Quantum or classical mechanical systems symmetric under $SU(2)$ are called spin systems. A $SU(2)$-equivariant map from $(n+1)$-square matrices to functions on the $2$-sphere S^2, satisfying some basic properties, is called a spin-$j$ symbol correspondence ($n = 2j \in \mathbb{N}$). Given a spin-$j$ symbol correspondence, the matrix algebra induces a twisted $j$-algebra of symbols. In the first part of this paper, we establish a more intuitive criterion for when the Poisson algebra of smooth functions on $S^2$ emerges asymptotically ($n \to \infty$) from the sequence of twisted $j$-algebras. This more geometric criterion, which in many cases is equivalent to the numerical criterion obtained in [20] for describing symbol correspondence sequences of (anti-)Poisson type, is now given in terms of a classical (asymptotic) localization of symbols of all projectors (quantum pure states) in a certain family. For some important kinds of symbol correspondence sequences, such a classical localization condition is equivalent to asymptotic emergence of the Poisson algebra. But in general, the classical localization condition is stronger than Poisson emergence. We thus also consider some weaker notions of asymptotic localization of projector-symbols. In the second part of this paper, for each sequence of symbol correspondences of (anti-)Poisson type, we define the sequential quantization of a smooth function on $S^2$ and its asymptotic operator acting on a ground Hilbert space. Then, after presenting some concrete examples of these constructions, we obtain some relations between asymptotic localization of a symbol correspondence sequence and the asymptotics of its sequential quantization of smooth functions on $S^2$.
For phase-space manifolds which are compact Kaehler manifolds relations between the Berezin-Toeplitz quantization and the quantization with the help of Berezin's coherent states and symbols are studied. First the results on the Berezin-Toeplitz quantization of arbitrary compact Kaehler manifolds due to Bordemann, Meinrenken and Schlichenmaier are recalled. It is shown that the covariant symbol map is adjoint to the Toeplitz map. The Berezin transform for compact Kaehler manifolds is discussed. (Talk presented at the XVII workshop on geometric methods in physics, Bialowieza, Poland, July 3 --Ju…
The spaces of higher-order differential operators (in Dimension 1|2), which are modules over the stringy Lie superalgebra K(2), are isomorphic to the corresponding spaces of symbols as orthosymplectic modules in non resonant cases. Such an osp (2|2)-equivariant quantization, which has been given in second-order differential operators case, keeps existing and unique. We calculate its explicit formula that provides extension in particular order cases.
Toeplitz quantization is defined in a general setting in which the symbols are the elements of a possibly non-commutative algebra with a conjugation and a possibly degenerate inner product. We show that the quantum group $SU_q(2)$ is such an algebra. Unlike many quantization schemes, this Toeplitz quantization does not require a measure. The theory is based on the mathematical structures defined and studied in several recent papers of the author; those papers dealt with some specific examples of this new Toeplitz quantization. Annihilation and creation operators are defined as densely defined …
In this paper we give a construction of Fedosov quantization incorporating the odd variables and an analogous formula to Getzler's pseudodifferential calculus composition formula is obtained. A Fedosov type connection is constructed on the bundle of Weyl tensor Clifford algebras over the cotangent bundle of a Riemannian manifold. The quantum algebra associated with this connection is used to define a deformation of the exterior algebra of Riemannian manifolds.
We define co-Toeplitz operators, a new class of Hilbert space operators, in order to define a co-Toeplitz quantization scheme that is dual to the Toeplitz quantization scheme introduced by the author in the setting of symbols that come from a possibly non-commutative algebra with unit. In the present dual setting the symbols come from a possibly non-co-commutative co-algebra with co-unit. However, this co-Toeplitz quantization is a usual quantization scheme in the sense that to each symbol we assign a densely defined linear operator acting in a fixed Hilbert space. Creation and annihilation op…
We consider the supercircle $S^{1|1}$ equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on $S^{1|1}$ as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra $osp(1|2)$. We study the space of linear differential operators on weighted densities as a module over $osp(1|2)$. We introduce the canonical isomorphism between this space and the corresponding space of symbols and find interesting resonant cases where such an isomorphism does not exist.
