Non-Abelian Gauge Fields. Commutation Relations

1979 ◽  
pp. 190-195
Author(s):  
Julian Schwinger
Keyword(s):  
Gauge Fields ◽  
Abelian Gauge ◽  
Commutation Relations

Non-Abelian Gauge Fields. Commutation Relations

1962 ◽  
Vol 125 (3) ◽  
pp. 1043-1048 ◽  
Author(s):  
Julian Schwinger
Keyword(s):  
Gauge Fields ◽  
Abelian Gauge ◽  
Commutation Relations

scholarly journals Non-Abelian generalizations of the Hofstadter model: spin–orbit-coupled butterfly pairs

2020 ◽  
Vol 9 (1) ◽  
Author(s):  
Yi Yang ◽  
Bo Zhen ◽  
John D. Joannopoulos ◽  
Marin Soljačić
Keyword(s):  
Quantum Hall ◽  
Building Blocks ◽  
Real Space ◽  
Gauge Fields ◽  
Spin Orbit ◽  
Abelian Gauge ◽  
Experimental Realization ◽  
Integer Quantum Hall Effect ◽  
Integer Quantum ◽  
Necessary And Sufficient

Abstract The Hofstadter model, well known for its fractal butterfly spectrum, describes two-dimensional electrons under a perpendicular magnetic field, which gives rise to the integer quantum Hall effect. Inspired by the real-space building blocks of non-Abelian gauge fields from a recent experiment, we introduce and theoretically study two non-Abelian generalizations of the Hofstadter model. Each model describes two pairs of Hofstadter butterflies that are spin–orbit coupled. In contrast to the original Hofstadter model that can be equivalently studied in the Landau and symmetric gauges, the corresponding non-Abelian generalizations exhibit distinct spectra due to the non-commutativity of the gauge fields. We derive the genuine (necessary and sufficient) non-Abelian condition for the two models from the commutativity of their arbitrary loop operators. At zero energy, the models are gapless and host Weyl and Dirac points protected by internal and crystalline symmetries. Double (8-fold), triple (12-fold), and quadrupole (16-fold) Dirac points also emerge, especially under equal hopping phases of the non-Abelian potentials. At other fillings, the gapped phases of the models give rise to topological insulators. We conclude by discussing possible schemes for experimental realization of the models on photonic platforms.

Non-Abelian gauge fields, charges and strings

1982 ◽  
Vol 70 (2) ◽  
pp. 180-189 ◽  
Author(s):  
G. Venturi
Keyword(s):  
Gauge Fields ◽  
Abelian Gauge

Symmetry conditions and non‐Abelian gauge fields

1978 ◽  
Vol 19 (2) ◽  
pp. 436-441 ◽  
Author(s):  
R. Maciejko
Keyword(s):  
Gauge Fields ◽  
Abelian Gauge

Non-Abelian gauge theory from the Poisson bracket

2018 ◽  
Vol 33 (30) ◽  
pp. 1850182
Author(s):  
Mu Yi Chen ◽  
Su-Long Nyeo
Keyword(s):  
Gauge Theory ◽  
Poisson Bracket ◽  
Commutation Relation ◽  
Maxwell Equations ◽  
Dynamical Variable ◽  
Gauge Fields ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Generalized Poisson ◽  
Lie Algebraic Structure

The Hamiltonian of a nonrelativistic particle coupled to non-Abelian gauge fields is defined to construct a non-Abelian gauge theory. The Hamiltonian which includes isospin as a dynamical variable dictates the dynamics of the particle and isospin according to the Poisson bracket that incorporates the Lie algebraic structure of isospin. The generalized Poisson bracket allows us to derive Wong’s equations, which describe the dynamics of isospin, and the homogeneous (sourceless) equations for non-Abelian gauge fields by following Feynman’s proof of the homogeneous Maxwell equations.It is shown that the derivation of the homogeneous equations for non-Abelian gauge fields using the generalized Poisson bracket does not require that Wong’s equations be defined in the time-axial gauge, which was used with the commutation relation. The homogeneous equations derived by using the commutation relation are not Galilean and Lorentz invariant. However, by using the generalized Poisson bracket, it can be shown that the homogeneous equations are not only Galilean and Lorentz invariant but also gauge independent. In addition, the quantum ordering ambiguity that arises from using the commutation relation can be avoided when using the Poisson bracket.From the homogeneous equations, which define the “electric field” and “magnetic field” in terms of non-Abelian gauge fields, we construct the gauge and Lorentz invariant Lagrangian density and derive the inhomogeneous equations that describe the interaction of non-Abelian gauge fields with a particle.

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