Narrow Results

A concise derivation of all uncertainty relations is given entirely within the context of phase-space quantization, without recourse to operator methods, to the direct use of Weyl's correspondence, or to marginal distributions of x and p.

A unifying perspective on the Moyal and Voros products and their physical meanings has been recently presented in the literature, where the Voros formulation admits a consistent physical interpretation. We define a star product ⋆, in terms of an antisymmetric fixed matrix Θ, and an arbitrary symmetric matrix Φ, that is a generalization of the Moyal and the Voros products. We discuss the quantum mechanics and the physical meaning of the generalized star product.

We derive an explicit expression for an associative star product on noncommutative versions of complex Grassmannian spaces, in particular for the case of complex two-planes. Our expression is in terms of a finite sum of derivatives. This generalizes previous results for complex projective spaces and gives a discrete approximation for the Grassmannians in terms of a noncommutative algebra, represented by matrix multiplication in a finite-dimensional matrix algebra. The matrices are restricted to have a dimension which is precisely determined by the harmonic expansion of functions on the commutative Grassmannian, truncated at a finite level. In the limit of infinite-dimensional matrices we recover the commutative algebra of functions on the complex Grassmannians.

We investigate the properties of κ-Minkowski space–time by using representations of the corresponding deformed algebra in terms of undeformed Heisenberg–Weyl algebra. The deformed algebra consists of κ-Poincaré algebra extended with the generators of the deformed Weyl algebra. The part of deformed algebra, generated by rotation, boost and momentum generators, is described by the Hopf algebra structure. The approach used in our considerations is completely Lorentz covariant. We further use an advantage of this approach to consistently construct a star product, which has a property that under integration sign, it can be replaced by a standard pointwise multiplication, a property that was since known to hold for Moyal but not for κ-Minkowski space–time. This star product also has generalized trace and cyclic properties, and the construction alone is accomplished by considering a classical Dirac operator representation of deformed algebra and requiring it to be Hermitian. We find that the obtained star product is not translationally invariant, leading to a conclusion that the classical Dirac operator representation is the one where translation invariance cannot simultaneously be implemented along with hermiticity. However, due to the integral property satisfied by the star product, noncommutative free scalar field theory does not have a problem with translation symmetry breaking and can be shown to reduce to an ordinary free scalar field theory without nonlocal features and tachyonic modes and basically of the very same form. The issue of Lorentz invariance of the theory is also discussed.

Generalized (f)-coherent state approach in deformation quantization framework is investigated by using a *-eigenvalue equation. For this purpose we introduce a new Moyal star product called f-star product, so that by using this *_f-eigenvalue equation one can obtain exactly the spectrum of a general Hamiltonian of a deformed system. Eventually the method is supported with some examples.

General realizations, star products and plane waves for κ-Minkowski space–time are considered. Systematic construction of general Hermitian realization is presented, with special emphasis on noncommutative plane waves and Hermitian star product. Few examples are elaborated and possible physical applications are mentioned.

Four formulations of quantum mechanics on noncommutative Moyal phase spaces are reviewed. These are the canonical, path-integral, Weyl–Wigner and systematic formulations. Although all these formulations represent quantum mechanics on a phase space with the same deformed Heisenberg algebra, there are mathematical and conceptual differences which we discuss.

By using the homological notion, we analyze the vertex and edge shift matrices in symbolic dynamics. The former is the mathematical basis of the general star product which is transformed into a star direct product of vertex shift matrices, the latter a basis of the calculation of topological entropy. We show that in a general case the first entropy invariant holds, but the second one is broken.

Within unfolded dynamics approach, we represent actions and conserved charges as elements of cohomology of the L_∞ algebra underlying the unfolded formulation of a given dynamical system. The unfolded off-shell constraints for symmetric fields of all spins in Minkowski space are shown to have the form of zero curvature and covariant constancy conditions for 1-forms and 0-forms taking values in an appropriate star product algebra. Unfolded formulation of Yang–Mills and Einstein equations is presented in a closed form.

It was shown in [1], only for scalar conformal fields, that the Moyal–Weyl star product can introduce the quantum effect as the phase factor to the ordinary product.

In this paper we show that, even on the same complex structure, the Moyal–Weyl star product of two j-differentials (conformal fields of weights (j, 0)) does not vanish but it generates the quantum effect at the first order of its perturbative series.

More generally, we get the explicit expression of the Moyal–Weyl star product of j-differentials defined on any complex structure of a bi-dimensional Riemann surface Σ. We show that the star product of two j-differentials is not a j-differential and does not preserve the conformal covariance character.

This can shed some light on the Moyal–Weyl deformation quantization procedure connection's with the deformation of complex structures on a Riemann surface. Hence, the situation might relate the star products to the Moduli and Teichmüller spaces of Riemann surfaces.

We present a brief outline of recent and new results on the mathematical structure underlying the κ-deformed space. We suggest to turn attention to the observable C*-algebra of κ-deformed coordinates and its Galilean symmetries.

We describe a natural construction of deformation quantization on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.

In this paper, we present a quantization of the functions of spacetime, i.e. a map, analog to Weyl map, which reproduces the $\kappa$-Minkowski commutation relations, and it has the desirable properties of mapping square integrable functions into Hilbert–Schmidt operators, as well as real functions into symmetric operators. The map is based on Mellin transform on radial and time coordinates. The map also defines a deformed ∗ product which we discuss with examples.

We derive an expression for the k-Minkowski star product in d space-time dimensions as a symplectic reduction of a normal ordered star product of the Wick type on the tangent bundle T^*ℝ^d.

After a brief introduction to the concept of formal Deformation Quantisation, we shall focus on general konwn constructions of star products, enhancing links with linear connections.

We first consider the symplectic context: we recall how any natural star product on a symplectic manifold determines a unique symplectic connection and we recall Fedosov's construction which yields a star product, given a symplectic connection.

In the more general context, we consider universal star products, which are defined by bidifferential operators expressed by universal formulas for any choice of a linear torsionfree connection and of a Poisson structure. We recall how formality implies the existence (and classification) of star products on a Poisson manifold. We present Kontsevich formality on ℝ^d and we recall how Cattaneo-Felder-Tomassini globalisation of this result proves the existence of a universal star product.