A New Quantization Map

Deformation quantization and the tomographic representation of quantum fields

The tomographic representation of quantum fields within the deformation quantization formalism is constructed. By employing the Wigner functional we obtain the symplectic tomogram associated with quantum fields. In addition, the tomographic version of the Wigner map allows us to compute the symbols corresponding to field operators. Finally, the functional integral representation of the tomographic star product is determined. Some possible applications of the formalism to loop quantum cosmology and loop quantum gravity are briefly discussed.

On positive maps, entanglement and quantization

We outline the scheme for quantization of classical Banach space results associated with some prototypes of dynamical maps and describe the quantization of correlations as well. A relation between these two areas is discussed.

A Uniqueness Theorem for Constraint Quantization

This work addresses certain ambiguities in the Dirac approach to constrained systems. Specifically, we investigate the space of so-called ``rigging maps'' associated with Refined Algebraic Quantization, a particular realization of the Dirac scheme. Our main result is to provide a condition under which the rigging map is unique, in which case we also show that it is given by group averaging techniques. Our results comprise all cases where the gauge group is a finite-dimensional Lie group.

A Parity-Conserving Canonical Quantization for the Baker's Map

We present here a complete description of the quantization of the baker's map. The method we use is quite different from that used in Balazs and Voros [BV] and Saraceno [S]. We use as the quantum algebra of observables the operators generated by {exp(2 Pi ix),exp (2 Pi ip)} and construct a unitary propagator such that as Planck's constant tends to zero,the classical dynamics is returned. For Planck's constant satisfying the integrality condition 1/N with N even, and for periodic boundary conditions for the wave functions on the torus, we show that the dynamics can be reduced to the dynamics on…

Branched Quantization

We propose a method for quantization of Lagrangians for which the Hamiltonian, as a function of momentum, is a branched function with cusps. Appropriate boundary conditions, which we identify, insure unitary time evolution. In special cases a dual (canonical) transformation maps the problem into a problem of quantum mechanics on singular spatial manifolds, which we also develop. Several possible applications are indicated.

A Canonical Quantization of the Baker's Map

We present here a canonical quantization for the baker's map. The method we use is quite different from that used in Balazs and Voros (ref. \QCITE{cite}{}{BV}) and Saraceno (ref. \QCITE{cite}{}{S}). We first construct a natural ``baker covering map'' on the plane $\QTO{mathbb}{\mathbb{R}}^{2}$. We then use as the quantum algebra of observables the subalgebra of operators on $L^{2}(\QTO{mathbb}{\mathbb{R}}) $ generated by $\left\{\exp (2πi\hat{x}) ,\exp (2πi\hat{p}) \right\} $ . We construct a unitary propagator such that as $\hbar \to 0$ the classical dynamics is returned. For Planck's constan…

利用微分几何结构对Kostant-Souriau量子化映射的重言式调谐

Tautological Tuning of the Kostant-Souriau Quantization Map with Differential Geometric Structures

针对规范量子化的特设性及现有几何/形变量子化的局限，本文提出仅用辛几何与黎曼几何微分结构（如重言式1-形式）的重言式调谐量子化，更贴近规范量子化且适用于协变哈密顿场论。

Notes on phase space quantization

We consider questions related to a quantization scheme in which a classical variable f:Ω\to R on a phase space Ωis associated with a semispectral measure E^f, such that the moment operators of E^f are required to be of the form Γ(f^k), with Γa suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in some Hilbert space. In particular, we investigate the situation where the map Γis implemented by the operator integral with respect to some fixed positive operator measure. The phase space Ωis first taken to be an abstract measurable space, then a lo…

Extended covariance under nonlinear canonical transformation in Weyl quantization

A theory of nonunitary-invertible as well as unitary canonical transformations is formulated in the context of Weyl's phase space representations. Exact solutions of the transformation kernels and the phase space propagators are given for the three fundamental canonical maps as fractional-linear, gauge and contact (point) transformations. Under the nonlinear maps a phase space representation is mapped to another phase space representation thereby extending the standard concept of covariance. This extended covariance allows Dirac-Jordan transformation theory to naturally emerge from the Hilbert…

Quantization and noiseless measurements

In accordance with the fact that quantum measurements are described in terms of positive operator measures (POMs), we consider certain aspects of a quantization scheme in which a classical variable $f:\R^2\to \R$ is associated with a unique positive operator measure (POM) $E^f$, which is not necessarily projection valued. The motivation for such a scheme comes from the well-known fact that due to the noise in a quantum measurement, the resulting outcome distribution is given by a POM and cannot, in general, be described in terms of a traditional observable, a selfadjoint operator. Accordingly,…

Quantization of Non-Hamiltonian Systems

In this paper a generalization of Weyl quantization which maps a dynamical operator in a function space to a dynamical superoperator in an operator space is suggested. Quantization of dynamical operator, which cannot be represented as Poisson bracket with some function, is considered. The usual Weyl quantization of observables can be derived as a specific case of suggested quantization if dynamical operator is an operator of multiplication on a function. This approach allows to define consistent Weyl quantization of non-Hamiltonian and dissipative systems. Examples of the harmonic oscillator w…

伪厄米系统的量子化

Quantization of pseudo-hermitian systems

本研究将现有成果推广至任意维Grassmann代数，提出面向非厄米量子力学伪经典理论的协变量子化方案，可实现正则关联伪经典理论到任意维等价量子实现的映射，并应用于带Heisenberg相互作用的双耦合自旋问题。

