Gauge field tensor

We briefly reformulate the above necessary and sufficient condition for the existence of a strictly diabatic basis in the context of gauge theory. In contrast to the gauge potential which transforms in a relatively complicated manner via Eq. (27b), the gauge field tensor simply transforms as... 

Can the nonremovable part also exhibit singularities as one approaches a conical intersection The nonremovable part of the derivative couplings originates from the fact that the gauge field tensor does not vanish [see Eqs. (34) and (35) and text]. Since by construction this tensor assumes only finite values, there is no reason to expect the nonremovable part to exhibit singularities at conical intersections, see also the discussion in Ref 32 and explicit examples in Ref 5. Let us briefly consult Eq. (40) in a conical... 

In conclusion, we have shown that the non-Abelian gauge-field intensity tensor fi sc(X) shown in Eq. (113) vanishes when... 

In general gauge field theory [6], the field tensor is proportional to the commutator of variant derivatives. This is the result of a round trip or closed... 

The development just given illustrates the fact that the topology of the vacuum determines the nature of the gauge transformation, field tensor, and field equations, as inferred in Section (I). The covariant derivative plays a central role in each case for example, the homogeneous field equation of 0(3) electrodynamics is a Jacobi identity made up of covariant derivatives in an internal 0(3) symmetry gauge group. The equivalent of the Jacobi identity in general relativity is the Bianchi identity. 

Recall that in general gauge field theory, for any gauge group, the field tensor is defined through the commutator of covariant derivatives. In condensed notation [6]... 

Potential differences are primary in gauge theory, because they define both the covariant derivative and the field tensor. In Whittaker s theory [27,28], potentials can exist without the presence of fields, but the converse is not true. This conclusion can be demonstrated as follows. Equation (479b) is invariant under... 

Therefore the fact that 9 is arbitrary in U(l) theory compels that theory to assert that photon mass is zero. This is an unphysical result based on the Lorentz group. When we come to consider the Poincare group, as in section XIII, we find that the Wigner little group for a particle with identically zero mass is E(2), and this is unphysical. Since 9 in the U(l) gauge transform is entirely arbitrary, it is also unphysical. On the U(l) level, the Euler-Lagrange equation (825) seems to contain four unknowns, the four components of , and the field tensor H v seems to contain six unknowns. This situation is simply the result of the term 7/MV in the initial Lagrangian (824) from which Eq. (826) is obtained. However, the fundamental field tensor is defined by the 4-curl ... 

It is also possible to consider the holonomy of the generic A in the vacuum. This is a round trip or closed loop in Minkowski spacetime. The general vector A is transported from point A, where it is denoted Aa 0 around a closed loop with covariant derivatives back to the point Aa () in the vacuum. The result [46] is the field tensor for any gauge group... 

Gauge covariance of the classical theory is due to the invariance of the field tensor F/I V under the local gauge transformation... 

Energy-momentum conservation is expressed by dvT = 0 for a closed system. If Tfi were a symmetric tensor (when converted to 7 /x"), this would be assured because i f Tfi = 0 by construction. Since the gauge field part of the tensor deduced from Noether s theorem is not symmetric, this requires special consideration, as discussed below. A symmetric energy-momentum tensor is required for any eventual unification of quantum field theory and general relativity [422], The fermion field energy and momentum are... 

The Lagrangian density for a massless fermion field interacting with the SU 2) gauge field defines the Noether energy-momentum tensor... 

Because 3MAdoes not reduce to terms that vanish even for a noninteracting field, this construction must be verified. The energy and 3-momentum of the gauge field derived from the resulting symmetric energy-momentum tensor are... 

At this point it is crucial to note that this latter form was chosen to obey the requirement of Lorentz invariance. Under a gauge transformation only the interaction term is modified as the electromagnetic field tensor is gauge invariant... 

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