Symmetry Breaking  

In Quantum Theory > s.a. modified lorentz symmetry.
* Idea: A quantum theory may break symmetries that hold classically, either at the level of the theory itself or its solutions (the latter is spontaneous symmetry breaking, see below), and then some symmetries are lost, or may replace the symmetry group by a deformed one.
@ References: Amelino-Camelia gq/02-in [quantum spacetime]; van Wezel & van den Brink AJP(07) [simple procedure].

Spontaneous Symmetry Breaking in Field Theory > s.a. critical phenomena; phase transitions.
* Idea: Suppose we have a Lagrangian which is invariant under a certain Lie group of transformations G; We say that the symmetry group G is broken to a subgroup G/H if the ground state of the theory is only invariant under the action of G/H; It will then also be degenerate, the subgroup H taking one vacuum into another.
* Mechanism: The way this usually happens is that the potential V in the contains a parameter, say m, such that for some values of m the ground state has the same symmetries as , and for some others it does not.
* Symmetry restoring: It can be usually achieved by adding energy to the system, through a phase transition.
* Restrictions: It cannot occur in 1+1 dimensions, since in this case there is no well-posed theory of a massless scalar field.

Goldstone Bosons / Theorem > s.a. Higgs Mechanism.
* Idea: Every generator of a spontaneously broken Lie symmetry group G G/H gives rise to a collective boson excitation, a (Nambu-)Goldstone mode described by a field with values in G/H, or a massless particle called a Goldstone boson.
* Conditions: A spontaneous symmetry breaking gives rise to Goldstone bosons only in the case of a global symmetry breaking; In the gauge symmetry case the degenerate directions are gauge and don't correspond to physical modes; Both types however give rise to topological defects.
@ General references: Goldstone NC(61); Goldstone, Salam & Weinberg PR(62); Burgess ht/98-ln [primer], PRP(00)ht/98 [nuclear, particle and condensed-matter physics]; Chodos & Gallatin JMP(01)mp/00; Smeenk PhSc(07) [meaning of breaking of gauge symmetry].
@ Known particles: Bjorken ht/01-in [photon]; Kraus & Tomboulis PRD(02)ht [photon and graviton, Lorentz symmetry].
@ Spacetime symmetries: Low & Manohar PRL(02)ht/01.
@ Related topics: Strocchi PLA(00) [classical counterpart]; Coleman CMP(73), Faber & Ivanov ht/02 [1+1]; Balachandran & Immirzi IJMPA(03)ht/02 [1+1, fuzzy]; Bluhm & Kostelecky PRD(05)ht/04 [for breaking of Lorentzian symmetry].

Models and Examples > s.a. decoherence; effective quantum field theory; electroweak; mass [origin]; Peccei-Quinn Mechanism.
* Applications: Used in unified theories of fields in the early universe; Can lead to inflation.
* Models: To write down a phenomenological model that realizes a spontaneous symmetry breaking, it is convenient to choose a field valued in the new vacuum; e.g., an SU(2)-valued field in the SU(2) × SU(2) SU(2) -model; One uses normally a spacetime scalar to break the symmetry, so that the vacuum expectation value will not break Poincaré invariance.
* Of Lorentz symmetry: Could arise in a theory with non-vanishing vacuum expectation values of vector fields.
* Physical examples: Crystals; Rotational levels of a deformed nucleus; The most deeply bound states of hadrons; Ferromagnets.
* Specific model: Consider the Lagrangian

= (a)(a) – m2* – (*)2 .

which has a global U(1) symmetry (x) exp{i} (x); For m2 0, the ground state is also symmetric with respect to U(1) transformations, while, for m < 0, the U(1) symmetry is spontaneously broken.
@ General: Cho PRL(85); Kerbrat et al RPMP(89) [electroweak, geometric]; Vachaspati hp/97-ln [early universe, rev]; Witten BAMS(07) [applications to superconductors, four-manifold theory, and particle physics]; Rabinovici a0708 [spacetime symmetries].
@ Lorentz group: Yokoi PLB(01)ht/00 [2+1]; Chkareuli et al NPB(01) [non-observability, and symmetry generation].
@ Diffeomorphism group: Giddings PLB(91); > s.a. Induced Gravity.
@ Other examples: Ashtekar GRG(78) [due to gravitational interaction]; Girotti et al PRD(03)ht/02 [non-commutative field theory]; Cseh & Tímár JPA(06) [2D interacting boson model, kinematics vs dynamics]; > s.a. Chiral Symmetry; conformal symmetry; inflationary scenarios; PT Symmetry; supersymmetry; Time Reversal Symmetry Violation; topological defects.

References
@ Pedagogical intros, reviews: Crone & Sher AJP(91); Haft ht/97; Tsou ht/98-in.
@ Historical: Brout & Englert ht/98-in; Straumann hp/98-in [including elasticity and hydrodynamics]; Brout ht/02-in; Englert ht/02-in, ht/04-in.
@ General: Nambu PRL(60); Nambu & Jona-Lasinio PR(61), PR(61); Abud & Sartori AP(83) [geometrical, classification]; Coleman 85; Giddings & Wilczek MPLA(90); Fujita et al ht/04 [criticism of Nambu & Jona-Lasinio]; Liu & Emch SHPMP(05) [2 accounts]; Fujita et al ht/05 [in quantum field theory, reinterpretation]; Birtea et al IJGMP(06).
@ Non-perturbative: Bender & Milton PRD(97)ht/96; Dzhunushaliev FPL(03)ht/02; Strocchi 05.
@ Michel's theorem: Michel CRAS(71); Gaeta & Morando AP(97)mp/02.
@ Related topics: Gaeta PRP(90) [and bifurcation]; D'Hoker & Weinberg PRD(94)hp [effective actions]; Lepora JHEP(99) [geometry of vacuum]; Beckwith mp/04 [vacuum decay by tunneling Hamiltonian].

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