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].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jul 2008