Gauge Theories of the Yang-Mills Type| Gauge Theories of the Yang-Mills Type |
In General
> s.a. gauge theory; Wilson Loop.
* Idea: Gauge theories with
a specific form for the action/equations of motion; The original type of
gauge theory, and still the most widely used one.
* Evidence: The first direct
evidence came with the 1973 discovery of flavor-changing neutral currents
at CERN.
Classical Dynamics
> s.a. Faddeev-Niemi Equations; gauge theory
[including loop-based variables]; lattice gauge theory; twistors.
* Action: In terms of the curvature
Fab of the Lie algebra-valued
connection Aa ("tr"
uses the metric on G),
S[A] = \(1\over2\)∫M dv gab gmn tr(Fam Fbn) = # ∫M tr(F ∧ *F).
* Field equations: They are the source equations, and
\(DF:= {\rm d}F + [A, F] = 0\;,\qquad D^*F = J\) ;
the first one is the Bianchi identity, and the second is conformally invariant
(masslessness; J is the source current).
* Matter fields: The coupling
of the gauge field to matter is usually taken to be minimal; The free term
"FF" is added to the matter Lagrangian, and all derivatives
∇ of matter fields are replaced by gauge-covariant derivatives D.
@ General references: Boozer AJP(11)sep [coupled particle-field system, 2D].
@ Action: Aldrovandi & Pereira RPMP(88) [existence of Lagrangians]; & Sengupta;
Fine JMP(00)m.DG [lower bound on a Riemann surface];
Tolksdorf JGP(07) [form linear in the curvature];
Escalante & Berra IJPAM-a1301 [as a constrained BF-like theory];
Brandt et al AP(19)-a1810 [first-order, in background field].
Hamiltonian Formulation and Evolution
> s.a. hamiltonian systems; solutions
[singularities]; types of yang-mills theories [1+1 dimensions].
* Hamiltonian / Energy:
H(A, E) = ∫ d3x tr(E2 + B2) = # ∫ dv gab gmn tr(Fam Fbn) .
@ General references: Cronstrom ht/98,
ht/99;
Śniatycki RPMP(99);
Ozaki IJMPA(01)ht/00 [QCD];
Vignolo et al IJGMP(05)mp;
Reinhardt et al a0807-conf,
PoS-a0911 [Coulomb gauge];
Gerhardt CMP(10)-a0908 [energy gap];
Prokhorov & Shabanov 11;
Balachandran et al MPLA(17)-a1704 [equations of motion as constraints].
@ Gauge-invariant variables:
Lunev TMP(93) [and Yang-Mills-Higgs];
Freidel ht/06;
Guo AP(16)-a1410 [metric-like variables].
@ Cauchy problem: Segal JFA(79);
Eardley & Moncrief CMP(82),
CMP(82).
@ With matter: Lusanna IJMPA(95),
IJMPA(95) [fermions].
@ Structure of configuration space:
Fuchs et al NPB(94);
Fuchs ht/95-conf;
Rajeev & Rossi JMP(96) [on a cylinder];
Pause & Heinzl NPB(98)ht;
Nair & Yelnikov NPB(04)ht/03 [measure];
Agarwal & Nair NPB(09) [on \(\mathbb R\) × S2].
Other Issues and Effects > s.a. chaos in field theories;
gauge theory solutions; QCD [confinement].
* Gauge choice:
See Axial, Coulomb, Lorenz gauge.
* Mass generation:
A possible mechanism is through a topological coupling of vector
and tensor fields; After integrating over the tensor degrees of
freedom, one arrives at an effective massive theory that is gauge
invariant but non-local.
@ Mass generation:
Flores-Baez et al IJMPA(06) [without Higgs];
Sorella AP(06)-a0704;
Savvidy PLB(10)-a1001 [topological];
Frasca a1007 [and supersymmetry].
@ Related topics: DeGrand et al NPB(98) [SU(2), topological susceptibility];
Edwards et al PLB(98) [fractional topological charge];
Gambini et al PRD(99)gq/98 [Immirzi-like parameter];
Bizoń & Tabor PRD(01) [singularities and critical phenomena];
Fischer et al AP(09) [infrared behavior, Landau gauge].
> Other effects: see higgs mechanism;
Memory Effects; thermodynamics;
theta sectors.
> Related topics: see boundaries;
duality; entropy bounds;
Makeenko-Migdal Equation.
References
> s.a. gauge theory / BRST; history
of physics; lattice gauge theory; QCD [including finite-temperature
theory]; string phenomenology.
@ Articles, I: 't Hooft SA(80)jun;
Wilczek PW(89)feb;
Barlow EJP(90).
@ Books, II: Aitchison & Hey 12.
@ Books and reviews: Abers & Lee PRP(73);
Coleman in(75);
Iliopoulos pr(76);
Mayer 77;
Yang NYAS(77);
O'Raifeartaigh RPP(79);
Bleecker 81;
Gaillard & Stora ed-83;
Chaichian & Nelipa 84;
Cheng & Li 84;
Bailin & Love 86;
O'Raifeartaigh 86;
Carmeli et al 89;
Mills AJP(89)jun;
Huang 92;
Cheng & Li 00 [problems];
Frampton 00;
Pokorski 00;
't Hooft ed-05;
Quigg 13.
@ As perturbed topological field theory: Cattaneo et al CMP(98) [deformed BF theory];
Kondo PRD(98)ht,
IJMPA(01);
Rovelli & Speziale GRG(07)gq/05 [equivalent to perturbed abelian theory].
@ Geometrical approaches:
Cianci et al JPA(03),
add JPA(04);
Catren SHPMP(08);
Weatherall a1411 [and general relativity].
@ Reformulations: Faddeev & Niemi PRL(99)ht/98 [partially dual variables];
Majumdar & Sharatchandra PLB(00)ht [as ADM-type theory of metrics];
Sevostyanov mp/04-wd [as integrable, near ground state];
Carta et al AdP(06)ht/05 [Koopman-von Neumann formulation];
Gorsky & Rosly JHEP(06)ht/05 [light-cone formalism];
Catren & Devoto CMP(08)-a0710 [extended connection];
Maraner & Pachos PLA(09) [from fermionic lattice models];
Ferreira & Luchini PRD(12)-a1205 [integral formulation];
Mitra & Sharatchandra AHEP-a1307 [dreibein and metric as prepotential];
Koenigstein et al IJMPE(16)-a1601 [from canonical transformations];
> s.a. gauge theory [variables]; Operad;
sheaf theory.
Other Topics
> s.a. connections; solutions; types
of yang-mills theories [including curved and higher-dimensional spacetimes, generalizations].
@ General references:
Śniatycki RPMP(88) [charges];
Wald in(88);
Henneaux & Teitelboim PLB(90);
Henneaux et al NPB(90);
Śniatycki CMP(91);
Shabanov PLB(01)ht [infrared limit];
Capri et al PRD(05)ht [non-local mass operator];
Marateck a0712 [rederivation].
@ Gauge-invariant calculations: Arnone et al PRD(03)ht/02;
Rosten JPA(06)ht/05;
> s.a. renormalization.
@ Symmetries: Marchildon JGTP(95)mp/03 [Lie symmetries];
Torre JMP(95);
Pons et al JMP(00)gq/99 [Einstein-Yang-Mills theory];
Pohjanpelto DG&A(04) [local, semi-simple, classification];
Strominger JHEP(14)-a1308 [asymptotic symmetries at future null infinity];
Tanzi & Giulini JHEP(20)-a2006 [asymptotic];
> s.a. conservation laws; solutions.
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