Geometry of 2d spacetime and quantization of particle dynamics

We analyze classical and quantum dynamics of a relativistic particle in 2d spacetimes with constant curvature. We show that global symmetries of spacetime specify the symmetries of physical phase-space and the corresponding quantum theory. To quantize the systems we parametrize the physical phase-space by canonical coordinates. Canonical quantization leads to unitary irreducible representations of SO_↑(2.1) group.     

Stochastic quantization of gauge theories

The equivalence between a scalar quantum field theory in D dimensions and its classical counterpart in D + 2 dimensions which is coupled to an external random source with Gaussian correlations was observed by previous authors. This stochastic quantization is extended to gauge theories. The proof exploits the supersymmetry formalism suggested by Parisi and Sourlas.     

Quantum Superconformal Symmetry in N = 2 Harmonic Superspace

In gauge theories, not all rigid symmetries of the classical action can be maintained manifestly in the quantization procedure, even in the absence of anomalies. If this occurs for an anomaly-free symmetry, the effective action is invariant under a transformation that differs from its classical counterpart by quantum corrections. In this note, we set up a harmonic superspace formalism for computing quantum deformations of superconformal symmetry in the N = 4 supersymmetric Yang–Mills theory.     

Rotational symmetry of classical orbits, arbitrary quantization of angular momentum and the role of the gauge field in two-dimensional space

We study quantum–classical correspondence in terms of the coherent wave functions of a charged particle in two- dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.     

The QCD Hamiltonian and nonlinear quantum mechanics

We study the canonical quantization of SU(N) gauge theory in linear, noncovariant gauges. The canonical formalism is first discussed for the classical theory, with special attention to the features involving nonlinearity and the gauge degrees of freedom. The transition to the quantum theory is then performed for an arbitrary linear gauge, using the covariant quantization rules of nonlinear quantum mechanics. When the quantum Hamiltonian is written in the Weyl-ordered form appropriate for the application of the usual Dyson-Wick perturbative techniques, additional ordering terms appear with respects to the classical Hamiltonian. We discuss the relation of our results to those of previous authors, and the relevance of the ordering terms in field theory.     

The Role of Time in Relational Quantum Theories

We propose a solution to the problem of time for systems with a single global Hamiltonian constraint. Our solution stems from the observation that, for these theories, conventional gauge theory methods fail to capture the full classical dynamics of the system and must therefore be deemed inappropriate. We propose a new strategy for consistently quantizing systems with a relational notion of time that does capture the full classical dynamics of the system and allows for evolution parametrized by an equitable internal clock. This proposal contains the minimal temporal structure necessary to retain the ordering of events required to describe classical evolution. In the context of shape dynamics (an equivalent formulation of general relativity that is locally scale invariant and free of the local problem of time) our proposal can be shown to constitute a natural methodology for describing dynamical evolution in quantum gravity and to lead to a quantum theory analogous to the Dirac quantization of unimodular gravity.     

The quantum vacuum at the foundations of classical electrodynamics

In the classical theory of electromagnetism, the permittivity ε ₀ and the permeability μ ₀ of free space are constants whose magnitudes do not seem to possess any deeper physical meaning. By replacing the free space of classical physics with the quantum notion of the vacuum, we speculate that the values of the aforementioned constants could arise from the polarization and magnetization of virtual pairs in vacuum. A classical dispersion model with parameters determined by quantum and particle physics is employed to estimate their values. We find the correct orders of magnitude. Additionally, our simple assumptions yield an independent estimate for the number of charged elementary particles based on the known values of ε ₀ and μ ₀ and for the volume of a virtual pair. Such an interpretation would provide an intriguing connection between the celebrated theory of classical electromagnetism and the quantum theory in the weak-field limit.     

Quantization of the Sphere with Coherent States

Current views link quantization with dynamics. The reason is that quantum mechanics or quantum field theories address to dynamical systems, i.e., particles or fields. Our point of view here breaks the link between quantization and dynamics: any (classical) physical system can be quantized. Only dynamical systems lead to dynamical quantum theories, which appear to result from the quantization of symplectic structures.     

Quantum dynamics of a massless relativistic string (II)

The massless relativistic free string is studied in the gauge x₀ = τ. It is found that the classical solutions include transverse and longitudinal vibrations. The problem is treated both in the Lagrangian and Hamiltonian formalism. Different ways of quantizing the system are investigated. The path integral quantization leads to a Poincaré invariant quantum theory in any number of dimensions.     

用圈量子引力解除Schwarichild-de Sitter黑洞的时空奇点

通过引入圈量子引力中holonomy基本变量的类比变量和采用相应的量子化方法,对Schwarichild-de Sitter黑洞中心附近的引力场进行量子化.分析和计算了黑洞中心附近的1r和曲率标量的谱分布,得到了它们均存在有限上界的结果.通过求解经典时空奇点r=0附近的量子哈密顿约束方程,给出了黑洞波函数在黑洞中心附近的时间演化行为,得到了该波函数可以通过经典奇点进行量子演化的结果.     

