Dualities in Field Theories| Dualities in Field Theories |
In Electromagnetic Theory
> s.a. electromagnetism / differential forms;
integrable systems; theory of physical
theories [equivalence].
\(\def\se{_{\rm e}}\def\sm{_{\rm m}}\)
* Hodge duality: The
transformation \(F^{~}_{ab} \mapsto {}^*F^{~}_{ab}\), defined by \({}^*F^{~}_{ab}
= {1\over2}\,\epsilon^{~}_{ab}{}^{cd}F^{~}_{cd}\), with \(\epsilon^{0123} = 1\).
* Continuous duality transformations:
The 1-parameter family of transformations of the electric and magnetic quantities
\(\{(\rho\se,\rho\sm),(j\se,j\sm),(E_i,H_i),(D_i,B_i)\}\); For a generic pair
\((x\se,x\sm)\) of quantities, the duality is defined by
for the electromagnetic field, where ξ is a constant parameter
(which cannot be promoted to a local one as in the gauge-theory trick).
* Properties: The values of
\({\bf E} \times {\bf H}\), \({\bf E}\cdot{\bf D} + {\bf B} \cdot {\bf H}\),
\(T_{ab}\), and the Maxwell equations, are left invariant.
* Applications:
Duality between the Aharonov-Bohm and
Aharonov-Casher effects.
@ General references:
Misner & Wheeler AP(57);
Montonen & Olive PLB(77) [monopole];
Zhu JPA(89) [complexions];
Anandan gq/95 [topological phases];
Deser & Sarioglu PLB(98)ht/97 [and Lorentz invariance];
Igarashi et al NPB(98)ht;
Hatsuda et al NPB(99)ht [invariant lagrangians];
Li & Naón MPLA(01)ht/00;
Przeszowski JPA(05)ht [light-front variables];
Julia ht/05-conf [generalization];
Barnich & Troessaert JMP(09)-a0812 [as bi-Hamiltonian system];
Bunster & Henneaux PRD(11)-a1011,
Deser CQG(11)-a1012,
Saa CQG(11)-a1101 [no local, gauged version];
Freidel & Pranzetti PRD(18)-a1806 [extension to the boundary];
Bunster et al PRD(20)-a1905 [and BMS invariance].
@ On a general manifold, and gravity: Witten SelMath(95)ht [on a general manifold];
in Garat JMP(15)gq/04;
Bakas a0910-proc [and gravity];
Agulló et al PRD(14)-a1409,
PRL(17)-a1607,
PRD(18)-a1810 [anomaly, in curved spacetime].
@ In quantum electrodynamics:
Buhmann & Scheel PRL(09)-a0806 [macroscopic];
Yang & Xu a2103 [no new conservation law].
@ In non-linear electrodynamics:
Gibbons & Rasheed NPB(95)ht;
Gaillard & Zumino LNP(98)ht/97,
ht/97 [non-linear].
@ From Lagrangian: Bhattacharyya & Gangopadhyay MPLA(00)ht/98;
Bliokh et al NJP(13)
[helicity, spin, momentum and angular momentum].
@ Related topics: Fayyazuddin a1608 [3D theory with massive photon];
Castellani & De Haro a1803-in [fundamentality and emergence].
Gauge-Gravity Duality
> s.a. AdS-cft correspondence; approaches to
quantum gravity; Double Copy; holography.
* Idea: A set of
relationships (for example the AdS-cft correspondence) which convert
difficult problems in certain types of gauge theories into (relatively)
simple geometric problems in gravity in one higher dimension.
@ General references:
't Hooft NPB(74) [precursor];
Polchinski a1010-ln;
Shuvaev a1106;
Maldacena a1106-ch;
Billó et al a1304-proc [non-perturbative aspects];
Ammon & Erdmenger 15;
De Haro SHPMP(15)-a1501 [and emergent gravity];
De Haro et al FP(16)-a1509 [rev];
Engelhardt & Fischetti CQG(16)-a1604 [boundary causality];
DeWolfe a1804-ln [intro];
Semenoff a1808-ln [holographic duality of gauge fields and strings].
@ And quantum gravity: Engelhardt & Horowitz IJMPD(16)-a1605-GRF;
Hanada & Romatschke JHEP(19)-a1808 [simulations and phases].
@ And condensed-matter physics: Sachdev ARCMP(12)-a1108;
> s.a. 2012 talk
by Gary Horowitz on using the duality to model high-T superconductors.
