33 research outputs found
Functional Representations for Fock Superalgebras
The Fock space of bosons and fermions and its underlying superalgebra are
represented by algebras of functions on a superspace. We define Gaussian
integration on infinite dimensional superspaces, and construct superanalogs of
the classical function spaces with a reproducing kernel -- including the
Bargmann-Fock representation -- and of the Wiener-Segal representation. The
latter representation requires the investigation of Wick ordering on Z2-graded
algebras. As application we derive a Mehler formula for the Ornstein-Uhlenbeck
semigroup on the Fock space.Comment: 33 pages, Late
Feynman formulae and phase space Feynman path integrals for tau-quantization of some L\'evy-Khintchine type Hamilton functions
This note is devoted to representation of some evolution semigroups. The
semigroups are generated by pseudo-differential operators, which are obtained
by different (parametrized by a number ) procedures of quantization from
a certain class of functions (or symbols) defined on the phase space. This
class contains functions which are second order polynomials with respect to the
momentum variable and also some other functions. The considered semigroups are
represented as limits of -fold iterated integrals when tends to infinity
(such representations are called Feynman formulae). Some of these
representations are constructed with the help of another pseudo-differential
operators, obtained by the same procedure of quantization (such representations
are called Hamiltonian Feynman formulae). Some representations are based on
integral operators with elementary kernels (these ones are called Lagrangian
Feynman formulae and are suitable for computations). A family of phase space
Feynman pseudomeasures corresponding to different procedures of quantization is
introduced. The considered evolution semigroups are represented also as phase
space Feynman path integrals with respect to these Feynman pseudomeasures. The
obtained Lagrangian Feynman formulae allow to calculate these phase space
Feynman path integrals and to connect them with some functional integrals with
respect to probability measures
Chernoff approximations of Feller semigroups in Riemannian manifolds
Chernoff approximations of Feller semigroups and the associated diffusion
processes in Riemannian manifolds are studied. The manifolds are assumed to be
of bounded geometry, thus including all compact manifolds and also a wide range
of non-compact manifolds. Sufficient conditions are established for a class of
second order elliptic operators to generate a Feller semigroup on a (generally
non-compact) manifold of bounded geometry. A construction of Chernoff
approximations is presented for these Feller semigroups in terms of shift
operators. This provides approximations of solutions to initial value problems
for parabolic equations with variable coefficients on the manifold. It also
yields weak convergence of a sequence of random walks on the manifolds to the
diffusion processes associated with the elliptic generator. For parallelizable
manifolds this result is applied in particular to the representation of
Brownian motion on the manifolds as limits of the corresponding random walks.Comment: 36 pages, no figures. Title and abstract changed. Minor corrections,
some proposition generalise
States of quantum systems and their liftings
Let H(1), H(2) be complex Hilbert spaces, H be their Hilbert tensor product
and let tr2 be the operator of taking the partial trace of trace class
operators in H with respect to the space H(2). The operation tr2 maps states in
H (i.e. positive trace class operators in H with trace equal to one) into
states in H(1). In this paper we give the full description of mappings that are
linear right inverse to tr2. More precisely, we prove that any affine mapping
F(W) of the convex set of states in H(1) into the states in H that is right
inverse to tr2 is given by the tensor product of W with some state D in H(2).
In addition we investigate a representation of the quantum mechanical state
space by probability measures on the set of pure states and a representation --
used in the theory of stochastic Schroedinger equations -- by probability
measures on the Hilbert space. We prove that there are no affine mappings from
the state space of quantum mechanics into these spaces of probability measures.Comment: 14 page
Transformations of Feynman path integrals and generalized densities of Feynman pseudomeasures
Applications of transformations of Feynman path integrals and Feynman pseudomeasures to explain arising quantum anomalies are considered. A contradiction in the literature is also explained
On spectral structure of bounded linear operators on reflexive Banach spaces
A descriptive characterization of point, continuous, and residual spectra of operators acting on a separable Hilbert space is obtained. The possible point spectra of bounded linear operators acting on lp, 1 < p < â are characterized.Ministerio de Ciencia y TecnologĂaJunta de AndalucĂ
Noetherâs theorem for dissipative quantum dynamical semi-groups
Noether's Theorem on constants of the motion of dynamical systems has
recently been extended to classical dissipative systems (Markovian semi-groups)
by Baez and Fong. We show how to extend these results to the fully quantum
setting of quantum Markov dynamics. For finite-dimensional Hilbert spaces, we
construct a mapping from observables to CP maps that leads to the natural
analogue of their criterion of commutativity with the infinitesimal generator
of the Markov dynamics. Using standard results on the relaxation of states to
equilibrium under quantum dynamical semi-groups, we are able to characterise
the constants of the motion under quantum Markov evolutions in the
infinite-dimensional setting under the usual assumption of existence of a
stationary strictly positive density matrix. In particular, the Noether
constants are identified with the fixed point of the Heisenberg picture
semigroup.Comment: 8 pages, no figure
Differentiability of measures and a Malliavin-Stroock Theorem
We compare different notions of differentiability of a measure along a vector field on a locally convex space. We consider in the -space of a differentiable measure the analoga of the classical concepts of gradient, divergence and Laplacian (which coincides with the Ornstein-Uhlenbeck
operator in the Gaussian case). We use these operators for the extension of the basic results of Malliavin and Stroock on the smoothness of finite dimensional image measures under certain nonsmooth mappings to the case of non-Gaussian measures. The proof of this extension is quite direct and does not use any Chaos-decomposition. Finally, the role of this Laplacian in the
procedure of quantization of anharmonic oscillators is discussed
Feynman path integrals and Lebesgue-Feynman measures
We call a Lebesgue-Feynman measure (LFM) any generalized measure
(distribution in the sense of Sobolev and Schwartz) on a locally convex
topological vector space E which is translation invariant. In the present
paper, we investigate transformations of the LFM generated by transformations
of the domain and also discuss the connections of these transformations of the
LFM with so-called quantum anomalies, improving some recent results of teh
authors and co-workers. We revisit the contradiction between the points of view
on quantum anomalies presented in the books of Fujikawa and Suzuki on the one
hand, and of Cartier and DeWitt-Morette on the other.Comment: 8 page