On a possible origin of quantum groups

A Poisson bracket structure having the commutation relations of the quantum group SL _q (2) is quantized by means of the Moyal star-product on C (²), showing that quantum groups are not exactly quantizations, but require a quantization (with another parameter) in the background. The resulting associative algebra is a strongly invariant nonlinear star-product realization of the q-algebra U _q (sl(2)). The principle of strong invariance (the requirement that the star-commutator is star-expressed, up to a phase, by the same function as its classical limit) implies essentially the uniqueness of the commutation relations of U _q (sl(2)).     

Berezin–Toeplitz Quantization and Invariant Symbolic Calculi

It is well known that one can often construct an invariant star-product by expanding the product of two Toeplitz operators asymptotically into a series of another Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin–Toeplitzquantization. We show that on bounded symmetric domains (Hermitian symmetric spaces of noncompact type), one can in fact obtain in a similar way any invariant star-productwhich is G-equivalent to the Berezin–Toeplitz star-product, by using, instead of Toeplitz operators, other suitable assignments fQ _f from compactly supported C functions f to bounded linear operators Q _f on the corresponding Hilbert spaces. (This procedureis referred to as prime quantization by some authors.) Along the way, we establish two technical results which are of interest in their own right, namely a controlled-growth parameter generalization of the classical theorem of Borel on the existence of a function with prescribed derivatives of all orders at a point, and the fact that any invariant bi-differential operator (Hochschild two-cochain) on a bounded symmetric domain automatically maps the Schwartz space into itself.     

Projectively and Conformally Invariant Star-Products

We consider the Poisson algebra S(M) of smooth functions on T ^* M which are fiberwise polynomial. In the case where M is locally projectively (resp. conformally) flat, we seek the star-products on S(M) which are SL(n+1,) (resp. SO(p+1,q+1))-invariant. We prove the existence of such star-products using the projectively (resp. conformally) equivariant quantization, then prove their uniqueness, and study their main properties. We finally give an explicit formula for the canonical projectively invariant star-product.     

A finite-dimensional canonical formalism in the classical field theory

A canonical formalism based on the geometrical approach to the calculus of variations is given. The notion of multi-phase space is introduced which enables to define whole the canonical structure (physical quantities, Poisson bracket, canonical fields) without use of functional derivatives. All definitions are of pure geometrical (finite dimensional) character.The observable algebra (physical quantities algebra) obtained here is much smaller then the algebra of all (sufficiently smooth) functionals on the space of states, derived from the standard infinite-dimensional formulation. As it is known, the latter is much too large for purposes of quantization. As the examples prove, our algebra could be an adequate start-point for quantization.For simplifying the language the notion of observable-valued distribution is introduced. Many concrete physical examples are given. E.g. it is shown that some problems connected with gauge in electrodynamics are automatically solved in this approach. The introduced language allows to obtain the Noether theorem in a most natural way.     

Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom

This paper gives a thorough investigation on formulating and solving quantum problems by extended analytical mechanics that extends canonical variables to complex domain. With this complex extension, we show that quantum mechanics becomes a part of analytical mechanics and hence can be treated integrally with classical mechanics. Complex canonical variables are governed by Hamilton equations of motion, which can be derived naturally from Schrödinger equation. Using complex canonical variables, a formal proof of the quantization axiom p → pˆ = −i, which is the kernel in constructing quantum-mechanical systems, becomes a one-line corollary of Hamilton mechanics. The derivation of quantum operators from Hamilton mechanics is coordinate independent and thus allows us to derive quantum operators directly under any coordinate system without transforming back to Cartesian coordinates. Besides deriving quantum operators, we also show that the various prominent quantum effects, such as quantization, tunneling, atomic shell structure, Aharonov–Bohm effect, and spin, all have the root in Hamilton mechanics and can be described entirely by Hamilton equations of motion.     

Poisson Reduction and Branes in Poisson–Sigma Models

We analyse the problem of boundary conditions for the Poisson–Sigma model and extend previous results showing that non-coisotropic branes are allowed. We discuss the canonical reduction of a Poisson structure to a submanifold, leading to a Poisson algebra that generalizes Diracs construction. The phase space of the model on the strip is related to the (generalized) Dirac bracket on the branes through a dual pair structure.Mathematics Subject Classifications (2000). 81T45, 53D17, 81T30, 53D55.     

