14269086
Hopf Algebras in General and in Combinatorial Physics: a practical introduction
This tutorial is intended to give an accessible introduction to Hopf algebras. The mathematical context is that of representation theory, and we also illustrate the structures with examples taken from combinatorics and quantum physics, showing that in this latter case the axioms of Hopf algebra arise naturally. The text contains many exercises, some taken from physics, aimed at expanding and exemplifying the concepts introduced.
27515895
Generalized Holstein-Primakoff squeezed states for SU(n).
We show how to define multiphoton, many-mode squeezed states for SU(n), using a generalized Holstein-Primakoff realization. We prove that for the class of realizations given, the resulting squeezing reduces to that of SU(2), and exemplify with a specific calculation for SU(3).
119711153
Dynamical group model of the CDW state
Abstract The dynamical group for a one-dimensional model of a many-electron system exhibiting a charge-density wave is obtained. The corresponding Lie algebra in a physical model is U(2); it is used to obtain the spectrum and coherent ground state, and to define a corresponding order parameter.
121551361
Chiral symmetry and the goldstone theorem
1803
A multipurpose Hopf deformation of the Algebra of Feynman-like Diagrams
We construct a three parameter deformation of the Hopf algebra $\mathbf{LDIAG}$. This new algebra is a true Hopf deformation which reduces to $\mathbf{LDIAG}$ on one hand and to $\mathbf{MQSym}$ on the other, relating $\mathbf{LDIAG}$ to other Hopf algebras of interest in contemporary physics. Further, its product law reproduces that of the algebra of polyzeta functions.
12865240
An interface between physics and number theory
We extend the Hopf algebra description of a simple quantum system given previously, to a more elaborate Hopf algebra, which is rich enough to encompass that related to a description of perturbative quantum field theory (pQFT). This provides a mathematical route from an algebraic description of non-relativistic, non-field theoretic quantum statistical mechanics to one of relativistic quantum field theory. Such a description necessarily involves treating the algebra of polyzeta functions, extensions of the Riemann Zeta function, since these occur naturally in pQFT. This provides a link between physics, algebra and number theory. As a by-product of this approach, we are led to indicate inter alia a basis for concluding that the Euler gamma constant γ may be rational.
119221920
Inflationary Instabilities of Einstein-Aether Cosmology
We examine the consequences of Lorentz violation during slow-roll inflation. We consider a canonical scalar inflaton coupled, through its potential, to the divergence of a fixed-norm timelike vector field, or "aether." The vector is described by Einstein-aether theory, a vector-tensor model of gravitational Lorentz violation. We derive and analyze the cosmological perturbation equations for the metric, inflaton, and aether. If the scale of Lorentz violation is sufficiently small compared to the Planck mass, and the strength of the scalar-aether coupling is suitably large, then the spin-0 and spin-1 perturbations grow exponentially and spoil the inflationary background. The effects of such a coupling on the CMB are too small to be visible to current or near-future CMB experiments; unusually, no isocurvature modes are produced at first order in a perturbative expansion around the aether norm. These results are discussed for both a general potential and a worked example, $m^2 \phi^2$ inflation with a quadratic scalar-aether coupling term.