Gauge Field Theories| Gauge Field Theories |
In General
> s.a. gauge symmetry [including emergence];
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 and fields:
Majumdar & Sharatchandra PRD(01)ht/98 [equivalent potentials];
Mulder FP-a2103 [are gauge potentials real?].
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];
Wetterich a1710 [and flow equations].
@ 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];
Meneses a1904 [holonomy approach, overview];
> s.a. knots in physics;
Nicolai Map.
> Online resources:
see Wikipedia page [loop representation in gauge theories and quantum gravity].
Features, Techniques
> s.a. constrained systems [including reduction];
fiber bundles; gauge choices.
* 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];
Gomes et al NPB(19)-a1808,
a1902 [unified geometric framework for boundary charges];
Corichi & Vukašinac a2001 [Maxwell + Pontryagin, canonical];
> s.a. quantum gauge theories.
@ Measure: Pickrell JGP(96);
Fleischhack mp/01,
mp/01;
> s.a. connection.
@ Perturbations:
Mišković & Pons JPA(06)ht/05 [dynamics and symmetries];
Chiaffrino et al a2012 [in terms of gauge invariants, at all orders].
@ 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];
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];
Balachandran & Reyes-Lega a1807 [role of the Gauss law];
Giddings JHEP(19)-a1907 [asymptotic boundary conditions].
> 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];
Deser PLB(19)-a1901
[no-go result on non-Lagrangian gauge fields];
Gording a2005 [new approach to particle content].
> 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.
Other References > s.a. physics teaching.
@ General:
Moriyasu 83 [primer];
Robinson et al a0810-ln [algebraic];
in Scheck 12;
Hamilton a1512-ln [intro, for mathematicians].
@ Conceptual: Healey 07;
Roberts et al a2105 [Noether-based argument].
@ 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.
main page
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