|Gauge Field Theories|
In General > s.a. gauge
transformations and gauge choices;
history of physics; lorentz
group phenomenology; symmetry.
* Motivation: Make a global symmetry into a local one (observers at different points can choose independently); Masslessness of gauge particles related to renormalizability (but see the Higgs mechanism); Can treat monopoles without singularities in potentials; Geometric picture of fields obtained using fiber bundle language.
* History: The principle was introduced by Weyl; The fiber bundle picture appeared in the late 1960s, but was accepted only around 1973.
* Approaches: Modern mathematical formulations include ordinary differential geometry of fiber bundles, compactified extra dimensions in Kaluza-Klein theories, Grassmanian models, non-commutative geometry, and transitive Lie algebroids.
* Idea: In the differential geometry approach, the basic objects are a semisimple (in order for it to have a non-degenerate metric) Lie group G, with Lie algebra g, and a principal G-bundle P over spacetime; The variables are a g-valued connection 1-form (i e) A on this principal fiber bundle (often used interchangeably with a gauge potential, the pullback of the connection 1-form), and possibly coupled matter fields (cross-sections φ of associated G-bundles); If (i e) F is the curvature of the connection, and D its associated covariant derivative, one field equation is the Bianchi identity,
DF := dF + [A, F] = 0 ;
Other field equations will depend on the form of the action chosen (careful,
F = dA + A ∧ A ≠ DA !).
@ Texts and reviews: Göckeler & Schücker 87; Cheng & Li AJP(88)jul [RL]; Chan & Tsou 93; Tsou ht/00-ln.
@ Texts, and differential geometry: Marathe & Martucci 92; Naber 00, 11.
@ Potentials generating the same fields: Majumdar & Sharatchandra PRD(01)ht/98.
Line / Loop and Other Variables > s.a. BF
theory; connection; Field
Line; holonomy; QCD;
quantum gauge theory; topological field theories.
@ Wilson loops: Mandelstam AP(62); Wu & Yang PRD(75), PRD(76), PRD(76); Kozameh & Newman PRD(85) [differential holonomies and Yang-Mills equations]; Chan et al AP(86); Gambini & Trias NPB(86); Diakonov & Petrov PLB(89); Bezerra & Letelier CQG(91)refs; Rajeev & Turgut JMP(96)ht/95.
@ Gauge-invariant: Newman & Rovelli PRL(92) [lines of force]; Loll CQG(93)gq [inequalities on traces of holonomies]; Armand-Ugón et al PRD(94)ht/93 [loop variables]; Frittelli et al PRD(94) [Faraday lines]; Chechelashvili et al TMP(96)ht/95; Ganor & Sonnenschein IJMPA(96)ht/95; Haagensen ht/95, et al NPB(96)ht/95; Kijowski et al RPMP(87); Zapata JMP(97)gq [graphs]; Faddeev & Niemi PRL(99)ht/98, PLB(99)ht/98, PLB(99)ht; Blaschke et al ht/00 [topological invariants for QCD]; Orland PRD(04)ht; Ferreira & Luchini a1109 [and global properties].
@ Fluxes: Dzhunushaliev et al PLB(00) [flux tubes]; Freed et al AP(07)ht/06, CMP(07) [non-commutativity]; > s.a. lattice gauge theory [flux and charge].
@ Related topics: Brambilla & Prosperi ht/94-conf [and potentials]; Watson PLB(94) [identities]; Gukov & Witten a0804 [surface operators]; Schroer FP(11)-a1012 [alternative setting, stringlike approach]; Ferreira & Luchini NPB(12) [integral formulation, in loop spaces]; Chung & Lu PRD(16)-a1609 [basis tensor fields]; > s.a. knots in physics; Nicolai Map.
Features, Techniques > s.a. constrained
systems [including reduction];
* Configuration space: The natural one is the moduli space of all gauge equivalence classes of connections on a principal G-bundle over the spatial manifold Σ (superspace) or connections over all such principal bundles over Σ (grand superspace); > see connections.
* Alice configurations: Fields in theories with disconnected groups such that the disconnectedness has physical effects; > s.a. monopoles.
@ With boundaries: Śniatycki et al CMP(96); Avramidi & Esposito CMP(99)ht/97, gq/99-conf; Ferrara & Frønsdal PLB(98)ht; Díaz-Marín Sigma(15)-a1407 [n-dimensional abelian gauge fields, general-boundary formulation]; Geiller NPB(17)-a1703 [edge modes and corner ambiguities]; > s.a. quantum gauge theories.
