Pseudodifferential Symbols on Riemann Surfaces and Krichever–Novikov Algebras

We define the Krichever-Novikov-type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic operators and symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. Very few of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples and KdV-type hierarchies, emphasizing the similarities and differences with the case of smooth symbols on the circle.     

Quantizer–dequantizer operators as a tool for formulating the quantization procedure

We consider the quantization procedure and investigate the application of the quantizer–dequantizer method and star-product technique to construct associative products and the associative algebras formed by the quantizer–dequantizer operators and their symbols. The corresponding Lie algebras are also constructed. We study the case where the quantizer–dequantizer operators form a self-dual system and show that the structure constants of the Lie algebras satisfy some identity, in addition to the Jacobi identity. Using tomographic quantizer–dequantizer operators and their symbols, we construct the continuous associative algebra and the corresponding Lie algebra.     

Semiclassical States Associated with Isotropic Submanifolds of Phase Space

We define classes of quantum states associated with isotropic submanifolds of cotangent bundles. The classes are stable under the action of semiclassical pseudo-differential operators and covariant under the action of semiclassical Fourier integral operators. We develop a symbol calculus for them; the symbols are symplectic spinors. We outline various applications.     

Quantization of Forms on the Cotangent Bundle

We consider the following construction of quantization. For a Riemannian manifold $M$ the space of forms on T ^{⋆ M} is made into a space of (full) symbols of operators acting on forms on M. This gives rise to the composition of symbols, which is a deformation of the (“super”)commutative multiplication of forms. The symbol calculus is exact for differential operators and the symbols that are polynomial in momenta. We calculate the symbols of natural Laplacians. (Some nice Weitzenb?ck like identities appear here.) Formulae for the traces corresponding to natural gradings of Ω (T ^{⋆ M} ) are established. Using these formulae, we give a simple direct proof of the Gauss–Bonnet–Chern Theorem. We discuss these results in connection with a general question of the quantization of forms on a Poisson manifold. Received: 12 November 1998 / Accepted: 1 March 1999     

Berezin-Toeplitz Operators, a Semi-Classical Approach

This article is devoted to the Toeplitz Operators [4] in the context of the geometric quantization [11, 15]. We propose an ansatz for their Schwartz kernel. From this, we deduce the main known properties of the principal symbol of these operators and obtain new results : we define their covariant and contravariant symbols, which are full symbols, and compute the product of these symbols in terms of the Kähler metric. This gives canonical star products on the Kählerian manifolds. This ansatz is also useful to introduce the notion of microsupport.     

Operator symbols in the description of observable-state systems

For an observable-state system with finite degrees of freedom N topological properties of the kernels and symbols belonging to the considered operators are investigated. For the operators of $\text{L}$⁺($\text{S}$) kernels and symbols are distributions and for density matrices o? they are smooth functions.     

Semiclassical Weyl Formula for a Class of Weakly Regular Elliptic Operators

We investigate the semiclassical Weyl formula describing the asymptotic behaviour of the counting function for the number of eigenvalues in the case of self-adjoint elliptic differential operators satisfying weak regularity hypotheses. We consider symbols with possible critical points and with coefficients which have Hölder continuous derivatives of first order. Mathematics Subject Classification (2000) 35P20.     

Projectively Equivariant Quantization and Symbol Calculus: Noncommutative Hypergeometric Functions

We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked out, one of them yielding a quantum length element on S ³.     

The method of covariant symbols in curved space-time

Diagonal matrix elements of pseudodifferential operators are needed in order to compute effective Lagrangians and currents. For this purpose the method of symbols is often used, which however lacks manifest covariance. In this work the method of covariant symbols, introduced by Pletnev and Banin, is extended to curved space-time with arbitrary gauge and coordinate connections. For the Riemannian connection we compute the covariant symbols corresponding to external fields, the covariant derivative and the Laplacian, to fourth order in a covariant derivative expansion. This allows one to obtain the covariant symbol of general operators to the same order. The procedure is illustrated by computing the diagonal matrix element of a nontrivial operator to second order. Applications of the method are discussed. PACS 04.62.+v; 11.15.-q; 11.15.Tk     

The Efimov Effect and an Extended Szegö–Kac Limit Theorem

Fredholm determinant asymptotics of convolution operators on large finite intervals with rational symbols having real zeros are studied. The explicit formulae obtained can be considered as a genuine generalization of Szegö–Kac's formula to symbols with real zeros. Connections with the Efimov effect are discussed.     

