Quantization and Dynamisation of Trace-Poisson Brackets

The quantization problem for the trace-bracket algebra, derived from double Poisson brackets, is discussed. We obtain a generalization of the boundary YBE (or so-called ABCD-algebra) for the quantization of quadratic trace-brackets. A dynamical deformation is proposed on the lines of Gervais–Neveu–Felder dynamical quantum algebras.     

Quantum deformations of the Heisenberg group obtained by geometric quantization

All multiplicative Poisson brackets on the Heisenberg group are classified and Manin groups [14] corresponding to a wide class of those brackets are constructed. A geometric quantization procedure is applied to the resulting symplectic pseudogroups yielding a wide class of pre-C^*-algebras with comultiplication, counit and coinverse, which provide quantum deformations of the Heisenberg group.     

Unambiguous Quantization from the Maximum Classical Correspondence that Is Self-consistent: The Slightly Stronger Canonical Commutation Rule Dirac Missed

Dirac’s identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac’s canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables, but does imply the quantum/classical correspondence of Poisson brackets between any linear function of phase space and the sum of an arbitrary function of only configuration space with one of only momentum space. Since every linear function of phase space is itself such a sum, it is worth checking whether the assumption of quantum/classical correspondence of Poisson brackets for all such sums is still self-consistent. Not only is that so, but this slightly stronger canonical commutation rule also unambiguously determines the order of noncommuting factors within quantized dynamical variables in accord with the 1925 Born-Jordan quantization surmise, thus replicating the results of the Hamiltonian path integral, a fact first realized by E.H. Kerner. Born-Jordan quantization validates the generalized Ehrenfest theorem, but has no inverse, which disallows the disturbing features of the poorly physically motivated invertible Weyl quantization, i.e., its unique deterministic classical “shadow world” which can manifest negative densities in phase space.     

Double Quantization on Some Orbits in the Coadjoint Representations of Simple Lie Groups

Let ? be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, g, with the Lie algebra ?. We study one and two parameter quantizations ? _h and ? _t,h of ? such that the multiplication on the quantized algebra is invariant under action of the Drinfeld–Jimbo quantum group, U _{h (?)}. In particular, the algebra ? _t,h specializes at h= 0 to a U(?)-invariant ($G$-invariant) quantization, %Ascr; _{t ,0}. We prove that the Poisson bracket corresponding to ? _h must be the sum of the so-called r-matrix and an invariant bracket. We classify such brackets for all semisimple orbits, M, and show that they form a dim H ²(M) parameter family, then we construct their quantizations. A two parameter (or double) quantization, $? _t,h , corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on M. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases. Received: 15 August 1998 / Accepted: 13 January 1999     

De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory

A quantization of field theory based on the De Donder-Weyl (DW) covariant Hamiltonian formulation is discussed. A hypercomplex extension of quantum mechanics, in which the space-time Clifford algebra replaces that of the complex numbers, appears as a result of quantization of Poisson brackets on differential forms which were put forward for the DW theory earlier. The proposed covariant hypercomplex Schrödinger equation is shown to lead in the classical limit to the DW Hamilton-Jacobi equation and to obey the Ehrenfest principle in the sense that the DW canonical field equations are satisfied for expectation values of properly chosen operators.     

BRST analysis of general mechanical systems

We study the groups of local BRST cohomology associated to the general systems of ordinary differential equations, not necessarily Lagrangian or Hamiltonian. Starting with the involutive normal form of the equations, we explicitly compute certain cohomology groups having clear physical meaning. These include the groups of global symmetries, conservation laws and Lagrange structures. It is shown that the space of integrable Lagrange structures is naturally isomorphic to the space of weak Poisson brackets. The last fact allows one to establish a direct link between the path-integral quantization of general not necessarily variational dynamics by means of Lagrange structures and the deformation quantization of weak Poisson brackets.     

Proper Dirac quantization of a free particle on a D-dimensional sphere

We show that an unambiguous and correct quantization of the second-class constrained system of a free particle on a sphere in D dimensions is possible only by converting the constraints to Abelian gauge constraints, which are of first class in Dirac's classification scheme. The energy spectrum is equal to that of a pure Laplace-Beltrami operator with no additional constant arising from the curvature of the sphere. A quantization of Dirac's modified Poisson brackets for second-class constraints is also possible and unique, but must be rejected since the resulting energy spectrum is physically incorrect.     

Deformation Quantization of Algebraic Varieties

The paper is devoted to peculiarities of the deformation quantization in the algebro-geometric context. A direct application of the formality theorem to an algebraic Poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. The global category is very degenerate in general. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry of an actual deformation (versus a formal one). Deformed algebras obtained by semiformal deformations are Noetherian and have polynomial growth. We propose constructions of semiformal quantizations of projective and affine algebraic Poisson manifolds satisfying certain natural geometric conditions. Projective symplectic manifolds (e.g. K3 surfaces and Abelian varieties) do not satisfy our conditions, but projective spaces with quadratic Poisson brackets and Poisson–Lie groups can be semiformally quantized.     

On the Gauge and BRST Invariance of the Chiral QED with Faddeevian Anomaly

Chiral Schwinger model with the Faddeevian anomaly is considered. It is found that imposing a chiral constraint this model can be expressed in terms of chiral boson. The model when expressed in terms of chiral boson remains anomalous and the Gauss law of which gives anomalous Poisson brackets between itself. In spite of that a systematic BRST quantization is possible. The Wess-Zumino term corresponding to this theory appears automatically during the process of quantization. A gauge invariant reformulation of this model is also constructed. Unlike the former one gauge invariance is done here without any extension of phase space. This gauge invariant version maps onto the vector Schwinger model. The gauge invariant version of the chiral Schwinger model for a=2 has a massive field with identical mass however gauge invariant version obtained here does not map on to that.     

