From Classical to Quantum Mechanics: "How to translate physical ideas into mathematical language"

In this paper, we investigate the connection between Classical and Quantum Mechanics by dividing Quantum Theory in two parts: - General Quantum Axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoint operators and so on) - Quantum Mechanics properly that specifies the Hilbert space, the Heisenberg rule, the free Hamiltonian... We show that General Quantum Axiomatics (up to a supplementary "axiom of classicity") can be used as a non-standard mathematical ground to formulate all the ideas and equations of ordinary Classical Statistical Mechanics. So the question of a "true quantization" with "h" must be seen as an independent problem not directly related with quantum formalism. Moreover, this non-standard formulation of Classical Mechanics exhibits a new kind of operation with no classical counterpart: this operation is related to the "quantization process", and we show why quantization physically depends on group theory (Galileo group). This analytical procedure of quantization replaces the "correspondence principle" (or canonical quantization) and allows to map Classical Mechanics into Quantum Mechanics, giving all operators of Quantum Mechanics and Schrodinger equation. Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We find also that this approach gives a natural semi-classical formalism: some exact quantum results are obtained only using classical-like formula. So this procedure has the nice property of enlightening in a more comprehensible way both logical and analytical connection between classical and quantum pictures.Comment: 47 page

On Foundation of the Generalized Nambu Mechanics

We outline the basic principles of canonical formalism for the Nambu mechanics---a generalization of Hamiltonian mechanics proposed by Yoichiro Nambu in 1973. It is based on the notion of Nambu bracket which generalizes the Poisson bracket to the multiple operation of higher order $n \geq 3$ on classical observables and is described by Hambu-Hamilton equations of motion given by $n-1$ Hamiltonians. We introduce the fundamental identity for the Nambu bracket which replaces Jacobi identity as a consistency condition for the dynamics. We show that Nambu structure of given order defines a family of subordinated structures of lower order, including the Poisson structure, satisfying certain matching conditions. We introduce analogs of action from and principle of the least action for the Nambu mechanics and show how dynamics of loops ($n-2$-dimensional objects) naturally appears in this formalism. We discuss several approaches to the quantization problem and present explicit representation of Nambu-Heisenberg commutation relation for $n=3$ case. We emphasize the role higher order algebraic operations and mathematical structures related with them play in passing from Hamilton's to Nambu's dynamical picture.Comment: 27 page

Landau Diamagnetism in Noncommutative Space and the Nonextensive Thermodynamics of Tsallis

We consider the behavior of electrons in an external uniform magnetic field B where the space coordinates perpendicular to B are taken as noncommuting. This results in a generalization of standard thermodynamics. Calculating the susceptibility, we find that the usual Landau diamagnetism is modified. We also compute the susceptibility according to the nonextensive statistics of Tsallis for (1-q)<<1, in terms of the factorization approach. Two methods agree under certain conditions.Comment: Clarifications and new references. Version to appear in Phys.Lett.

Wigner Trajectory Characteristics in Phase Space and Field Theory

Exact characteristic trajectories are specified for the time-propagating Wigner phase-space distribution function. They are especially simple---indeed, classical---for the quantized simple harmonic oscillator, which serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field theory is thus reformulated in terms of distributions in field phase space. Applications to duality transformations in field theory are discussed.Comment: 9 pages, LaTex2

Weyl-Underhill-Emmrich quantization and the Stratonovich-Weyl quantizer

Weyl-Underhill-Emmrich (WUE) quantization and its generalization are considered. It is shown that an axiomatic definition of the Stratonovich-Weyl (SW) quantizer leads to severe difficulties. Quantization on the cylinder within the WUE formalism is discussed.Comment: 15+1 pages, no figure

A Path Integral Approach To Noncommutative Superspace

A path integral formula for the associative star-product of two superfields is proposed. It is a generalization of the Kontsevich-Cattaneo-Felder's formula for the star-product of functions of bosonic coordinates. The associativity of the star-product imposes certain conditions on the background of our sigma model. For generic background the action is not supersymmetric. The supersymmetry invariance of the action constrains the background and leads to a simple formula for the star-product.Comment: Latex 13 pages. v2: references and footnotes adde

Toeplitz operators on symplectic manifolds

We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz operators in terms of their asymptotic expansion. The semi-classical limit properties of the Berezin-Toeplitz quantization for non-compact manifolds and orbifolds are also established.Comment: 40 page

Perturbative Construction of Models of Algebraic Quantum Field Theory

We review the construction of models of algebraic quantum field theory by renormalized perturbation theory.Comment: 38 page