Remarks on the geometric quantization of a class of harmonic oscillator type potentials

The conditions that must be fulfilled by a certain physical system to apply geometric quantization prescription on it are investigated. These terms are sought as mathematical requirements, which can be traced in an analysis of integrable systems, from the perspective of both potential function and Hamiltonian vector field. The answer is found in momentum map critical points. Basically, a certain disposal of points that allow a momentum map $C^*$ isomorphism, of observables, with harmonic oscillator momentum map enforce geometric quantization rules. Following the general theory, two newly prese…

Comment on ``Symplectic quantization, inequivalent quantum theories, and Heisenberg's principle of uncertainty''

In Phys. Rev. A 70, 032104 (2004), M. Montesinos and G. F. Torres del Castillo consider various symplectic structures on the classical phase space of the two-dimensional isotropic harmonic oscillator. Using Dirac's quantization condition, the authors investigate how these alternative symplectic forms affect this system's quantization. They claim that these symplectic structures result in mutually inequivalent quantum theories. In fact, we show here that there exists a unitary map between the two representation spaces so that the various quantizations are equivalent.

Berezin-Toeplitz quantization and Berezin symbols for arbitrary compact Kaehler manifolds

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…

量子化：历史与问题

Quantization: History and Problems

本文梳理了量子化从经典相空间函数到量子算符映射的历史，重点分析狄拉克方案的缺陷、格罗内沃尔德-范霍夫定理的意义，并介绍外尔量子化与几何量子化等替代方案及其与狄拉克纲领的关系。

最大混沌量子映射中的谱异常与对称性破缺

Spectral anomalies and broken symmetries in maximally chaotic quantum maps

本文研究最大混沌量子映射中的谱统计，发现短程统计可能误判宏观对称性，而类Berry相位会破坏SFF长程统计与对称性的对应，并验证谱刚性与量子遍历性相关。

Geometric (pre)quantization in the polysymplectic approach to field theory

The prequantization map for a Poisson-Gerstenhaber algebra of dynamical variables represented by differential forms within the polysymplectic formulation of the De Donder--Weyl covariant Hamiltonian field theory is presented and the corresponding prequantum Schroedinger equation for a non-homogeneous form valued wave function is derived. This is the first step toward understanding the procedures of covariant precanonical field quantization from the point of view of geometric quantization.

Quantization of non-Hamiltonian and Dissipative Systems

A generalization of canonical quantization which maps a dynamical operator to a dynamical superoperator is suggested. Weyl quantization of dynamical operator, which cannot be represented as Poisson bracket with some function, is considered. The usual Weyl quantization of observables is a specific case of suggested quantization. This approach allows to define consistent quantization procedure for non-Hamiltonian and dissipative systems. Examples of the harmonic oscillator with friction (generalized Lorenz-Rossler-Leipnik-Newton equation), the Fokker-Planck-type system and Lorenz-type system are…

Geometric quantization and the generalized Segal-Bargmann transform for Lie groups of compact type

Let K be a connected Lie group of compact type and let T*(K) be its cotangent bundle. This paper considers geometric quantization of T*(K), first using the vertical polarization and then using a natural Kahler polarization obtained by identifying T*(K) with the complexified group K_C. The first main result is that the Hilbert space obtained by using the Kahler polarization is naturally identifiable with the generalized Segal-Bargmann space introduced by the author from a different point of view, namely that of heat kernels. The second main result is that the pairing map of geometric quantizati…

Unitarity in "quantization commutes with reduction"

Let M be a compact Kahler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M//G. Guillemin and Sternberg [Invent. Math. 67 (1982), 515--538] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M//G and the G-invariant subspace of the quantum Hilbert space over M. We prove that in general the natural map of Guillemin and Sternberg is not unitary, \textit{even to leading order in Planck's constant}. We then modify the quan…

复合费米子态的反交换化用于核反应的首次量化映射量子模拟

Antisymmetrization of composite fermionic states for quantum simulations of nuclear reactions in first-quantization mapping

本文提出了一种首次量化的确定性算法，用于反交换核反应中复合费米子态，通过高效构造完全反称波函数，扩展了量子算法的应用范围。

单层石墨烯中通过电压驱动电荷密度的狄拉克动力学量子化映射：克莱因悖论与熵控波矢力学研究

Quantization Mapping on Dirac Dynamics via Voltage-Driven Charge Density in Monolayer Graphene: A Klein Paradox and Entropy-Ruled Wavevector Mechanics Study

本研究通过电压驱动电荷密度的经验关系，利用微分熵控波矢力学对狄拉克材料进行能量量子化映射，揭示了克莱因悖论下的电子动力学，并发现量子态跃迁遵循N(k)=N(U)^3关系。

Cahill-Glauber对应中乘积的正则量子化

Canonical quantization of a product within the Cahill-Glauber correspondence

提出Cahill-Glauber s参数化框架下乘积量子化的微分形式，称为“hatted star product”，其特例为Moyal star product。

Quantum Dynamical Entropies and Complexity in Dynamical Systems

We analyze the behaviour of two quantum dynamical entropies in connection with the classical limit. Using strongly chaotic classical dynamical systems as models (Arnold Cat Maps and Sawtooth Maps), we also propose a discretization procedure that resembles quantization; even in this case, studies of quantum dynamical entropy production are carried out and the connection with the continuous limit is explored. In both case (quantization and discretization) the entropy production converge to the Kolmogorov-Sinai invariant on time-scales that are logarithmic in the quantization (discretization) par…