arXiv subjects
Publications and source records attributed to Cosmas K. Zachos.
A general group element for the fundamental representation of SU(3) is expressed as a second order polynomial in the hermitian generating matrix H, with coefficients consisting of elementary trigonometric functions dependent on the sole invariant det(H), in addition to the group parameter.
Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group. The simple explicit result exhibits connections between group theory, combinatorics, and Fourier analysis, especially in the large spin limit. Salient intuitive features of the formula are illustrated and discussed.
The visualization advantages of ternary plots are illustrated for the PMNS neutrino mixing matrix. Unitarity constraints are incorporated automatically, in part, since barycentric plots of this type allow three variables with a fixed sum to be plotted as mere points inside an equilateral triangle on a plane.
Functional conjugation methods are used to analyze the global structure of various renormalization group trajectories, and to gain insight into the interplay between continuous and discrete rescaling. With minimal assumptions, the methods produce continuous flows from step-scaling σ functions, and lead to exact functional relations for the local flow β functions, whose solutions may have novel, exotic features, including multiple branches. As a result, fixed points of σ are sometimes not true fixed points under continuous changes in scale, and zeroes of β do not necessarily signal fixed points of the flow, but instead may only indicate turning points of the trajectories.
Holographic functional methods are introduced as probes of discrete time-stepped maps that lead to chaotic behavior. The methods provide continuous time interpolation between the time steps, thereby revealing the maps to be quasi-Hamiltonian systems underlain by novel potentials that govern the motion of a perceived point particle. Between turning points, the particle is strictly driven by Hamiltonian dynamics, but at each encounter with a turning point the potential changes abruptly, loosely analogous to the switchbacks on a mountain road. A sequence of successively deepening switchback potentials explains, in physical terms, the frequency cascade and trajectory folding that occur on the particular route to chaos revealed by the logistic map.
Given a minimum measurable length underlying spacetime, the latter may be effectively regarded as discrete, at scales of order the Planck length. A systematic discretization of continuum physics may be effected most efficiently through the umbral deformation. General functionals yielding such deformations at the level of solutions are furnished and illustrated, and broad features of discrete oscillations and wave propagation are outlined.
The WZW functional in D=4 can be derived directly from the Chern-Simons functional of a compactified D=5 gauge theory and the boundary fermions it supplants. A simple pedagogical model based on U(1) gauge groups illustrates how this works. A bulk-boundary system with the fermions eliminated manifestly evinces anomaly cancelations between CS and WZW terms.
The set of points of a one-dimensional cut-and-project quasicrystal or model set, while not additive, is shown to be multiplicative for appropriate choices of acceptance windows. This leads to the definition of an associative additive graded composition law and permits the introduction of Lie algebras over such aperiodic point sets. These infinite dimensional Lie algebras are shown to be representatives of a new type of semi-direct product induced Lie algebras.
A new technique is developed for the derivation of the Wess-Zumino-Witten terms of gauged chiral lagrangians. We start in D=5 with a pure (mesonless) Yang-Mills theory, which includes relevant gauge field Chern-Simons terms. The theory is then compactified, and the effective D=4 lagrangian is derived using lattice techniques, or ``deconstruction,'' where pseudoscalar mesons arise from the lattice Wilson links. This yields the WZW term with the correct Witten coefficient by way of a simple heuristic argument. We discover a novel WZW term for singlet currents, that yields the full Goldstone-Wilczek current, and a U(1) axial current for the skyrmion, with the appropriate anomaly structures. A more detailed analysis is presented of the dimensional compactification of Yang-Mills in D=5 into a gauged chiral lagrangian in D=4, heeding the consistency of the D=4 and D=5 Bianchi identities. These dictate a novel covariant derivative structure in the D=4 gauge theory, yielding a field strength modified by the addition of commutators of chiral currents. The Chern-Simons term of the pure D=5 Yang-Mills theory then devolves into the correct form of the Wess-Zumino-Witten term with an index (the analogue of N_{colors}=3) of N=D=5. The theory also has a Skyrme term with a fixed coefficient.
Phase space is a framework ideally suited for quantizing superintegrable systems through the use of deformation methods, as illustrated here by applications to de Sitter and chiral particles. Within this framework, Nambu brackets elegantly incorporate the additional quantum invariants of such models. New results are presented for the non-Abelian quantization of these brackets.