Jordanian twist quantization of D=4 Lorentz and Poincaré algebras and D=3 contraction limit

We describe in detail the two-parameter nonstandard quantum deformation of the D=4 Lorentz algebra , linked with a Jordanian deformation of . Using the twist quantization technique we obtain the explicit formulae for the deformed co-products and antipodes. Further extending the considered deformation to the D=4 Poincaré algebra we obtain a new Hopf-algebraic deformation of four-dimensional relativistic symmetries with a dimensionless deformation parameter. Finally, we interpret as the D=3 de Sitter algebra and calculate the contraction limit (R is the de Sitter radius) providing an explicit Hopf algebra structure for the quantum deformation of the D=3 Poincaré algebra (with mass-like deformation parameters), which is the two-parameter light-cone κ-deformation of the D=3 Poincaré symmetry.     

Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the Formality conjecture), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.     

Generalized Hitchin Systems and the Knizhnik–zamolodchikov–bernard Equation on Ellipic Curves

The Knizhnik–Zamolodchikov–Bernard (KZB) equation on an elliptic curve with a marked point is derived by classical Hamiltonian reduction and further quantization. We consider classical Hamiltonian systems on a cotangent bundle to the loop group L(GL(N, C)) extended by the shift operators, to be related to the elliptic module. After reduction, we obtain a Hamiltonian system on a cotangent bundle to the moduli of holomorphic principle bundles and an elliptic module. It is a particular example of generalized Hitchin systems (GHS) which are defined as Hamiltonian systems on cotangent bundles to the moduli of holomorphic bundles and to the moduli of curves. They are extensions of the Hitchin systems by the inclusion the moduli of curves. In contrast with the Hitchin systems, the algebra of integrals are noncommutative on GHS. We discuss the quantization procedure in our example. The quantization of the quadratic integral leads to the KZB equation. We present an explicit form of higher quantum Hitchin integrals which, upon reducing from GHS phase space to the Hitchin phase space, gives a particular example of the Beilinson–Drinfeld commutative algebra of differential operators on the moduli of holomorphic bundles.     

Self-consistent quantization of a non-abelian theory in an axial gage

Self-consistent quantization of a non-Abelian theory in an axial gage is described. This gage is obtained from the Coulomb gage by a point operator gage transformation within the framework of minimal canonical quantization based on the explicit solution of the coupling equation. Here the author succeeds in uniquely identifying true transverse variables in whose terms the Hamiltonian of the system is finite and defined.Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, 19–23, October, 1989.The authors would like to thank B. M. Barbashov, A. V. Efremov, V. N. Pervushin, and Yu. L. Kalinovskii for discussing the results of this work.     

Noncommutative Deformation Theory

In Gerstenhaber's classical theory of deformations, the deformation parameter commutes with the original algebra. Motivated by some non classical deformations which recently appeared for quantization of Nambu mechanics, we introduce new deformations where the parameter no longer commutes with the original algebra. We find the associated cohomology and Gerstenhaber algebra and give rigidity and integrability criterions. We show that the Weyl algebra (though rigid in classical theory) can be nontrivially deformed, in super-commutative theory, to the supersymmetry enveloping algebra      

Towards Drinfeld–Sokolov reduction for quantum groups

In this paper we study the Poisson–Lie version of the Drinfeld–Sokolov reduction defined in [E. Frenkel, N. Reshetikhin, M.A. Semenov-Tian-Shansky, Drinfeld–Sokolov reduction for difference operators and deformations of W-algebras. I. The case of Virasoro algebra, Comm. Math. Phys. 192 (1998) 605; M.A. Semenov-Tian-Shansky, A.V. Sevostyanov, Drinfeld–Sokolov reduction for difference operators and deformations of W-algebras. II. General semisimple case, Comm. Math. Phys. 192 (1998) 631]. Using the bialgebra structure related to the new Drinfeld realization of affine quantum groups we describe reduction in terms of constraints. This realization of reduction admits direct quantization.As a byproduct we obtain an explicit expression for the symplectic form associated to the twisted Heisenberg double and calculate the moment map for the twisted dressing action. For some class of infinite-dimensional Poisson–Lie groups we also prove an analogue of the Ginzburg–Weinstein isomorphism.     

Stochastic and quantum space-time metrics and the weak-field limit

In this review we present a simple method of introducing stochastic and quantum metrics into gravitational theory at short distances in terms of small fluctuations around a classical background space-time. We consider only residual effects due to the stochastic (or quantum) theory of gravity and use a perturbative stochastization (or quantization) method. By using the general covariance and correspondence principles, we reconstruct the theory of gravitational, mechanical, electromagnetic, and quantum mechanical processes and tensor algebra in the space-time with stochastic and quantum metrics. Some consequences of the theory are also considered, in particular, it indicates that the value of the fundamental lengthl lies in the interval 10^–23l10^–22 cm.     

