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Propagation and Energy Transfer
Published in James D. Taylor, Introduction to Ultra-Wideband Radar Systems, 2020
Robert Roussel-Dupré, Terence W. Barrett
The conventional theory of electromagnetism is the simplest example of a gauge theory in (3 + 1) dimensions and belongs to the gauge group U(1). This involves only linear Maxwell’s equations which do not yield solitary waves or solitons. If, however, the electromagnetic field is generalized to a non-Abelian gauge group, e.g., SU(2), then a triplet of gauge fields known as the Yang-Mills fields is created.256 If, then, these Yang-Mills fields are coupled to a triplet of scalar fields a nonsingular localized static soliton — or, at least, a solitary wave — solution is created257-259 with which magnetic monopoles can be associated.260
Supersymmetry and Superstring Theory with Engineering Applications
Published in Harish Parthasarathy, Supersymmetry and Superstring Theory with Engineering Applications, 2023
is the covariant derivative of λA in the adjoint representation of the gauge group. Explain this result in the context of a supersymmetric generalization of the electromagnetic field Lagrangian or more generally in the context of the non-supersymetric Yang-Mills gauge field Lagrangian with anomalous terms.
The Weak Interaction in the Framework of Grand Unification Theories
Published in K Grotz, H V Klapdor, S S Wilson, The Weak Interaction in Nuclear, Particle and Astrophysics, 2020
K Grotz, H V Klapdor, S S Wilson
In addition to the divergences, the so-called anomalies are another problem in quantum field theory. As previously mentioned in Chapter 4, a situation is said to be anomalous if an initially present symmetry of the Lagrange function is destroyed by pure quantum effects, so that the conservation of the corresponding quantum number is no longer guaranteed. Whether or not a gauge theory contains such anomalies depends on the corresponding gauge group and the particle multiplets. Green and Schwarz (1985) discovered that a type I superstring theory is anomaly-free only for the gauge group SO(32). If we consider only closed strings, the second possible gauge group is E8⨂ E8 (E8 is the largest finite exceptional group). Both possible gauge groups, SO(32) and E8 ⨂ E8, are enormous with 496 generators, and both contain the 24 generators of the minimal SU(5) unification group. In some, as yet unclear, way the SO(32) or the E8 ⨂ E8 symmetry must be broken down in stages to the SU(5) or the SO(10) symmetries, so that for lower energies (E ≲ 1015 GeV) superstring models are indistinguishable from the normal GUT models. In this case, the whole of ‘low-energy’ physics would be contained in only one of the E8 factors (Gross et al (1985)). The second E8 factor has no effect here. In a first symmetry breaking step, the low-energy E8 factor might be broken into the E6 group of Subsection 6.3.2. An E8 ⨂ E8 superstring theory (Green and Schwarz (1985)) involves yet another complication. It is only possible in a 10-dimensional geometric space. This must then compactify into the observable four space–time dimensions (for further details, see Witten (1981), Lee (1984), Derendinger (1986), Shafi (1986), Valle (1986), Dragon et al (1987), Mohapatra (1988) and in particular Green et al (1986)).
Diagonal Born–Oppenheimer correction for coupled-cluster wave-functions
Published in Molecular Physics, 2018
Therefore, the change of normalisation by different amounts at every geometry as in (12) can be viewed as a kind of gauge transformation [12,33,36]. Here, it arises from the fact that the adiabatic separation ansatz (7) is invariant to arbitrary geometry-dependent normalisation changing transformations [33] (from now onwards, we will refer to them simply as gauge transformations). This shows that the symmetry is a result of the decision to separate the electronic and nuclear degrees of freedom and to consider the electronic motion first. The gauge group which includes all possible gauge transformations at a geometry is the general linear group GL(1).