Lagrangian and Hamiltonian formalisms for relativistic dynamics of a charged particle with dipole moment

The Lagrangian and Hamiltonian formulations for the relativistic classical dynamics of a charged particle with dipole moment in the presence of an electromagnetic field are given. The differential conservation laws for the energy-momentum and angular momentum tensors of a field and particle are discussed. The Poisson brackets for basic dynamic variables, which form a closed algebra, are found. These Poisson brackets enable us to perform the canonical quantization of the Hamiltonian equations that leads to the Dirac wave equation in the case of spin 1/2. It is also shown that the classical limit of the squared Dirac equation results in equations of motion for a charged particle with dipole moment obtained from the Lagrangian formulation. The inclusion of gravitational field and non-Abelian gauge fields into the proposed formalism is discussed.Received: 4 June 2005, Published online: 27 July 2005     

Singular Lagrangian for the polaron

A Lagrangian which describes the electron-phonon interaction is proposed, starting from a simple model of interacting string and charged particle. The Lagrangian is singular, containing both bosonic variables (i.e., phonons) and fermionic ones (i.e., the electron). Symmetry Dirac brackets are used to obtain the BCS Hamiltonian. The addition of a total derivative of a scalar function to the Lagrangian density doesnot alter the quantization procedure.     

Quantum orbits of theR-matrix type

Given a simple Lie algebra g, we consider the orbits in g^* which are of theR-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-calledR-matrix bracket. We call an algebra quantizing the latter bracket a quantum orbit of theR-matrix type. We describe some orbits of this type explicitly and we construct a quantization of the whole Poisson pencil on these orbits in a similar way. The notions ofq-deformed Lie brackets, braided coadjoint vector fields, and tangent vector fields are discussed as well.     

Poisson brackets and clebsch representations for magnetohydrodynamics,multifluid plasmas,and elasticity

Poisson brackets are constructed by the same mathematical procedure for three physical theories: ideal magnetohydrodynamics, multifluid plasmas, and elasticity. Each of these brackets is given a simple Lie-algebraic interpretation. Moreover, each bracket is induced to physical space by use of a canonical Poisson bracket in the space of Clebsch potentials, which are constructed for each physical theory by the standard procedure of constrained Lagrangians.     

Symplectic Geometry of Quantum Noise

We discuss a quantum counterpart, in the sense of the Berezin–Toeplitz quantization, of certain constraints on Poisson brackets coming from “hard” symplectic geometry. It turns out that they can be interpreted in terms of the quantum noise of observables and their joint measurements in operational quantum mechanics. Our findings include various geometric mechanisms of quantum noise production and a noise-localization uncertainty relation. The methods involve Floer theory and Poisson bracket invariants originated in function theory on symplectic manifolds.     

A consistent formulation of the null-plane quantum field theory

It is shown in the usual framework of quantum field theory that the null-plane restriction of a field operator is not a well-defined operator-valued distribution. The cause of the trouble is the so-called P⁺ = 0 mode; in order to make it harmless, it is almost inevitable to violate Lorentz invariance. A consistent formulation of the null-plane quantization, which is supposed to be the simplest possible one, is proposed by modifying the definition of Poisson brackets. This theory is invariant under a Poincaré subalgebra containing seven generators. It is also shown that the absence of vacuum polarization is realized consistently in this formalism.     

Hamiltonian Formulation and Action Principle for the Lorentz-Dirac System

The possibility of constructing a Lagrangian and Hamiltonian formulation is examined for a radiating point-like charge usually described by the classical Lorentz-Dirac equation. It turns out that the latter equation cannot be obtained from the variational principle, and, furthermore, has nonphysical solutions. It is proposed to consider a physically equivalent set of reduced equations which admit a Hamiltonian formulation with non-canonical Poisson brackets. As an example, the effective dynamics of a non-relativistic particle moving in a homogeneous magnetic field is considered. The proposed Hamiltonian formulation may be considered as a first step to a consistent quantization of the Lorentz-Dirac system.     

Superconformal algebras and their relation to integrable nonlinear systems

We relate the manifold of periodic functions on a circle with values in the Grassmann algebra to extended superconformal algebras. The graded Poisson brackets of these functions give the classical realization of the corresponding superconformal algebras and determine the hamiltonian structure for a class of integrable nonlinear equations. A super-generalization of the Korteweg-de Vries equation is found among these equations. In this way an important step in the program of the quantization of the Liouville equation is realized for the supersymmetric cases which are crucial in constructing a consistent quantum string theory. The construction of Miura transformations is outlined and the results for the N = 1,2 supersymmetric cases are presented.     

Weakly nonlinear dynamics in noncanonical Hamiltonian systems with applications to fluids and plasmas

A method, called beatification, is presented for rapidly extracting weakly nonlinear Hamiltonian systems that describe the dynamics near equilibria of systems possessing Hamiltonian form in terms of noncanonical Poisson brackets. The procedure applies to systems like fluids and plasmas in terms of Eulerian variables that have such noncanonical Poisson brackets, i.e., brackets with nonstandard and possibly degenerate form. A collection of examples of both finite and infinite dimensions is presented.     

Noncommutative Differential Forms and Quantization of the Odd Symplectic Category

There is a simple and natural quantization of differential forms on odd Poisson supermanifolds, given by the relation [f,dg]={f,g} for all functions f and g. We notice that this noncommutative differential algebra has a geometrical realization as a convolution algebra of the symplectic groupoid integrating the Poisson manifold. This quantization is just a part of a quantization of the odd symplectic category (where objects are odd symplectic supermanifolds and morphisms are Lagrangian relations) in terms of ₂-graded chain complexes. It is a straightforward consequence of the theory of BV operator acting on semidensities, due to H. Khudaverdian.