Tomographic probability representation for quantum fermion fields

Tomographic probability representation is introduced for fermion fields. The states of the fermions are mapped onto the probability distribution of discrete random variables (spin projections). The operators acting on the fermion states are described by fermionic tomographic symbols. The product of the operators acting on the fermion states is mapped onto the star-product of the fermionic symbols. The kernel of the star-product is obtained. The antisymmetry of the fermion states is formulated as a specific symmetry property of the tomographic joint probability distribution associated with the states.     

Decomposition of symmetric tensor fields in the presence of a flat contact projective structure

Abstract

Let M be an odd-dimensional Euclidean space endowed with a contact 1-form α. We investigate the space of symmetric contravariant tensor fields over M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up of those vector fields that preserve the contact structure defined by a. If we consider symmetric tensor fields with coefficients in tensor densities (also called symbols), the vertical cotangent lift of the contact form a defines a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symbols. This generalized Hamiltonian operator on the space of symbols is invariant with respect to the action of the projective contact algebra sp(2n+2) the algebra of vector fields which preserve both the contact structure and the projective structure of the Euclidean space. These two operators lead to a decomposition of the space of symbols, except for some critical density weights, which generalizes a splitting proposed by V. Ovsienko in [18].     

Feynman path integral and ordering rules on discrete finite space

The Feynman path integral is constructed for systems whose configuration space is a discrete finite set. The construction is based on the operator formulation of quantum mechanics on a finite discrete space. We derive connections between operators and introduce the analogue of the^*-multiplication for discrete symbols.     

Stationary perturbation theory in the probability representation of quantum mechanics

In the probability representation of quantum mechanics, the eigenvalue problems in Hilbert space appear as *-genvalue equations. We show the possibility of employing the nondegenerate stationary perturbation method in the probability representation of quantum mechanics. The perturbed eigentomograms and the eigenvalues of energy are shown to be computed ab initio in terms of tomographic symbols of the operators involved.     

Nonnegative Discrete Symbols and Their Probabilistic Interpretation

We review the quantizer–dequantizer formalism of constructing symbols of the density operators and quantum observables, such as Wigner functions and tomographic-probability distributions. We present a tutorial consideration of the technique of obtaining minimal sets of dequantizers (quorum) related to the observable eigenvalues for one-qubit states. We discuss a generalization of the quantizer–dequantizer scheme on the example of spin-1/2 states. We consider the possibilities of extending the results to two-qubit systems using spin tomograms of the state density matrix.     

Asymptotic quantization for solution manifolds of some infinite dimensional Hamiltonian systems

The solution manifolds of some classes of Hamiltonian systems in Hilbert phase spaces are considered. Pseudodifferential operators with symbols on these manifolds are defined.     

Extended Higher-Spin Superalgebras and Their Realizations in Terms of Quantum Operators

The realization of the N = 1 higher-spin superalgebra, proposed earlier by E. S. FRADKIN and the author, is found in terms of bosonic quantum operators. The extended higher-spin super-algebras, generalizing ordinary extended supersymmetry with arbitrary N > 1, are constructed by adding fermion quantum operators. Automorphisms, real forms, subalgebras, contractions and invariant forms of these infinite-dimensional superalgebras are studied. The formulation of the higher-spin superalgebras is described in terms of symbols of operators by BEREZIN. We hope that this formulation will provide in future the powerful tool for constructing the complete solution of the higher-spin problem, the problem of introducing a consistent gravitational interaction for massless higher-spin fields (S > 2).