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Radon Transform for Rotation Estimation, Interaction Picture in Quantum Mechanics, Curvature Tensor in GTR, Gauge Fields
Published in Harish Parthasarathy, Electromagnetics, Control and Robotics, 2023
[ 122] Gauge fields and field equations associated to a matrix Lie group. Let G be a matrix Lie group with ta, a = 1, 2, ...,...N a basis for its Lie algebra g. The commutation relations are [ta,tb]=iC(abc)tc summation over the repeated index c being implied. C(abc) are the structure constants of the Lie algebra associated with the basis {ta} . Let Aaμ (x) be the gauge fields and ψ(x) the matter field. The gauge covariant derivative is given by ∇μ=∂μ+ieAμ where is a constant and Aμ(x)=Aμa(x)ta
Propagation and Energy Transfer
Published in James D. Taylor, Introduction to Ultra-Wideband Radar Systems, 2020
Robert Roussel-Dupré, Terence W. Barrett
where Dµ is a generalized form of the gauge covariant derivative describing the changes in both the external and internal parts of ψ(x) () Dμψβ=∑α[δβα∂μ−iq(Aμ)βα]ψα
The Atiyah-Singer Index Theorem and Its Application to Anomalies in Quantum Field Theory
Published in Harish Parthasarathy, Supersymmetry and Superstring Theory with Engineering Applications, 2023
where Dx is the gauge covariant derivative γ.(∂ + ieA) with A the non-Abelian gauge potential. In the limit as M →∞, this becomes Tr(f(0)I)which corresponds exactly to what we want. The additional feature of making such an approximation to the trace of the δ function is that that it is a gauge invariant expression since its is built out of the gauge covariant derivative. The only term that contributes to the above trace is (1/2)f”(0)Tr((γ.Dx))). Now, Dx2=(γ.(∂+ieA))2=γμγνDμDν=ημνDμDν+(1/2)[γμ,γν]Fμν where Fμν=[Dμ,Dν]
The Dirac equation as a model of topological insulators
Published in Philosophical Magazine, 2020
Xiao Yuan, M. Bowen, P. S. Riseborough
If the parameters of a Hamiltonian are adiabatically changed along a closed path in parameter space, Berry [16] showed that in addition to the dynamical phase, the phase of an (initial) energy eigenstate of acquires an additive time-independent contribution, γ, which has geometric significance. An infinitesimal change of a Hamiltonian parameter could be expected to yield a change in the state given by . However, the derivative does not transform well under a change of gauge, such assince the gauge transformation leads to a term which is ‘parallel’ to The Berry connection is required to turn the -gradient of into a gauge-covariant derivative, analogous to the way that the electromagnetic vector potential is required to yield a gauge-invariant momentum operator in quantum mechanics . For this reason, the gauge-covariant derivative is defined bywhich is chosen to be orthogonal to , ieHence, the Berry connection is determined asThe Berry connection can be shown to be real, since the states for all are normalised to unity. Thus,which yieldsTherefore, like the electromagnetic vector potential , the Berry connection is real.