Formal Deformation Quantization Research Articles

Strict quantization of coadjoint orbits

For every semisimple coadjoint orbit $\\hat{\\mathcal{O}}$ of a complex connected semisimple Lie group $\\hat{G}$, we obtain a family of $\\hat{G}$-invariant products $\\hat{}\_\\hbar$ on the space of holomorphic functions on $\\hat{\\mathcal{O}}$. For every semisimple coadjoint orbit $\\mathcal{O}$ of a real connected semisimple Lie group $G$, we obtain a family of $G$-invariant products $\\hbar$ on a space $\\mathcal{A}(\\mathcal{O})$ of certain analytic functions on $\\mathcal{O}$ by restriction. $\\mathcal{A}(\\mathcal{O})$, endowed with one of the products $\*\\hbar$, is a $G$-Fréchet algebra, and the formal expansion of the products around $\\hbar=0$ determines a formal deformation quantization of $\\mathcal{O}$, which is of Wick type if $G$ is compact. Our construction relies on an explicit computation of the canonical element of the Shapovalov pairing between generalized Verma modules and complex analytic results on the extension of holomorphic functions.

Group actions in deformation quantization

This set of notes corresponds to a mini-course given in September 2018 in Będlewo; it does not contain any new result; it complements—with intersection—the introduction to formal deformation quantization and group actions published in 2017, correspond

Canonical quantization of constants of motion

We develop a quantization method, that we name decomposable Weyl quantization, which ensures that the constants of motion of a prescribed finite set of Hamiltonians are preserved by the quantization.Our method is based on a structural analogy between the notions of reduction of the classical phase space and diagonalization of selfadjoint operators. We obtain the spectral decomposition of the emerging quantum constants of motion directly from the quantization process.If a specific quantization is given, we expect that it preserves constants of motion exactly when it coincides with decomposable Weyl quantization on the algebra of constants of motion. We obtain a characterization of when such property holds in terms of the Wigner transforms involved. We also explain how our construction can be applied to spectral theory.Moreover, we discuss how our method opens up new perspectives in formal deformation quantization and geometric quantization.

The star product in interacting quantum field theory

We propose a new formula for the star product in deformation quantization of Poisson structures related in a specific way to a variational problem for a function S, interpreted as the action functional. Our approach is motivated by perturbative algebraic quantum field theory (pAQFT). We provide a direct combinatorial formula for the star product, and we show that it can be applied to a certain class of infinite-dimensional manifolds (e.g. regular observables in pAQFT). This is the first step towards understanding how pAQFT can be formulated such that the only formal parameter is hbar , while the coupling constant can be treated as a number. In the introductory part of the paper, apart from reviewing the framework, we make precise several statements present in the pAQFT literature and recast these in the language of (formal) deformation quantization. Finally, we use our formalism to streamline the proof of perturbative agreement provided by Drago, Hack, and Pinamonti and to generalize some of the results obtained in that work to the case of a nonlinear interaction.

Formal oscillatory integrals and deformation quantization

Following [14] and [12], we formalize the notion of an oscillatory integral interpreted as a functional on the amplitudes supported near a fixed critical point $x_0$ of the phase function with zero critical value. We relate to an oscillatory integral two objects, a formal oscillatory integral kernel and the full formal asymptotic expansion at $x_0$. The formal asymptotic expansion is a formal distribution supported at $x_0$ which is applied to the amplitude. In [12] this distribution itself is called a formal oscillatory integral (FOI). We establish a correspondence between the formal oscillatory integral kernels and the FOIs based upon a number of axiomatic properties of a FOI expressed in terms of its formal integral kernel. Then we consider a family of polydifferential operators related to a star product with separation of variables on a pseudo-K\"ahler manifold. These operators evaluated at a point are FOIs. We completely identify their formal oscillatory kernels.