@ Other applications:
Wadia MPLA(10);
Hossenfelder PRD(15)-a1412 [analog systems].
In Other Theories
> s.a. hamiltonian systems; higher-order
lagrangians; lagrangian dynamics; M-theory.
* In quantum mechanics:
What Isidro calls duality is in reality an ambiguity in the choice of
complex structure used in quantizing a classical theory.
@ General references: Savit RMP(80);
Banerjee & Ghosh JPA(98)
[chiral oscillator model]; Olive ht/02-proc;
De Haro et al SHPMP-a1603 [and gauge symmetries];
McInnes NPB(16)-a1606 [field theories with no holographic dual];
De Haro & Butterfield a1707-in [schema, and bosonization example];
De Haro Syn(19)-a1801 [theoretical and heuristic roles];
Butterfield a1806-in [in physics vs philosophy];
Thompson a1904-proc
[generalised dualities and their applications];
De Haro & Butterfield Syn-a1905 [and symmetries];
Turner PoS(19)-a1905 [in 2+1 dimensions];
De Haro a2004 [empirical equivalence].
@ Quantum mechanics: Isidro MPLA(03)qp,
PLA(03)qp,
qp/03-in [projective phase space],
MPLA(04)qp/03 [torus phase space];
> s.a. coherent states.
@ Non-abelian gauge theory:
Mohammedi ht/95;
Duff IJMPA(96) and
IJMPD(96)
[in supersymmetric gauge theory, from strings];
Martín MPLA(99) [in path space];
Chan & Tsou IJMPA(99)ht;
Tsou ht/00-ln,
ht/00-conf;
Faddeev & Niemi PLB(02)ht/01 [SU(2) Yang-Mills theory];
Majumdar & Sharatchandra IJMPA(02);
Deser & Seminara PLB(05)ht [duality invariance for free bosonic and fermionic gauge fields];
Kihara JMP(11) [generalized self-duality equations];
Ho & Ma NPB(16)-a1507.
@ Sigma-models:
Mohammedi et al ZPC(97)ht/95;
Mohammedi PLB(96)ht/95,
PLB(96).
@ Linearized gravity: Henneaux & Teitelboim PRD(05)gq;
Barnich & Troessaert JMP(09)-a0812 [as bi-Hamiltonian system];
Bakas a0910-proc;
Troessaert a1312-PhD.
@ General relativity: Hawking & Ross PRD(95)ht [electric and magnetic black holes];
Maartens & Bassett CQG(98)gq/97;
Nouri-Zonoz et al CQG(99)gq/98 [NUT];
Dadhich MPLA(99)gq/98,
MPLA(99),
GRG(00)gq/99;
Abramo et al MPLA(03) [with scalar field];
Deser & Seminara PRD(05)ht [failure in non-linear case];
Julia ht/05-conf;
da Rocha & Rodrigues JPA(10)-a0910 [in gravitational theories];
Dehouck NPPS(11)-a1101 [and supergravity].
@ Scale factor duality in cosmology: Clancy et al CQG(98)gq;
Di Pietro gq/01/MPLA [quintessence];
Harlow & Susskind a1012 [general criteria].
@ In string theory:
Rickles SHPMP(11);
Polchinski SHPMP(17)-a1412;
Huggett & Wüthrich a2005-ch [meaning and significance];
> s.a. strings.
@ Related topics: Gaona & García IJMPA(07) [first-order actions];
Lindström et al JHEP(08)-a0707 [T-duality for generalized Kähler geometries];
Barnich & Troessaert JHEP(09)-a0812 [for spin-2 fields in Minkowski space];
Nussinov et al NPB(15)-a1311 [dualities as conformal transformations, and practical consequences];
Miyaji & Takayanagi PTEP(15)-a1503 [codimension-two spacelike surfaces and states in dual Hilbert spaces];
Sourlas a1907 [fermionic theories].
> Related topics:
see gravitomagnetism; self-dual fields
[connections] and self-dual solutions in general relativity [Weyl tensor].
> Other dualities:
see Galerkin Duality.
Dual Mass in Gravitation
@ General references:
Lubkin IJTP(77);
Magnon JMP(87),
NCA(88); Torre CQG(95)gq/94.
@ Phenomenology: Cates et al GRG(88);
Rahvar & Habibi ApJ(04)ap/03 [microlensing signatures];
Danehkar HEPGC(17)-a0707.
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