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In this paper, we define Berezin-type and Odzijewicz-type quantizations on compact smooth manifolds. The method is as follows: we embed the smooth manifold of real dimension $n$ into $ℂP^n$ and induce the quantizations from there. The standard way by which reproducing kernel Hilbert spaces are defined on submanifolds gives a way to define the pullback coherent states. In Berezin-type quantization, the Hilbert space of quantization is the pullback (by the embedding) of the Hilbert space of geometric quantization of $ℂP^n$. In the Odzijewicz-type quantization, one has to consider a tensor product of the geometric quantization line bundle with holomorphic $n$-forms. In the Berezin case, the operators that are quantized are those induced from the ambient space $ℂP^n$. The Berezin-type quantization exhibited here is a generalization of an earlier work of the author and Ghosh. In both Berezin- and Odzijewicz-type quantizations, we first exhibit this quantization explicitly on $ℂP^n$ and we induce the quantization on the smooth compact embedded manifold from $ℂP^n$.

We study a quantum moment map and propose an invariant for G-invariant star products on a G-transitive symplectic manifold. We start by describing a new method to construct a quantum moment map for G-invariant star products of Fedosov type. We use it to obtain an invariant that is invariant under G-equivalence. In the last section we give two simple examples of such invariants, which involve non-classical terms and provide new insights into the classification of G-invariant star products.

The characteristic class of a star product on a symplectic manifold appears as the class of a deformation of a given symplectic connection, as described by Fedosov. In contrast, one usually thinks of the characteristic class of a star product as the class of a deformation of the Poisson structure (as in Kontsevich's work). In this paper, we present, in the symplectic framework, a natural procedure for constructing a star product by directly quantizing a deformation of the symplectic structure. Basically, in Fedosov's recursive formula for the star product with zero characteristic class, we replace the symplectic structure by one of its formal deformations in the parameter ℏ. We then show that every equivalence class of star products contains such an element. Moreover, within a given class, equivalences between such star products are realized by formal one-parameter families of diffeomorphisms, as produced by Moser's argument.

In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the ad-invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on.

In this review we discuss various aspects of representation theory in deformation quantization starting with a detailed introduction to the concepts of states as positive functionals and the GNS construction. Rieffel induction of representations as well as strong Morita equivalence, Dirac monopole and strong Picard Groupoid are also discussed.

This survey is an overview of some of the better known quantization techniques (for systems with finite numbers of degrees-of-freedom) including in particular canonical quantization and the related Dirac scheme, introduced in the early days of quantum mechanics, Segal and Borel quantizations, geometric quantization, various ramifications of deformation quantization, Berezin and Berezin–Toeplitz quantizations, prime quantization and coherent state quantization. We have attempted to give an account sufficiently in depth to convey the general picture, as well as to indicate the mutual relationships between various methods, their relative successes and shortcomings, mentioning also open problems in the area. Finally, even for approaches for which lack of space or expertise prevented us from treating them to the extent they would deserve, we have tried to provide ample references to the existing literature on the subject. In all cases, we have made an effort to keep the discussion accessible both to physicists and to mathematicians, including non-specialists in the field.

We show that the cross Wigner function can be written in the form where is the Fourier transform of ϕ and Ŝ is a metaplectic operator that projects onto a linear symplectomorphism S consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl–Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from deformation quantization. Using the "metaplectic formulation" of the Wigner transform, we construct a complete set of intertwiners relating the Weyl and the Bopp pseudo-differential operators. This is an important result that allows us to prove the spectral and dynamical equivalence of the Schrödinger and the Bopp representations of quantum mechanics.

We reconsider the (non-relativistic) quantum theory of indistinguishable particles on the basis of Rieffel’s notion of $C^{\ast }$-algebraic (“strict”) deformation quantization. Using this formalism, we relate the operator approach of Messiah and Greenberg (1964) to the configuration space approach pioneered by Souriau (1967), Laidlaw and DeWitt-Morette (1971), Leinaas and Myrheim (1977), and others. In dimension $d>2$, the former yields bosons, fermions, and paraparticles, whereas the latter seems to leave room for bosons and fermions only, apparently contradicting the operator approach as far as the admissibility of parastatistics is concerned. To resolve this, we first prove that in $d>2$ the topologically non-trivial configuration spaces of the second approach are quantized by the algebras of observables of the first. Secondly, we show that the irreducible representations of the latter may be realized by vector bundle constructions, among which the line bundles recover the results of the second approach. Mathematically speaking, representations on higher-dimensional bundles (which define parastatistics) cannot be excluded, which render the configuration space approach incomplete. Physically, however, we show that the corresponding particle states may always be realized in terms of bosons and/or fermions with an unobserved internal degree of freedom (although based on non-relativistic quantum mechanics, this conclusion is analogous to the rigorous results of the Doplicher–Haag–Roberts analysis in algebraic quantum field theory, as well as to the heuristic arguments which led Gell-Mann and others to qcd (i.e. Quantum Chromodynamics)).

Increasing tensor powers of the $k\times k$ matrices $M_k(ℂ)$ are known to give rise to a continuous bundle of $C^{\ast }$-algebras over $I=\{0\}\cup 1/\mathbb{N}\subset [0,1]$ with fibers $A_{1/N}=M_k{(ℂ)}^{\otimes N}$ and $A_0=C(X_k)$, where $X_k=S(M_k(ℂ))$, the state space of $M_k(ℂ)$, which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of $X_k$ à la Rieffel, defined by perfectly natural quantization maps $Q_{1/N}:{Ã}_0\to A_{1/N}$ (where ${Ã}_0$ is an equally natural dense Poisson subalgebra of $A_0$).