Affine connection in Hilbert space

A concise geometrodynamic approach to quantum theory is introduced, via a quo;quantum connectionQ _µ , which is the affine connection in Hilbert space. It is emphasized that this is the simplest and most natural interpretation of quantum mechanics in general relativity and yet has been largely neglected, so that much work remains to be done on it. The generalized Hilbert space has a simple Hermitian metric, but the precise form ofQ _µ remains to be determined. The quantum connection is matheamtically analogous to the spinor connection, which is discussed here for that reason, although the spinor connection arises in the first quantization, whereasQ _µ geometrizes the second quantization.     

Deformations of compact quantum groups via Rieffel's quantization

It is shown that compact quantum groups containing torus subgroups can be deformed into new compact quantum groups under Rieffel's quantization. This is applied to showing that the two classes of compact quantum groupsK _q ^u andK _q studied by Levendorkii and Soibelman are strict deformation quantization of each other, and that the quantum groupsA _u (m) have many deformations.     

On a possible origin of quantum groups

A Poisson bracket structure having the commutation relations of the quantum group SL _q (2) is quantized by means of the Moyal star-product on C (²), showing that quantum groups are not exactly quantizations, but require a quantization (with another parameter) in the background. The resulting associative algebra is a strongly invariant nonlinear star-product realization of the q-algebra U _q (sl(2)). The principle of strong invariance (the requirement that the star-commutator is star-expressed, up to a phase, by the same function as its classical limit) implies essentially the uniqueness of the commutation relations of U _q (sl(2)).     

Static structure factor of electrons around a heavy positively charged impurity in a one-component quantum rare plasma

An expression for the static structure factor,g ⁺⁻ (r), of electrons at a distancer from an infinitely heavy positively charged particle in a one component quantum rare plasma has been obtained in linear response theory using an appropriate quantum dielectric function of the rare plasma. The expression is a complicated function of the electron plasma frequency, Debye screening length andr, but reduces to that of classical plasma when quantum corrections are neglected. Forr<r _s (2r _s being the mean distance between two electrons), the temperature dependentg ⁺⁻ (r) has larger values in quantum case in comparison to that in classical situation and keeps increasing with decrease inr, more so at low temperatures when de-Broglie wavelength becomes larger and a considerable fraction ofr _s.     

A new method of quantization of classical solutions

The stochastic quantization method of Parisi and Wu is used to derive exact equations for the correlators of quantum fluctuations around the classical solution in the massless φ ⁴ theory. The equations obtained are then solved in the lowest orders of perturbation theory, and the first correction to the free propagator of a quantum fluctuation is calculated. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 5, 381–386 (10 March 1996) Publsihed in English in the original Russian journal. Edited by Steve Torstveit.     

(T * G) t : A toy model for conformal field theory

We study a chiral operator algebra of conformal field theory and quantum deformation of the finite-dimensional Lie group to obtain the definition of (T ^* G) _t and its representation.The closeness of the Ka-Moody algebras, constituting the chiral operator algebra of a typical (and generic) conformal field theory model, namely the WZNW model, and quantum deformation of corresponding finite-dimensional Lie groupG has become more and more evident in recent years [1–5]. This in particular prompts further investigation of the differential geometry of such deformations. The notion of tangent and cotangent bundles is basic in classical differential geometry. It is only natural that the quantum deformations ofTG andT ^* G are to be introduced alongside those forG itself. Physical ideas could be useful for this goal.Indeed, theT ^* G can be interpreted as a phase space for a kind of a top, generalizing the usual top associated withG=SO(3). The classical mechanics is a natural language to describe differential geometry, whereas the usual quantization is nothing but the representation theory.In this paper we put corresponding formulas in such a fashion that their deformation becomes almost evident, given the experience in this domain. As a result we get the definition of (T ^* G) _t and its representation (t is the deformation parameter).To make the exposition most simple and formulas transparent we shall work on an example ofG=sl(2) and present results in such a way that the generalizations become evident. We shall stick to generic complex versions, real and especially compact forms requiring some additional consideration, not all of which are self-evident.This work was supported in part by a grant provided by the Academy of Finland, and the U.S. Department of Energy (DOE) under contract DE-AC02-76ER03069     

An Axiomatization for Quantum Processes to Unifying Quantum and Classical Computing

We establish an axiomatization for quantum processes, which is a quantum generalization of process algebra ACP (Algebra of Communicating Processes). We use the framework of a quantum process configuration 〈p, ϱ〉, but we treat it as two relative independent part: the structural part p and the quantum part ϱ, because the establishment of a sound and complete theory is dependent on the structural properties of the structural part p. We let the quantum part ϱ be the outcomes of execution of p to examine and observe the function of the basic theory of quantum mechanics. We establish not only a strong bisimilarity for quantum processes, but also a weak bisimilarity to model the silent step and abstract internal computations in quantum processes. The relationship between quantum bisimilarity and classical bisimilarity is established, which makes an axiomatization of quantum processes possible. An axiomatization for quantum processes called qACP is designed, which involves not only quantum information, but also classical information and unifies quantum computing and classical computing. qACP can be used easily and widely for verification of most quantum communication protocols.

     

Quantum geometry and quantum mechanics of integrable systems

Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter ħ to produce a new (classical) integrable system. The new tori selected by the ħ-equidistance rule represent the spectrum of the quantum system up to O(ħ ^∞) and are invariant under quantum dynamics in the long-time range O(ħ ^−∞). The quantum diffusion over the deformed tori is described. The analytic apparatus uses quantum action-angle coordinates explicitly constructed by an ħ-deformation of the classical action-angles.