Physical Properties of Quantum Field Theory Measures

Well known methods of measure theory on infinite dimensional spaces are used to study physical properties of measures relevant to quantum field theory. The difference of typical configurations of free massive scalar field theories with different masses is studied. We apply the same methods to study the Ashtekar-Lewandowski (AL) measure on spaces of connections. We prove that the diffeomorphism group acts ergodically, with respect to the AL measure, on the Ashtekar-Isham space of quantum connections modulo gauge transformations. We also prove that a typical, with respect to the AL measure, quantum connection restricted to a (piecewise analytic) curve leads to a parallel transport discontinuous at every point of the curve.Comment: 24 pages, LaTeX, added proof for section 4.2, added reference

Pseudoconvex Domains in Almost Complex Abstract Wiener Spaces

AbstractThe ∂-operator on an almost complex abstract Wiener space (B, H, μ, J) is defined by making use of the Malliavin calculus. The authors then study pseudoconvex domains in B, domains where the ∂-equations ∂u = ƒ are solvable. As an application, they establish an approximation theorem of holomorphic forms and a Dolbeault type theorem. Examples of such domains, obtained through SDE, one also discussed

Upper estimate of martingale dimension for self-similar fractals

We study upper estimates of the martingale dimension $d_m$ of diffusion processes associated with strong local Dirichlet forms. By applying a general strategy to self-similar Dirichlet forms on self-similar fractals, we prove that $d_m=1$ for natural diffusions on post-critically finite self-similar sets and that $d_m$ is dominated by the spectral dimension for the Brownian motion on Sierpinski carpets.Comment: 49 pages, 7 figures; minor revision with adding a referenc

Transition density of diffusion on Sierpinski gasket and extension of Flory's formula

Some problems related to the transition density u(t,x) of the diffusion on the Sierpinski gasket are considerd, based on recent rigorous results and detailed numerical calculations. The main contents are an extension of Flory's formula for the end-to-end distance exponent of self-avoiding walks on the fractal spaces, and an evidence of the oscillatory behavior of u(t,x) on the Sierpinski gasket.Comment: 11 pages, REVTEX, 2 postscript figure

Fredholm Modules on P.C.F. Self-Similar Fractals and their Conformal Geometry

The aim of the present work is to show how, using the differential calculus associated to Dirichlet forms, it is possible to construct Fredholm modules on post critically finite fractals by regular harmonic structures. The modules are d-summable, the summability exponent d coinciding with the spectral dimension of the generalized laplacian operator associated with the regular harmonic structures. The characteristic tools of the noncommutative infinitesimal calculus allow to define a d-energy functional which is shown to be a self-similar conformal invariant.Comment: 16 page

Hydrodynamic limit for a zero-range process in the Sierpinski gasket

We prove that the hydrodynamic limit of a zero-range process evolving in graphs approximating the Sierpinski gasket is given by a nonlinear heat equation. We also prove existence and uniqueness of the hydrodynamic equation by considering a finite-difference scheme.Comment: 24 pages, 1 figur

Measures on Banach Manifolds and Supersymmetric Quantum Field Theory

We show how to construct measures on Banach manifolds associated to supersymmetric quantum field theories. These measures are mathematically well-defined objects inspired by the formal path integrals appearing in the physics literature on quantum field theory. We give three concrete examples of our construction. The first example is a family $\mu_P^{s,t}$ of measures on a space of functions on the two-torus, parametrized by a polynomial $P$ (the Wess-Zumino-Landau-Ginzburg model). The second is a family \mu_\cG^{s,t} of measures on a space \cG of maps from $\P^1$ to a Lie group (the Wess-Zumino-Novikov-Witten model). Finally we study a family $\mu_{M,G}^{s,t}$ of measures on the product of a space of connection s on the trivial principal bundle with structure group $G$ on a three-dimensional manifold $M$ with a space of \fg-valued three-forms on $M.$ We show that these measures are positive, and that the measures \mu_\cG^{s,t} are Borel probability measures. As an application we show that formulas arising from expectations in the measures \mu_\cG^{s,1} reproduce formulas discovered by Frenkel and Zhu in the theory of vertex operator algebras. We conjecture that a similar computation for the measures $\mu_{M,SU(2)}^{s,t},$ where $M$ is a homology three-sphere, will yield the Casson invariant of $M.$Comment: Minor correction

The Vlasov continuum limit for the classical microcanonical ensemble

For classical Hamiltonian N-body systems with mildly regular pair interaction potential it is shown that when N tends to infinity in a fixed bounded domain, with energy E scaling quadratically in N proportional to e, then Boltzmann's ergodic ensemble entropy S(N,E) has the asymptotic expansion S(N,E) = - N log N + s(e) N + o(N); here, the N log N term is combinatorial in origin and independent of the rescaled Hamiltonian while s(e) is the system-specific Boltzmann entropy per particle, i.e. -s(e) is the minimum of Boltzmann's H-function for a perfect gas of "energy" e subjected to a combination of externally and self-generated fields. It is also shown that any limit point of the n-point marginal ensemble measures is a linear convex superposition of n-fold products of the H-function-minimizing one-point functions. The proofs are direct, in the sense that (a) the map E to S(E) is studied rather than its inverse S to E(S); (b) no regularization of the microcanonical measure Dirac(E-H) is invoked, and (c) no detour via the canonical ensemble. The proofs hold irrespective of whether microcanonical and canonical ensembles are equivalent or not.Comment: Final version; a few typos corrected; minor changes in the presentatio

Time separation as a hidden variable to the Copenhagen school of quantum mechanics

The Bohr radius is a space-like separation between the proton and electron in the hydrogen atom. According to the Copenhagen school of quantum mechanics, the proton is sitting in the absolute Lorentz frame. If this hydrogen atom is observed from a different Lorentz frame, there is a time-like separation linearly mixed with the Bohr radius. Indeed, the time-separation is one of the essential variables in high-energy hadronic physics where the hadron is a bound state of the quarks, while thoroughly hidden in the present form of quantum mechanics. It will be concluded that this variable is hidden in Feynman's rest of the universe. It is noted first that Feynman's Lorentz-invariant differential equation for the bound-state quarks has a set of solutions which describe all essential features of hadronic physics. These solutions explicitly depend on the time separation between the quarks. This set also forms the mathematical basis for two-mode squeezed states in quantum optics, where both photons are observable, but one of them can be treated a variable hidden in the rest of the universe. The physics of this two-mode state can then be translated into the time-separation variable in the quark model. As in the case of the un-observed photon, the hidden time-separation variable manifests itself as an increase in entropy and uncertainty.Comment: LaTex 10 pages with 5 figure. Invited paper presented at the Conference on Advances in Quantum Theory (Vaxjo, Sweden, June 2010), to be published in one of the AIP Conference Proceedings serie