The author has introduced in a recent paper a new class of operators, called co-Toeplitz operators, with symbols in a co-algebra. This is the categorical dual to Toeplitz operators which have symbols in an algebra. The mapping from a symbol to its co-Toeplitz operator gives a quantization scheme, called co-Toeplitz quantization. A new, quite simple particular case of co-Toeplitz quantization is introduced in this note. Examples are given in order to show some of its properties.
In this paper, we give a correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the torus $ \mathbb{T}^2$. To achieve this, we use the correspondence between the Berezin-Toeplitz and the complex Weyl quantizations of the complex plane and a relation between the Berezin-Toeplitz quantization of a periodic symbol on the real phase space $\mathbb{R}^2$ and the Berezin-Toeplitz quantization of a symbol on the torus $ \mathbb{T}^2 $.
We apply Shubin's theory of global symbol classes $Γ_ρ^{m}$ to the Born-Jordan pseudodifferential calculus we have previously developed. This approach has many conceptual advantages, and makes the relationship between the conflicting Born-Jordan and Weyl quantization methods much more limpid. We give, in particular, precise asymptotic expansions of symbols allowing to pass from Born-Jordan quantization to Weyl quantization, and vice-versa. In addition we state and prove some regularity and global hypoellipticity results.
Parallel to the quantization of the complex plane, using the canonical coherent states of a right quaternionic Hilbert space, quaternion field of quaternionic quantum mechanics is quantized. Associated upper symbols, lower symbols and related quantities are analysed. Quaternionic version of the harmonic oscillator and Weyl-Heisenberg algebra are also obtained.
The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.
There has recently been evidence for replacing the usual Weyl quantization procedure by the older and much less known Born-Jordan rule. In this paper we discuss this quantization procedure in detail and relate it to recent results of Boggiato, De Donno, and Oliaro on the Cohen class. We begin with a discussion of some properties of Shubin's $τ$-pseudo-differential calculus, which allows us to show that the Born-Jordan quantization of a symbol $a$ is the average for $τ\in\lbrack0,1]$ of the $τ$-operators with symbol $a$. We study the properties of the Born-Jordan operators, including their symp…
Using the orthonormality of the 2D-Zernike polynomials, reproducing kernels, reproducing kernel Hilbert spaces, and ensuring coherent states attained. With the aid of the so-obtained coherent states, the complex unit disc is quantized. Associated upper symbols, lower symbols and related generalized Berezin transforms also obtained. A number of necessary summation formulas for the 2D-Zernike polynomials proved.
In this work we give a deformation theoretical approach to the problem of quantization. First the notion of a deformation of a noncommutative ringed space over a commutative locally ringed space is introduced within a language coming from Algebraic Geometry and Complex Analysis. Then we define what a Dirac quantization of a commutative ringed space with a Poisson structure, the space of classical observables, is. Afterwards the normal order quantization of the Poisson space of classical polynomial observables on a cotangent bundle is constructed. By using a complete symbol calculus on manifold…
This article is concerned with compositions in the context of three standard quantizations in the Fock space framework, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for time dynamics issues. Anti-Wick quantization and Segal Bargmann transforms are implied there for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick symbol of the composition of two operat…
We define a generalization of the T\''oplitz quantization, suitable for operators whose T\''oplitz symbols are singular. We then show that singular curve operators in Topological Quantum Fields Theory (TQFT) are precisely generalized T\''oplitz operators of this kind and we compute for some of them, and conjecture for the others, their main symbol, determined by the associated classical trace function.
In this work various symbol spaces with values in a sequentially complete locally convex vector space are introduced and discussed. They are used to define vector-valued oscillatory integrals which allow to extend Rieffel's strict deformation quantization to the framework of sequentially complete locally convex algebras and modules with separately continuous products and module structures, making use of polynomially bounded actions of $\mathbb{R}^n$. Several well-known integral formulas for star products are shown to fit into this general setting, and a new class of examples involving compactl…
A bosonic-fermionic correspondence allows an analytic definition of functional super derivative, in particular, and a bosonic functional calculus, in general, on Bargmann- Gelfand triples for the second super quantization. A Feynman integral for the super transformation matrix elements in terms of bosonic anti-normal Berezin symbols is rigorously constructed.
