No abstract

Let p be a polynomial in the non-commuting variables (a, x) = (a 1 , . . . , a ga , x 1 , . . . , x gx ). If p is convex in the variables x, then p has degree two in x and moreover, p has the formwhere L has degree at most one in x and Λ is a (column) vector which is linear in x, so that Λ T Λ is a both sum of squares and homogeneous of degree two. Of course the converse is true also. Further results involving various convexity hypotheses on the x and a variables separately are presented.

In a prequel to this article, we used abstract Wiener measure to define the Chern–Simons path integral over ℝ3. In this sequel, we compute the Wilson Loop observable for the non-abelian gauge group and compare with current knot literature.

A hyperlink is a finite set of non-intersecting simple closed curves in R × R 3 . Let S be an orientable surface in R 3 . The dynamical variables in General Relativity are the vierbein e and a su(2) × su(2)-valued connection ω. Together with Minkowski metric, e will define a metric g on the manifold. Denote AS(e) as the area of S, for a given choice of e.The Einstein-Hilbert action S(e, ω) is defined on e and ω. We will quantize the area of the surface S by integrating AS(e) against a holonomy operator of a hyperlink L, disjoint from S, and the exponential of the Einstein-Hilbert action, over the space of vierbeins e and su(2) × su(2)-valued connections ω. Using our earlier work done on Chern-Simons path integrals in R 3 , we will write this infinite dimensional path integral as the limit of a sequence of Chern-Simons integrals. Our main result shows that the area operator can be computed from a link-surface diagram between L and S. By assigning an irreducible representation of su(2) × su(2) to each component of L, the area operator gives the total net momentum impact on the surface S.MSC 2010: 83C45, 81S40, 81T45, 57R56 Quantization of AreaUsing canonical quantization of Ashtekar variables, the authors in [12] were able to quantize the area of a surface S in R 3 . Their procedure was to promote the variables into operators, which act on a quantum state, defined by a spin network.The idea of a spin network was first introduced by Penrose, in an attempt to construct a quantum mechanical description of the geometry of space. A spin network in R 3 is a graph, each vertex has valency 3, with a positive integer assigned to every edge in the graph, satisfying certain conditions at each vertex.What they showed was that a spin network T is an eigenstate of the area operator and the eigenvalues of the area operator, is proportional to l

A hyperlink is a finite set of non-intersecting simple closed curves in R × R 3 . Let S be an orientable surface in R×R 3 . The Einstein-Hilbert action S(e, ω) is defined on the vierbein e and a su(2) × su(2)-valued connection ω, which are the dynamical variables in General Relativity. Define a functional FS(ω), by integrating the curvature dω + ω ∧ ω over the surface S, which is su(2) × su(2)-valued. We integrate FS(ω) against a holonomy operator of a hyperlink L, disjoint from S, and the exponential of the Einstein-Hilbert action, over the space of vierbeins e and su(2) × su(2)-valued connections ω. Using our earlier work done on Chern-Simons path integrals in R 3 , we will write this infinite dimensional path integral as the limit of a sequence of Chern-Simons integrals. Our main result shows that the quantized curvature can be computed from the linking number between L and S.MSC 2010: 83C45, 81S40, 81T45, 57R56