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A gauged SU q (2) theory is characterized by two dual algebras, the first lying close to the Lie algebra of SU(2) while the second introduces new degrees of freedom that may be associated with nonlocality or solitonic structure. The first and second algebras, here called the external and internal algebras respectively, define two sets of fields, also called external and internal. The gauged external fields agree with the Weinberg–Salam model at the level of the doublet representation but differ at the level of the adjoint representation. For example, the g-factor of the charged W-boson differs in the two models. The gauged internal fields remain speculative but are analogous to color fields.

The connection between spin and statistics implied by the continuous Lorentz group together with strong reflection (TCP) is shown to hold also for the q-Lorentz group.

Examples are given of q-deformed systems that may be interpreted by the standard rules of quantum theory in terms of new degrees of freedom and supplementary quantum numbers.

The standard transform with a twisted q -kernel is proposed for connecting q -amplitudes in three-dimensional configuration space with those in three-dimensional momentum space.

We obtain the complete set of states of theq-oscillator in both configuration space and momentum space as well as the transformation between these spaces. The states as well as the matrix elements lie in the SU q (2) algebra. To obtain transition probabilities, one must take the ‘Woronowicz square.’

Although there is no empirical motivation for replacing the commutators of dynamically conjugate operators in quantum mechanics byq-commutators, it appears possible to construct a consistent mathematical formulism based on this idea. To examine such a possibility further, we have studied the relation of this proposal to the Schwinger action principle, since the entire quantum mechanical formulism may be inferred from this principle. In particular, we have discussed the quantum transformation theory within this framework.

One may attempt to implementq-symmetry in field theory by replacing commutators byq-commutators. The quantized field is then described byq-oscillators. This program encounters a basic difficulty, however, when one demands either causality or consistency between the twin requirements of causality and positive energy. Independent of the preceding remark, one notes that the Einstein-Bose and Fermi-Dirac thermal distributions appear as limiting cases of the quantizedq-field whenq = ± 1.

We calculate the Clebsch-Gordan coefficients of SUq(2) by a Woronowicz integration over the group manifold and obtain a representation differing from that reached by working with theq-group algebra. These apparently different results must agree, however, and their equivalence implies aq-identity. On lettingq = 1, we shall obtain two results of different structures for the Clebsch-Gordan coefficients of SU(2) and their equivalence similarly implies an identity among the usual binomial coefficients. With the same approach, one may extend the Woronowicz integral of the product of two irreducible representations to products of many irreducible representations.

We present some partial results on theq-deformation of the GL(2) Yang-Mills theory. In particular, the irreducible representations needed to describe the complete set of physical states are obtained by a simple procedure.

We examine the propagators associated with particles characterized by an internal symmetry. Of particular interest is the case of spontaneously broken Kac-Moody algebra.

It is possible to construct non-Abelian field theories by gauging Kac-Moody algebras. Here we discuss the spontaneous symmetry breaking of such theories via the Higgs mechanism. If the Higgs particle lies in the Cartan subalgebra of the Kac-Moody algebra, the previously massless vectors acquire a mass spectrum that is linear in the Kac-Moody index and has additional fine structure depending on the associated Lie algebra.

We construct a non-Abelian field theory by gauging a Kac-Moody algebra. One obtains an infinite tower of interacting vector fields and associated ghosts obeying slightly modified Feynman rules.