We apply this quantization formalism to the Curie–Weiss model (an exemplary quantum spin with long-range forces) in the parameter domain where its ${\mathbb{Z}}_2$ symmetry is spontaneously broken in the thermodynamic limit $N\to \infty$. If this limit is taken with respect to the macroscopic observables of the model (as opposed to the quasi-local observables), it yields a classical theory with phase space $X_2\cong B^3$ (i.e. the unit three-ball in ${ℝ}^3$). Our quantization map then enables us to take the classical limit of the sequence of (unique) algebraic vector states induced by the ground state eigenvectors ${\mathrm{\Psi }}_N^{(0)}$ of this model as $N\to \infty$, in which the sequence converges to a probability measure $\mu$ on the associated classical phase space $X_2$. This measure is a symmetric convex sum of two Dirac measures related by the underlying ${\mathbb{Z}}_2$-symmetry of the model, and as such the classical limit exhibits spontaneous symmetry breaking, too. Our proof of convergence is heavily based on Perelomov-style coherent spin states and at some stage it relies on (quite strong) numerical evidence. Hence the proof is not completely analytic, but somewhat hybrid.

We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from ${ℂ}^{1+n}$ with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space $ℂ{\mathbb{P}}^n$ and the complex hyperbolic disc ${𝔻}^n$. We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of “polynomial” functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures, linking, e.g., the star products on $ℂ{\mathbb{P}}^n$ and ${𝔻}^n$. More precisely, we describe an isomorphism between the (polynomial or analytic) function algebras that is compatible with Poisson brackets and the convergent star products. This isomorphism is essentially given by Wick rotation, i.e. holomorphic extension of analytic functions and restriction to a new domain. It is not compatible with the ${}^{\ast }$-involution of pointwise complex conjugation.

A continuous bundle of $C^{\ast }$-algebras provides a rigorous framework to study the thermodynamic limit of quantum theories. If the bundle admits the additional structure of a strict deformation quantization (in the sense of Rieffel) one is allowed to study the classical limit of the quantum system, i.e. a mathematical formalism that examines the convergence of algebraic quantum states to probability measures on phase space (typically a Poisson or symplectic manifold). In this manner, we first prove the existence of the classical limit of Gibbs states illustrated with a class of Schrödinger operators in the regime where Planck’s constant $\hslash$ appearing in front of the Laplacian approaches zero. We additionally show that the ensuing limit corresponds to the unique probability measure satisfying the so-called classical or static KMS-condition. Subsequently, we conduct a similar study on the free energy of mean-field quantum spin systems in the regime of large particles, and discuss the existence of the classical limit of the relevant Gibbs states. Finally, a short section is devoted to single site quantum spin systems in the large spin limit.

We compute an explicit algebraic deformation quantization for an affine Poisson variety described by an ideal in a polynomial ring, and inheriting its Poisson structure from the ambient space.

We show that two approaches to equivariant strict deformation quantization of C${}^{\ast }$-algebras by actions of negatively curved Kählerian Lie groups, one based on oscillatory integrals and the other on quantizations maps defined by dual 2-cocycles, are equivalent.

We show that two approaches to equivariant deformation of C${}^{\ast }$-algebras by actions of negatively curved Kählerian Lie groups, one based on oscillatory integrals and the other on quantizations maps defined by dual $2$-cocycles, are still equivalent despite the nonunitarity of our $2$-cocycles.

The necessary and sufficient conditions are established for the second-class constraint surface to be (an almost) Kähler manifold. The deformation quantization for such systems is mentioned resulting in the Wick-type symbols for the respective Dirac brackets.

It is shown that noncommutative spaces, which are quotients of associative algebras by ideals generated by highly nonlinear relations of a particular type, admit extremely simple formulae for deformed or star products. Explicit construction of these star products is carried out. Quantum gauge theories are formulated on these spaces, and the Seiberg–Witten map is worked out in detail.

We study the fuzzy structure of the general Kähler coset space G/S ⊗ U(1)^k deformed by the Fedosov formalism. It is shown that the Killing potentials satisfy the fuzzy algebras working in the Darboux coordinates.

Deformation quantization is a powerful tool to quantize some classical systems especially in noncommutative space. In this work we first show that for a class of special Hamiltonian one can easily find relevant time evolution functions and Wigner functions, which are intrinsic important quantities in the deformation quantization theory. Then based on this observation we investigate a two-coupled harmonic oscillators system on the general noncommutative phase space by requiring both spatial and momentum coordinates do not commute each other. We derive all the Wigner functions and the corresponding energy spectra for this system, and consider several interesting special cases, which lead to some significant results.

In order to quantize systems involving second-class constraints, one should use Dirac bracket instead of Poisson bracket. Furthermore, one can specify a star product in which the term linear in $\hslash$ is proportional to the Dirac bracket. In this way an oscillator system in a deformed fermionic phase space is analyzed and the corresponding energy level and Wigner functions are evaluated according to scheme of deformation quantization. We also study the entanglement entropy induced by the deformation of the fermionic phase space.

We study the deformation quantization of scalar and Abelian gauge classical free fields. Stratonovich–Weyl quantizer, star products and Wigner functionals are obtained in field and oscillator variables. The Abelian gauge theory is particularly intriguing since the Wigner functional is factorized into a physical part and the other one containing the constraints only. Some effects of nontrivial topology within the deformation quantization formalism are also considered.