@ Measure: Pickrell JGP(96); Fleischhack mp/01, mp/01; > s.a. connection.
@ Related topics: Gomis et al PRP(95) [antibrackets]; Loll et al JGP(96) [complexification]; Lenz et al AP(00)ht [residual symmetries]; McInnes JPA(98) [Alice configurations]; Stoilov ht/05 [Lagrange multipliers]; Mišković & Pons JPA(06)ht/05 [dynamics and symmetries of perturbations]; Feng et al JHEP(07)ht [counting gauge invariants]; Kubyshin 89 [dimensional reduction]; Anderson CQG(08)-a0711 [new interpretation of variational principle]; Pommaret JModP(14)-a1310-talk [formal theory of systems of partial differential equations and Lie pseudogroups]; Berger et al a1806 [complete set of invariant tensors].
> Features, effects: see Gribov Effect; instantons; monopoles; phase transitions; Reference Frames [accelerated]; solutions.
> Techniques, tools: see homology [chain complexes]; manifold types [gauge orbit stratification]; Moduli Space; Seiberg-Witten Theory.
Types of Theories and Related Concepts > s.a. types
of gauge theories.
* Applications: They are very useful (especially the non-Abelian ones) in mathematics, to get insights on 4D differential topology; In condensed-matter physics, gauge fields provide the only means of describing the long-range interactions of vortices or defects in terms of local fields, rendering them accessible to standard field theoretic techniques.
@ References: Kleinert 89 [in condensed-matter physics]; García del Moral a1107 [new gauging mechanism]; Pivovarov PPN(13)-a1209-conf [inaction approach]; Margalli & Vergara PLA(15)-a1507 [hidden gauge symmetry in complex holomorphic systems].
> Theories: see lattice gauge theory; non-commutative gauge theories; yang-mills theories [including hamiltonian formulation].
> Related concepts: see BRST transformations; charge; energy-momentum tensor; noether symmetries; particle models.
@ General: Moriyasu 83 [primer]; Healey 07 [conceptual]; Robinson et al a0810-ln [algebraic]; in Scheck 12; Hamilton a1512-ln [intro, for mathematicians].
@ Geometric picture, approaches: Lubkin AP(63); Hermann 70, 78; Trautman RPMP(70), CzJP(79); Wu & Yang PRD(75); Atiyah 79; Daniel & Viallet RMP(80); Eguchi et al PRP(80); Balachandran et al 83, update a1702; Svetlichny ht/99-ln; Aldrovandi & Barbosa IJTP(00)mp/01 [non-bundle structure], IJTP(00)mp/01 [as optical medium]; Ferrantelli MSc(02)-a1002 [gauge-natural formulation, including suypersymmetries]; Harikumar et al PLB(03)ht/02 [topology]; Kubyshin mp/03-conf; Robinson et al a0908-ln; Alsid & Serna FP(15)-a1308, Jordan et al a1404-ch [approaches]; Zharinov TMP(14) [algebraic and geometric methods]; Mielke 17; > s.a. 2-spinors.
@ Origin, gauge symmetry as emergent: 't Hooft AIP(07)-a0707; Donoghue et al proc(10)-a1007 [and violations]; Bjorken a1008-conf [vacuum condensate and QED]; Freund a1008 [extension of Verlinde's entropic gravity proposal]; Zaanen & Beekman AP(12)-a1108; Kirillov et al PLB(12)-a1205; Chkareuli PLB(13) [from spontaneously broken supersymmetry]; Levin & Wen PRB(05), RMP(05)cm/04 [gauge bosons and fermions from "string-net condensation" in condensed-matter theory]; Canarutto IJGMP(14)-a1404-conf [from the geometry of Weyl spinors]; Arias et al PRB(15)-a1511 [elastic deformations in graphene]; Urrutia a1607-conf [from Nambu models]; Wetterich NPB(17)-a1608 [from decoupling]; Barceló et al JHEP(16)-a1608 [systematic study].
@ Meaning: de Souza ht/98, ht/99, ht/99 [discrete]; Guttmann & Lyre phy/00 [physics vs math]; Gubarev et al PRL(01)hp/00 [of A2]; Lyre PhSc(01)qp-conf [conceptual]; Healey PhSc(01)dec [reality of A]; Sánchez a0803; Rovelli FP(14)-a1308 [why gauge?]; Weatherall a1411, a1505-conf; Afriat a1706; Nguyen et al a1712 [the need for surplus structure].
– journals – comments
– other sites – acknowledgements
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