Gauge groups

Recently in [6] we constructed effective Lagrangians of the Veneziano-Yankielowicz (VY) type for two non-supersymmetric but strongly interacting theories with a Dirac fermion either in the two index symmetric or two index antisymmetric representation of the gauge group. These theories are planar equivalent, at N —> oo to SYM [7], In this limit the non-supersymmetric effective Lagrangians coincide with the bosonic part of the VY Lagrangian. 

At nonzero temperature the center of the SU(N) gauge group becomes a relevant symmetry [11], However except for mathematically defined objects such as Polyakov loops the physical states of the theory are neutral under the center group symmetry. 

A new class of effective Lagrangians have been constructed to show how the information about the center group symmetry is efficiently transferred to the actual physical states of the theory [12-15] and will be reviewed in detail elsewhere. Via these Lagrangians we were also able to have a deeper understanding of the relation between chiral restoration and deconfinement [15] for quarks in the fundamental and in the adjoint representation of the gauge group. 

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]... 

This result is true for all matter waves and also in the Michelson-Gale experiment, where it has been measured to a precision of one part in 1023 [49]. Hasselbach et al. [51] have demonstrated it in electron waves. We have therefore shown that the electrodynamic and kinematic explanation of the Sagnac effect gives the same result in a structured vacuum described by 0(3) gauge group symmetry. 

Another example of a physical effect of this type is the Aharonov-Bohm effect, which is supported by a multiply connected vacuum configuration such as that described by the 0(3) gauge group [6]. The Aharonov-Bohm effect is a gauge transform of the true vacuum, where there are no potentials. In our notation, therefore the Aharonov-Bohm effect is due to terms such as (1/ )8 , depending on the geometry chosen for the experiment. It is essential for the Aharonov-Bohm effect to exist such that (1/ )8 be physical, and not random. It follows therefore that the vacuum configuration defined by the... 

In Section I, it was argued that 0(3) electrodynamics on the classical level emerges from a vacuum configuration that can be described with an 0(3) symmetry gauge group. On the QED level, this concept is developed by considering higher-order terms in the Hamiltonian... 

The principle behind this derivation is the gauge principle, and so is the same for all gauge groups. The equivalence (456) was first demonstrated on the 0(3) level [15], but evidently exists for all gauge group symmetries. The gauge principle in electrodynamics therefore leads to the energy and momentum of the photon and classical field. The 4-current J appears in both Eqs. (443) and (444) and is self-dual, a result that is echoed in the self-duality of the vacuum field equations ... 

The non-Abelian Stokes theorem is a relation between covariant derivatives for any gauge group symmetry ... 

In gauge theory, for any gauge group, however, a rotation... 

The field equations of electrodynamics for any gauge group are obtained from the Jacobi identity of Poincare group generators [42,46] ... 

The Jacobi identity (40) means that the homogeneous field equation of electrodynamics for any gauge group is... 

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