; Chitaia
et al PRD(97), PRD(97);
> s.a. 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
[susy and others, rev]; > s.a. sigma-models.
> Types of theories:
see fields
with higher spin; gauge theory; integrable
system.
References > s.a. lie
algebra; physics
teaching.
@ Books, I: Lederman & Hill 04 [and physics]; Zee 07 [and physics,
r
JPA(08)#1].
@ Books: Weyl 52; Takens 77; Elliot & Dawber 79; Rosen 83; Froggatt & Nielsen
85; Doncel et al 87; Yaglom 88; Bunch 89; Rosen 95; Stewart & Golubitsky
93 [I]; Singer 01 [II/III]; Brading & Castellani ed-03 [philosophical];
Prakash 03 [including super].
@ General articles: Feynman TPT(66);
Rosen FP(90);
Sudarshan FP(95)
[classifying systems ito symmetry groups]; Gières ht/97-in;
Kosso BJPS(00)
[observation]; Suppes FP(00)
[invariance and covariance]; Brading & Castellani qp/03-in;
Esposito & Marmo mp/05-in
[rev]; Hon & Goldstein PhSc(06)
[types of symmetry arguments]; Giulini a0802 [types, in Pauli's work].
@ And group theory: Rosen AJP(81)RL.
@ History:
Brading SHPMP(02)
[Noether & Weyl]; Katzir HSPBS(04)
[origin]; Hon & Goldstein SHPSA(05)
[evolution].
@ And teaching: Mangelsdorf & Heald AJP(90);
Bloembergen AJP(90);
Rosen AJP(90);
Hill & Lederman phy/00.
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20 jul 2008