Mini-Workshop: Deformation quantization: between formal to strict

The philosophy of deformation was proposed by Bayen, Flato, Fronsdal, Lichnerowicz, and Sternheimer in the seventies and since then, many developments occurred. Deformation quantization is based on such a philosophy in order to provide a mathematical procedure to pass from classical mechanics to quantum mechanics. Basically, it consists in deforming the pointwise product of functions to get a non-commutative one, which encodes the quantum mechanics behaviour. In formal deformation quantization, the non-commutative product (also said, star product) is given by a formal deformation of the pointwise product, i.e. by a formal power series in the deformation parameter which physically play the role of Planck’s constant $\hbar$. From a physical point of view this is clearly not sufficient to provide a reasonable quantum mechanical description and hence one needs to overcome the formal power series aspects in some way. One option is strict deformation quantization, which produces quantum algebras not just in the space of formal power series but in terms of $C$\*-algebras, as suggested by Rieffel, with e.g. a continuous dependence on $\hbar$. There are several other options in between continuous and formal dependence on $\hbar$ like analytic or smooth deformations. The Oberwolfach workshop Deformation quantization: between formal to strict consolidated, continued, and extended these research activities with a focus on the study of the connection between formal and strict deformation quantization in their various flavours and their applications in particular those in quantum physics and non-commutative geometry. It brought together specialists in differential geometry, operator algebras, non-commutative geometry, and quantum field theory with research interests in the mentioned quantization procedures. The aim of the workshop was to develop a coherent viewpoint of the many recent diverse developments in the field and to initiate new lines of research.

The Necessity of Wheels in Universal Quantization Formulas

In the context of formal deformation quantization, we provide an elementary argument showing that any universal quantization formula necessarily involves graphs with wheels.

Quantization of Poisson Hamiltonian systems

In this paper we introduce the concept of Hamiltonian system in the canonical and Poisson settings. We will discuss the quantization of the Hamiltonian systems in the Poisson context, using formal deformation quantization and quantum group theories.

A universal formula for deformation quantization on Kähler manifolds

We give an explicit local formula for any formal deformation quantization, with separation of variables, on a Kähler manifold. The formula is given in terms of differential operators, parametrized by acyclic combinatorial graphs.

Fréchet algebraic deformation quantization of the Poincaré disk

Abstract. Starting from formal deformation quantization we use an explicit formula for a star product on the Poincaré disk 𝔻 n $\mathbb {D}_n$ to introduce a Fréchet topology making the star product continuous. To this end a general construction of locally convex topologies on algebras with countable vector space basis is introduced and applied. Several examples of independent interest are investigated as, e.g., group algebras over finitely generated groups and infinite matrices. In the case of the star product on 𝔻 n $\mathbb {D}_n$ the resulting Fréchet algebra is shown to have many nice features: it is a strongly nuclear Köthe space, the symmetry group SU ( 1 , n ) $\mathrm {SU}(1, n)$ acts smoothly by continuous automorphisms with an inner infinitesimal action, and evaluation functionals at all points of 𝔻 n $\mathbb {D}_n$ are continuous positive functionals.

Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes

In this article we study the quantization of a free real scalar field on a class of noncommutative manifolds, obtained via formal deformation quantization using triangular Drinfel’d twists. We construct deformed quadratic action functionals and compute the corresponding equation of motion operators. The Green’s operators and the fundamental solution of the deformed equation of motion are obtained in terms of formal power series. It is shown that, using the deformed fundamental solution, we can define deformed *-algebras of field observables, which in general depend on the spacetime deformation parameter. This dependence is absent in the special case of Killing deformations, which include in particular the Moyal-Weyl deformation of the Minkowski spacetime.

Deformation Quantization of a Certain Type of Open Systems

We give an approach to open quantum systems based on formal deformation quantization. It is shown that classical open systems of a certain type can be systematically quantized into quantum open systems preserving the complete positivity of the open time evolution. The usual example of linearly coupled harmonic oscillators is discussed.

Cyclic cocycles on deformation quantizations and higher index theorems

We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We give a new proof of Nest–Tsygan's algebraic higher index theorem by computing the pairing between such cyclic cocycles and the K-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes–Moscovici and its extension to orbifolds.

Bohr–Sommerfeld star products

We relate the Bohr–Sommerfeld conditions to formal deformation quantization of symplectic manifolds by classifying star products adapted to some Lagrangian submanifold L , i.e. products preserving the classical vanishing ideal I L of L up to I L -preserving equivalences.

An algebraic index theorem for orbifolds

Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann–Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the Kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem.

Bimodule Deformations, Picard Groups and Contravariant Connections

We study deformations of invertible bimodules and the behavior of Picard groups under deformation quantization. While K0-groups are known to be stable under formal deformations of algebras, Picard groups may change drastically. We identify the semiclassical limit of bimodule deformations as contravariant connections and study the associated deformation quantization problem. Our main focus is on formal deformation quantization of Poisson manifolds by star products.