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.     

Has Quantum Field Theory the Standard Transition from Poisson Brackets?

This paper is devoted to the problem of the validity of claims that commutation rules of quantum field theories have its origin in Poisson brackets of classical mechanics.     

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.     

Nonassociativity,Malcev algebras and string theory

Nonassociative structures have appeared in the study of D‐branes in curved backgrounds. In recent work, string theory backgrounds involving three‐form fluxes, where such structures show up, have been studied in more detail. We point out that under certain assumptions these nonassociative structures coincide with nonassociative Malcev algebras which had appeared in the quantum mechanics of systems with non‐vanishing three‐cocycles, such as a point particle moving in the field of a magnetic charge. We generalize the corresponding Malcev algebras to include electric as well as magnetic charges. These structures find their classical counterpart in the theory of Poisson‐Malcev algebras and their generalizations. We also study their connection to Stueckelberg's generalized Poisson brackets that do not obey the Jacobi identity and point out that nonassociative string theory with a fundamental length corresponds to a realization of his goal to find a non‐linear extension of quantum mechanics with a fundamental length. Similar nonassociative structures are also known to appear in the cubic formulation of closed string field theory in terms of open string fields, leading us to conjecture a natural string‐field theoretic generalization of the AdS/CFT‐like (holographic) duality.     

Poisson brackets in condensed matter physics

A general method of deriving nonlinear equations of hydrodynamics for both normal liquid and superfluid ⁴He and ³He, equations of the elasticity theory, equations for spin waves in magnets and spin glasses, liquid crystals, and so on is described. The method is based on the use of the Poisson “hydrodynamic” brackets. Hydrodynamic brackets are on the one hand, a classical limit of quantum commutators, on the other hand, Poisson brackets of certain symmetry groups inherent in the given problem: groups of general coordinate transformations for hydrodynamics and elasticity theory, groups of local spin rotations for spin waves, etc. Along with well-known examples nonlinear equations of the elasticity theory for bodies with impurities, dislocations and disclinations, and equations of motion for spin glasses and multisublattice magnets are studied.     

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.     

Euler–Poincaré flows and leibniz structure of nonlinear reaction–diffusion type systems

In this paper we present Euler–Poincaré formulation of the Fisher, Fitzhugh–Nagumo, Burgers–Huxley and extended Fitzhugh–Nagumo and extended Burgers–Huxley type nonlinear reaction–diffusion systems. All these flows are related to infinite dimensional almost Poisson manifolds and the corresponding Lie–Poisson structures yield Leibniz brackets, a bracket endowed with both symmetric and skewsymmetric parts. The symmetric part contributes the diffusion part of the ssystem. The properties exhibited by the reaction–diffusion systems defined in this way are in general very different from the standard Hamiltonian mechanics since the dynamics are controlled by the standard Poisson brackets. Moreover, all the nonlinear reaction–diffusion systems under consideration are Euler–Poincaré flows on the dual of Kirillov’s superalgebra associated to the Bott–Virasoro group.     

Covariant Poisson Brackets of Classical Fields

The covariant Poisson brackets of classical fields are defined in terms of the fields covariant canonical variables. These are then consistent with the causality principle and the quantum fields covariant commutation relations.     

Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics and that of classical statistical mechanics, this paper finds that there should be an analogy between geometrical optics and classical statistical mechanics instead of geometrical mechanics and classical mechanics. Furthermore, by comparison between the classical limit of quantum mechanics and classical statistical mechanics, it finds that classical limit of quantum mechanics is classical statistical mechanics not classical mechanics, hence it demonstrates that quantum mechanics is a natural generalization of classical statistical mechanics instead of classical mechanics. Thence quantum mechanics in its true appearance is a wave statistical mechanics instead of a wave mechanics.     

Wave Mechanics or Wave Statistical Mechanics

By comparison between equations of motion of geometrical optics and that of classical statistical mechanics, this paper finds that there should be an analogy between geometrical optics and classical statistical mechanics instead of geometrical mechanics and classical mechanics. Furthermore, by comparison between the classical limit of quantum mechanics and classical statistical mechanics, it finds that classical limit of quantum mechanics is classical statistical mechanics not classical mechanics, hence it demonstrates that quantum mechanics is a natural generalization of classical statistical mechanics instead of classical mechanics. Thence quantum mechanics in its true appearance is a wave statistical mechanics instead of a wave mechanics.     

An operator approach to Poisson brackets

The aim of this paper is to suggest a general approach to Poisson brackets, based on the study of the Lie algebra of potential operators with respect to closed skew-symmetric bilinear forms. This approach allows to extend easily to infinite-dimensional spaces the classical Cartan geometrical approach developed in the phase space. It supplies a simple, unified, and general formalism to deal with such brackets, which contains, as particular cases, the classical and the quantum treatments.     

On non-singular generalised dynamical formalisms

This work shows that a certain class of classical dynamical formalisms, characterised by non-singular Lie structures more general than the usual (Poisson) one, are derivable from ordinary constrained dynamical formalisms. As a consequence, the Lie brackets considered are special cases of suitably chosen Dirac brackets. Both unconstrained and constrained generalised dynamical formalisms are considered. The relations of our results with the problem of constructing classical analogues of generalised quantum systems are stressed.     

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.     

Some stochastic aspects of quantization

From the advent of quantum mechanics, various types of stochastic-dynamical approach to quantum mechanics have been tried. We discuss how to utilize Nelson’s stochastic quantum mechanics to analyze the tunneling phenomena, how to derive relativistic field equations via the Poisson process and how to describe a quantum dynamics of open systems by the use of quantum state diffusion, or the stochastic Schrödinger equation.     

Synthetic Hamiltonian mechanics

This paper deals with some infinitesimal aspects of Hamiltonian mechanics from the standpoint of synthetic differential geometry. Fundamental results concerning Hamiltonian vector fields, Poisson brackets, and momentum mappings are discussed. The significance of the Lie derivative in the synthetic context is also consistently stressed. In particular, the notion of an infinitesimally Euclidean space is introduced, and the Jacobi identity of vector fields with respect to Lie brackets is established naturally for microlinear, infinitesimally Euclidean spaces by using Lie derivatives instead of a highly combinatorial device such as P. Hall's 42-letter identity.     

Simple New Axioms for Quantum Mechanics

The space of pure states of any physical system,classical or quantum, is identified as a Poisson spacewith a transition probability. These two structures areconnected through unitarity. Classical and quantum mechanics are each characterized by asimple axiom on the transition probability p. Unitaritythen determines the Poisson bracket of quantum mechanicsup to a multiplicative constant (identified with Planck's constant).     

Probability operator of a harmonic oscillator

A unique linear rule of constructing quantum operators defined by the probability operator for coordinates and momenta, is considered. is assumed to be a normalized, positive definite operator, establishing a dynamical correspondence between the classical and quantum Poisson brackets. It is shown that such an operator exists in the case of a harmonic oscillator. The principal implications of the suggested rule of constructing the operators of physical quantities are determined, in comparison with the corresponding results of conventional quantum mechanics.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 89–93, October, 1982.