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-conf [quantum spacetime];
van Wezel & van den Brink AJP(07)jul [simple procedure];
Baker PhSc(11)jan [and spacetime];
Landsman SHPMP(13)-a1305.

**Spontaneous Symmetry Breaking in Field Theory** > s.a. critical
phenomena; Mermin-Wagner Theorem; phase transitions;
vector fields.

* __Idea__: Suppose we have
a Lagrangian density \(\cal L\) 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.

* __Rem__: It must be then that the symmetry group
*G* is not implementable by operators that establish the unitary equivalence between
representations; Rather, the unitary operator that implements the symmetry (by Wigner's theorem)
connects the folia of unitarily inequivalent representations.

* __Mechanism__: The way this usually happens is
that the potential *V* in \(\cal L\) contains a parameter, say *m*, such that for some
values of *m* the ground state has the same symmetries as \(\cal L\), and for some others
it does not; if at high temperatures the effective potential *V* has a vacuum with symmetry
group *G* and at low temperatures one with symmetry group *G*/*H*, as the system
cools it may stay for some time in the symmetric, false vacuum, and then decay to the less symmetric, true vacuum.

* __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.

@ __General references__:
Cianciaruso et al a1408, a1604 [classical nature of ordered phases].

@ __Vacuum decay__: Beckwith mp/04 [by tunneling Hamiltonian];
Czech PLB(12)-a1112 [by "barnacle" instability]; > s.a. vacuum.

**Nambu-Goldstone Bosons / Theorem** > s.a. higgs mechanism.

* __Idea__: Every generator of a spontaneously broken Lie symmetry group
*G* \(\mapsto) *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)dec
[meaning of breaking of gauge symmetry]; Guralnik MPLA(11)-a1107 [and gauge invariance]; Friederich EJPS-a1107 [breaking of gauge symmetry]; Kartavtsev a1404-wd [proof revisited].

@ __Known particles__: Bjorken ht/01-conf
[photon]; Kraus & Tomboulis
PRD(02)ht [photon
and graviton, Lorentz symmetry].

@ __Spacetime symmetries__: Low & Manohar PRL(02)ht/01; Kobayashi & Nitta PRL(14)-a1402 [and internal symmetries].

@ __In 1+1 dimensions__:
Coleman CMP(73);
Faber & Ivanov ht/02;
Balachandran & Immirzi IJMPA(03)ht/02 [fuzzy].

@ __Related topics__: Strocchi PLA(00)
[classical counterpart]; Bluhm &
Kostelecký PRD(05)ht/04,
Berezhiani & Kancheli a0808 [for
breaking of Lorentzian symmetry]; Watanabe et al PRL(13)-a1303 [massive Nambu-Goldstone bosons]; news ns(14) sep [observation in superconductors].

**Models and Examples** > s.a. decoherence;
effective quantum field theory; electroweak
theory; 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) \(\mapsto\) SU(2) *σ*-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__:
It 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

\(\cal L\) = (∂_{a}*φ*)(∂^{a}*φ*)
– *m*^{2}*φφ** – *λ* (*φφ**)^{2} .

which has a global U(1) symmetry *φ*(*x*) \(\mapsto\) exp{i*θ*}
*φ*(*x*); For *m*^{2} ≥ 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 references__: 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 LNP(08)-a0708 [spacetime
symmetries]; Jona-Lasinio PTP(10)-a1010.

@ __Lorentz group__: Yokoi PLB(01)ht/00 [2+1];
Chkareuli et al NPB(01) [non-observability,
and symmetry generation].

@ __Diffeomorphism group__: Giddings PLB(91);
Requardt a1203 [with gravitons as Goldstone modes];
Lin & Labun a1501 [low-energy effective theory];
Bluhm a1601-MG14 [gravity theories with background fields];
> s.a. Effective Field Theory; Induced Gravity.

@ __And gravity__: Ashtekar GRG(78)
[due to gravitational interaction]; Helesfai CQG(08)-a0806 [in loop quantum gravity];
Meierovich JETP(09)-a0910 [2
extra dimensions, vector order parameter], PRD(10) [vector order parameter,
covariant equations]; Wise JPCS(12)-a1112 [spontaneous breaking of Lorentz symmetry by an observer and Hamiltonian gravity]; Krasnov PRD(12)-a1112; > s.a. conformal gravity.

@ __Other examples__: Girotti et
al PRD(03)ht/02 [non-commutative field theory];
Cseh & Tímár JPA(06)
[2D interacting boson model, kinematics vs dynamics]; Petersen et al JHEP(09)-a0907 [U(1)^{N} theories,
patterns of remnant discrete symmetries];
Muñoz et al a1111 [classical model for spontaneous symmetry breaking];
Muñoz et al AJP(12)oct-a1205 [toy model]; Birman et al PRP(13) [in finite quantum systems]; Odagiri FP(14) [dilatation symmetry breaking and standard model gauge couplings];
Yoshimura PTEP(16)-a1604 [lepton-number violation]; Dong et al a1609 [many-body systems, quantitative]; > s.a. Chiral Symmetry; conformal symmetry;
crystals [time crystals]; inflationary scenarios;
PT Symmetry; supersymmetry;
Time Reversal Symmetry Violation; topological defects.

**References**

@ __Pedagogical intros, reviews__: Crone & Sher AJP(91)jan;
Haft ht/97;
Tsou ht/98-conf;
Bernstein AJP(11)jan [and Higgs mechanism];
Strocchi a1201 [Scholarpedia];
Hamilton a1512-ln [for mathematicians].

@ __Historical__: Brout & Englert ht/98-conf;
Straumann hp/98-in
[including
elasticity and hydrodynamics]; Brout ht/02-conf;
Englert ht/02-conf,
ht/04-conf;
Shirkov PU(09)-a0903,
MPLA(09);
Guralnik IJMPA(09)-a0907, a1110-proc;
Sardella a1012/NCC [spontaneous breaking].

@ __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);
Pérez & Sudarsky IJMPA(11)-a0811 [symmetries
of the vacuum]; Nambu IJMPA(09);
Jona-Lasinio PTPS(10)-a1003-conf [conceptual, as analogy]; Baker & Halvorson SHPMP(13)-a1103 [unitarily inequivalent representations and unitary operators];
Fraser PhSc(12) [conceptual, quantum statistical mechanics vs quantum field theory];
Hamma et al PRA(16)-a1501 [and quantum mutual information]; Becchi a1607 [renormalizable theories].

@ __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]; Tomasello et al EPL(11)-a1012 [and quantum correlations]; Bogoslovsky IJGMP-a1201 [phase transitions and Finslerian event space]; > s.a. Emergence.

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