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* := d*F* + [*A*,* F*] = 0
;

Other field equations will depend on the form of the action chosen
(careful, *F* = d*A* + *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];
fiber bundles.

* __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 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].

> __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.

**Other References**

@ __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 a1007-conf
[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 *A*^{2}];
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].

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2017