arXiv subjects
Publications and source records attributed to R. J. Finkelstein.
The q-deformation of the Lie algebras underlying the standard field theories leads to a pair of dual algebras. We describe a simple choice of possible field theories based on these derived algebras. One of these approximates the standard Lie theory of point particles, while the other is proposed as a field theory of knotted solitons.
If the Lie group of a non-Abelian theory is replaced by the corresponding q-group, one is led to replace the Lie algebra by two dual algebras. The first of these lies close to the Lie algebra that it is replacing while the second introduces new degrees of freedom. We interpret the theory based on the first algebra as a modification of standard field theory while we propose that the new degrees of freedom introduced by the second algebra describe solitonic rather than point particle sources. We have earlier found that the modified q-electroweak theory differs very little from the standard theory. Here we find a similar result for q-gravity. Both of the modified theories are incomplete, however, and must be completed by the solitonic sector. We propsoe that the solitonic sector of both q-electroweak and q-gravity have the symmetry of knots associated with SU_q(2). Since the Lorentz group is here deformed, there is no longer the standard classification of particles described by mass and spin. There is instead a classification of irreducible structures determined by SU_q(2).
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 non-locality 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.
We have examined the deformation of a generic non-Abelian gauge theory obtained by replacing its Lie group by the corresponding quantum group. This deformed gauge theory has more degrees of freedom than the theory from which it is derived. By going over from point particles in the standard theory to solitonic particles in the deformed theory, it is proposed to interpret the new degrees of fredom as descriptive of a non-locality of the deformed theory. It also turns out that the original Lie algebra gets replaced by two dual algebras, one of which lies close to and approaches the original Lie algebra in a correspondence limit, while the second algebra is new and disappears in this same correspondence limit. The exotic field particles associated with the second algebra can be interpreted as quark-like constituents of the solitons, which are themselves described as point particles in the first algebra. These ideas are explored for q-deformed SU(2) and $GL_q(3)$.
The q-electroweak theory obtained by replacing SU(2) by $SU_q(2)$ in the Weinberg-Salam model is experimentally not distinguishable from the standard model at the level of the doublet representation. However, differences between the two theories should be observable when higher dimensional representations are taken into account. In addition the possibility of probing non-local structure may be offered by the q-theory.
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.
The q-field theories are constructed by substituting quantum groups for the usual Lie groups. In earlier papers this construction was carried out for the quantum group SU_q(2). Here the investigation is extended to SL_q(3). The resulting theory describes two sectors, one sector lying close to the standard theory and accessible by perturbation theory, while the second sector describes particles that should be difficult to detect and become invisible in the q=1 limit. In this note we discuss these hypothetical particles: three quark-like spinor particles coupled to three gluon-like vector particles.
When the symmetry of a physical theory describing a finite system is deformed by replacing its Lie group by the corresponding quantum group, the operators and state function will lie in a new algebra describing new degrees of freedom. If the symmetry of a field theory is deformed in this way, the enlarged state space will again describe additional degrees of freedom, and the energy levels will acquire fine structure. The massive particles will have a stringlike spectrum lifting the degeneracy of the point-particle theory, and the resulting theory will have a non-local description. Theories of this kind naturally contain two sectors with one sector lying close to the standard theory while the second sector describes particles that should be more difficult to observe.
In non-Abelian field theories with q-symmetry groups the massive particles have a non-local interpretation with a stringlike spectrum. It is shown that a massless vector similarly acquires a tower of masses by spontaneous symmetry breaking.
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 straightforward description of q-deformed systems leads to transition amplitudes that are not numerically valued. To give physical meaning to these expressions without introducing {\it ad hoc} remedies, one may exploit an "internal" Fock space already defined by the q-algebra. This internal space may be interpreted in terms of internal degrees of freedom of the deformed system or alternatively in terms of non-locality. It is shown that the $q$-deformation may give stringy characteristics to a Yang-Mills theory.
There is a growing literature on dyonic black holes as they appear in string theory. Here we examine the correspondence limit of a dyonic black hole which is not supersymmetric. Assuming the existence of a dyon with non-supersymmetric Kerr-Schild structure, we calculate its gravitational and electromagnetic fields and compute its mass and angular momentum to obtain a modified B.P.S. relation. The contributionn of the angular momentum to the mass appears in the condition for the appearance of a horizon.