Deformation Quantization with Traces

In this Letter we prove a statement closely related to the cyclic formality conjecture. In particular, we prove that for a constant volume form and a Poisson bivector field on ^d such that div=0, the Kontsevich star product with the harmonic angle function is cyclic, i.e. (f*g)·h·= (g*h)·f· for any three functions f,g,h on (for which the integrals make sense). We also prove a globalization of this theorem in the case of arbitrary Poisson manifolds and an arbitrary volume form, and prove a generalization of the Connes–Flato–Sternheimer conjecture on closed star products in the Poisson case.     

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.     

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.     

On the Dequantization of Fedosov's Deformation Quantization

To each natural deformation quantization on a Poisson manifold M we associate a Poisson morphism from the formal neighborhood of the zero section of T ^* M to the formal neighborhood of the diagonal of the product M× , where is a copy of M with the opposite Poisson structure. We call it dequantization of the natural deformation quantization. Then we 'dequantize' Fedosov's quantization.     

Deformation Quantization of Nonregular Orbits of Compact Lie Groups

In this paper we construct a deformation quantization of the algebra of polynomials of an arbitrary (regular and nonregular) coadjoint orbit of a compact semisimple Lie group. The deformed algebra is given as a quotient of the enveloping algebra by a suitable ideal.     

Non-Abelian Reduction in Deformation Quantization

We consider a G-invariant star-product algebra A on a symplectic manifold (M,) obtained by a canonical construction of deformation quantization. Under assumptions of the classical Marsden–Weinstein Theorem we define a reduction of the algebra A with respect to the G-action. The reduced algebra turns out to be isomorphic to a canonical star-product algebra on the reduced phase space B. In other words, we show that the reduction commutes with the canonical G-invariant deformation quantization. A similar statement within the framework of geometric quantization is known as the Guillemin–Sternberg conjecture (by now, completely proved).     

Generalized Wigner Functions for Damped Systems in Deformation Quantization

Quantization of damped systems usually gives rise to complex spectra and corresponding resonant states, which do not belong to the Hilbert space. Therefore, the standard form of calculating Wigner function （WF） does not work for these systems. In this paper we show that in order to let WF satisfy a ,-genvalue equation for the damped systems, one must modify its standard form slightly, and this modification exactly coincides with the results derived from a ＊-Exponential expansion in deformation quantization.     

Generalized Wigner Functions for Damped Systems in Deformation Quantization

Quantization of damped systems usually gives rise to complex spectra and corresponding resonant states, which do not belong to the Hilbert space. Therefore, the standard form of calculating Wigner function (WF) does not work for these systems. In this paper we show that in order to let WF satisfy a *-genvalue equation for the damped systems, one must modify its standard form slightly, and this modification exactly coincides with the resu lts derived from a *-Exponential expansion in deformation quantization.     

Quantum Integrable Toda-Like Systems in Deformation Quantization

We define and discuss the notion of quantum integrability of a classically integrable system within the framework of deformation quantization, i.e. the question whether the classical conserved quantities (which are already in involution with respect to the Poisson bracket) commute with respect to some star product on the phase space after possible quantum corrections. As an example of this method, we show by means of suitable 2 by 2 quantum R-matrices that a list of Toda-like classical integrable systems given by Y. B. Suris is quantum integrable with respect to the usual star product of the Weyl type in flat 2n-dimensional space.     

Deformation Quantization of Algebraic Varieties

The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semiformal deformations are Noetherian and have polynomial growth. We propose constructions of semiformal quantizations of projective and affine algebraic Poisson manifolds satisfying certain natural geometric conditions. Projective symplectic manifolds (e.g. K3 surfaces and Abelian varieties) do not satisfy our conditions, but projective spaces with quadratic Poisson brackets and Poisson–Lie groups can be semiformally quantized.     

Operads and Motives in Deformation Quantization

The algebraic world of associative algebras has many deep connections with the geometric world of two-dimensional surfaces. Recently, D. Tamarkin discovered that the operad of chains of the little discs operad is formal, i.e. it is homotopy equivalent to its cohomology. From this fact and from Deligne's conjecture on Hochschild complexes follows almost immediately my formality result in deformation quantization. I review the situation as it looks now. Also I conjecture that the motivic Galois group acts on deformation quantizations, and speculate on possible relations of higher-dimensional algebras and of motives to quantum field theories.     

Deformation Quantization of Hermitian Vector Bundles

Motivated by deformation quantization, we consider in this paper *-algebras over rings = (i), where is an ordered ring and I²=–1, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) -valued inner product. For A=C (M), M a manifold, these modules can be identified with Hermitian vector bundles E over M. We show that for a fixed Hermitian star product on M, these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that there is a natural bijection between the sets of equivalence classes of local Hermitian deformations of C (M) and ( (E)) and that the corresponding deformed algebras are formally Morita equivalent, an algebraic generalization of strong Morita equivalence of C ^*-algebras. We also discuss the semi-classical geometry arising from these deformations.     

On the Inverse Mapping of the Formal Symplectic Groupoid of a Deformation Quantization

To each natural star product on a Poisson manifold M we associate an antisymplectic involutive automorphism of the formal neighborhood of the zero section of the cotangent bundle of M. If M is symplectic, this mapping is shown to be the inverse mapping of the formal symplectic groupoid of the star product. The construction of the inverse mapping involves modular automorphisms of the star product.     

On the Canonical Normalization of a Trace Density of Deformation Quantization

We describe a procedure of the canonical normalization of a formal trace density of an arbitrary deformation quantization on a symplectic manifold. We apply this procedure to give an explicit expression of the canonical formal trace density of deformation quantization with separation of variables on a pseudo-Käahler manifold.     

The Picard Groupoid in Deformation Quantization

In this Letter we give an overview on recent developments in representation theory of star product algebras. In particular, we relate the *-representation theory of *-algebras over rings C = R(i) with an ordered ring R and i²=–1 to the *-representation theory of *-algebras over and point out some properties of the Picard groupoid corresponding to the notion of strong Morita equivalence. Some Morita invariants are interpreted as arising from actions of this groupoid     

A Covariant Poisson Deformation Quantization with Separation of Variables up to the Third Order

We give a simple formula for the operator C ₃ of the standard deformation quantization with separation of variables on a Kähler manifold M. Unlike C ₁ and C ₂, this operator cannot be expressed in terms of the Kähler–Poisson tensor on M. We modify C ₃ to obtain a covariant deformation quantization with separation of variables up to the third order which is expressed in terms of the Poisson tensor on M and can thus be defined on an arbitrary complex manifold endowed with a Poisson bivector field of type (1,1).     

Poisson Bracket,Deformed Bracket and Gauge Group Actions in Kontsevich Deformation Quantization

We express the difference between the Poisson bracket and a deformed bracket for Kontsevich deformation quantization on any Poisson manifold by means of the second derivative of the formality quasi-isomorphism. The counterpart in star products of the action of formal diffeomorphisms on Poisson formal bivector fields is also investigated.     

参数空间十六极振动的量子化

利用核十六极形变最普遍的参数化形式,通过正则量子化程序,导出了参数空间十六极振动的量子化哈密顿量,分析了十六极形变和振动对于核结构研究的重要意义.     

A q-Analogue of the Berezin Quantization Method

We use Berezin's quantization procedure to obtain a formal -invariant deformation of the quantum disc. Explicit formulae for the associated bidifferential operators are produced.     

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.