389 research outputs found
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 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 for natural diffusions on post-critically finite self-similar sets
and that 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 of measures on a
space of functions on the two-torus, parametrized by a polynomial (the
Wess-Zumino-Landau-Ginzburg model). The second is a family \mu_\cG^{s,t} of
measures on a space \cG of maps from to a Lie group (the
Wess-Zumino-Novikov-Witten model). Finally we study a family
of measures on the product of a space of connection s on the trivial principal
bundle with structure group on a three-dimensional manifold with a
space of \fg-valued three-forms on
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
where is a homology three-sphere, will yield the
Casson invariant of 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
- …