A Formal Model of Berezin-Toeplitz Quantization

We give a new construction of symbols of the differential operators on the sections of a quantum line bundle L over a Kähler manifold M using the natural contravariant connection on L. These symbols are the functions on the tangent bundle TM polynomial on fibres. For high tensor powers of L, the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on TM corresponding to some pseudo-Kähler structure on TM. Surprisingly, this star product is intimately related to the formal symplectic groupoid with separation of variables over M. We extend the star product on TM to generalized functions supported on the zero section of TM. The resulting algebra of generalized functions contains an idempotent element which can be thought of as a natural counterpart of the Bergman projection operator. Using this idempotent, we define an algebra of Toeplitz elements and show that it is naturally isomorphic to the algebra of Berezin-Toeplitz deformation quantization on M.     

Deformation theory applied to quantization and statistical mechanics

After a review of the deformation (star product) approach to quantization, treated in an autonomous manner as a deformation (with parameter ) of the algebraic composition law of classical observables on phase-space, we show how a further deformation (with parameter ) of that algebra is suitable for statistical mechanics. In this case, the phase-space is endowed with what we call a conformal symplectic (or conformal Poisson) structure, for which the bracket is the Poisson bracket modified by terms of order (1, 0) and (0, 1). As an application, one sees that the KMS states (classical or quantum) are those that vanish on the modified (Poisson or Moyal-Vey) bracket of any two observables, multiplied by a conformal factor.     

ADM-like Hamiltonian formulation of gravity in the teleparallel geometry

We present a new Hamiltonian formulation of the teleparallel equivalent of general relativity (TEGR) meant to serve as the departure point for canonical quantization of the theory. TEGR is considered here as a theory of a cotetrad field on a spacetime. The Hamiltonian formulation is derived by means of an ADM-like $3+1$ decomposition of the field and without any gauge fixing. A complete set of constraints on the phase space and their algebra are presented. The formulation is described in terms of differential forms.     

Some Infinite-Dimensional Algebras Arising in Spin Systems and in Particle Physics and their Grand Algebra

We indicate similarities in the structure of two types of infinite-dimensional algebras, one introduced 28 years ago in connection with the mass problem of elementary particles and the other seven years ago in connection with spin systems (XY models). We show that these algebras can be considered as representations of a single Grand Algebra, the enveloping algebra of an affine Kac–Moody algebra built on the Poincaré Lie algebra. As an associative and coassociative bialgebra of operators, the latter representation of the grand algebra is a preferred nontrivial deformation of the Ising case bialgebra.     

A Remark on Formal KMS States in Deformation Quantization

Within the framework of deformation quantization, we define formal KMS states on the deformed algebra of power series of functions with compact support in phase space as C[[]]-linear functionals obeying a formal variant of the usual KMS condition known in the theory of C^*-algebras. We show that for each temperature KMS states always exist and are up to a normalization equal to the trace of the argument multiplied by a formal analogue of the usual Boltzmann factor, a certain formal star exponential.     

Deformation Quantizations of the Poisson Algebra of Laurent Polynomials

It is well known that the Moyal bracket gives a unique deformation quantization of the canonical phase space R2n up to equivalence. In his presentation of an interesting deformation quantization of the Poisson algebra of Laurent polynomials, Ovsienko discusses the equivalences of deformation quantizations of these algebras. We show that under suitable conditions, deformation quantizations of this algebra are equivalent. Though Ovsienko showed that there exists a deformation quantization of the Poisson algebra of Laurent polynomials which is not equivalent to the Moyal product, this is not correct. We show this equivalence by two methods: a direct construction of the intertwiner via the star exponential and a more standard approach using Hochschild 2-cocycles.     

Vanishing of the Kontsevich Integrals of the Wheels

We prove that the Kontsevich integrals (in the sense of the formality theorem) of all even wheels are equal to zero. These integrals appear in the approach to the Duflo formula via the formality theorem. The result means that for any finite-dimensional Lie algebra g, and for invariant polynomials f, g [S ^·(g)]^g one has f · g = f * g, where * is the Kontsevich star product, corresponding to the Kirillov–Poisson structure on g^*. We deduce this theorem form the result contained in math.QA/0010321 on the deformation quantization with traces.