In the theory of Toeplitz quantization of algebras, as developed by the second author, coherent states are defined as eigenvectors of a Toeplitz annihilation operator. These coherent states are studied in the case when the algebra is the generically non-commutative Manin plane. In usual quantization schemes one starts with a classical phase space, then quantizes it in order to produce annihilation operators and then their eigenvectors and eigenvalues. But we do this in the opposite order, namely the set of the eigenvalues of the previously defined annihilation operator is identified as a gener…
Quantization procedures play an essential role in microlocal analysis, time-frequency analysis and, of course, in quantum mechanics. Roughly speaking the basic idea, due to Dirac, is to associate to any symbol, or observable, $a(x,ξ)$ an operator $\mathrm{Op}(a)$, according to some axioms dictated by physical considerations. This led to the introduction of a variety of quantizations. They all agree when the symbol $a(x,ξ)=f(x)$ depends only on $x$ or $a(x,ξ)=g(ξ)$ depends only on $ξ$: \[ \mathrm{Op}(f\otimes1)u=fu,\quad\mathrm{Op}(1\otimes g)u=\mathcal{F}% ^{-1}(g\mathcal{F}u) \] where $\mathc…
Using geometric quantization, we represent curve operators in the TQFT of Witten-Reshetikhin-Turaev with jauge group SU_2 as Toeplitz operators with symbols corresponding to trace functions. As an application, we show that eigenvectors of these operators are concentrated near the level sets of these trace functions, and obtain asymptotic estimates of pairings of such eigenvectors. This yields an asymptotic for some matrix coefficients of the image of mapping classes by quantum representations.
One can argue that on flat space $\mathbb{R}^d$ the Weyl quantization is the most natural choice and that it has the best properties (e.g. symplectic covariance, real symbols correspond to Hermitian operators). On a generic manifold, there is no distinguished quantization, and a quantization is typically defined chart-wise. Here we introduce a quantization that, we believe, has the best properties for studying natural operators on pseudo-Riemannian manifolds. It is a generalization of the Weyl quantization - we call it the balanced geodesic Weyl quantization. Among other things, we prove that …
As a consequence of the Schwartz kernel Theorem, any linear continuous operator $\widehat{A}:$ $\mathcal{S}(\mathbb{R}^{n})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n})$ can be written in Weyl form in a unique way, namely it is the Weyl quantization of a unique symbol $a\in\mathcal{S}^{\prime}(\mathbb{R}^{2n})$. Hence, dequantization can always be performed, and in a unique way. Despite the importance of this topic in Quantum Mechanics and Time-frequency Analysis, the same issue for the Born-Jordan quantization seems simply unexplored, except for the case of polynomial symbols, which we …
A normal form transformation is carried out on the operators of a complete set of commuting observables in a multidimensional, integrable quantum system, mapping them by unitary conjugation into functions of the harmonic oscillators in the various degrees of freedom. The transformation works at the level of the Weyl symbols of the operators, which are manipulated as formal power series in hbar by use of the Moyal star product. It is assumed that the Weyl symbol of one of the operators (the Hamiltonian) has a generic, stable fixed point in phase space. The normal form transformation takes place…
Using an abstract notion of semiclassical quantization for self-adjoint operators, we prove that the joint spectrum of a collection of commuting semiclassical self-adjoint operators converges to the classical spectrum given by the joint image of the principal symbols, in the semiclassical limit. This includes Berezin-Toeplitz quantization and certain cases of $\hbar$-pseudodifferential quantization, for instance when the symbols are uniformly bounded, and extends a result by L. Polterovich and the authors. In the last part of the paper we review the recent solution to the inverse problem for q…
Kontsevich's 1997 formula for the deformation quantization of Poisson brackets is a Feynman expansion involving volume integrals over moduli spaces of marked disks. We develop a systematic theory of integration on these moduli spaces via suitable algebras of polylogarithms, and use it to prove that Kontsevich's integrals can be expressed as integer-linear combinations of multiple zeta values. Our proof gives a concrete algorithm for calculating the integrals, which we have used to produce the first software package for the symbolic calculation of Kontsevich's formula.
