Related papers: The Geometry of a $q$-Deformed Phase Space
We prove that any complete metric space has a unique decomposition as a direct product of a possibly finite or zero-dimensional Hilbert space and a space that does not split off lines.
A q-deformed version of classical analysis is given to quantum spaces of physical importance, i.e. Manin plane, q-deformed Euclidean space in three or four dimensions, and q-deformed Minkowski space. The subject is presented in a rather…
On logarithmic paper some real algebraic curves look like smoothed broken lines. Moreover, the broken lines can be obtained as limits of those curves. The corresponding deformation can be viewed as a quantization, in which the broken line…
The introduction of a metric onto the space of parameters in models in Statistical Mechanics and beyond gives an alternative perspective on their phase structure. In such a geometrization, the scalar curvature, R, plays a central role. A…
The algebra of diffeomorphisms derived from general coordinate transformations on commuting coordinates is represented by differential operators on noncommutative spaces. The algebra remains unchanged, the comultiplication however is…
This paper presents an alternative approach to geometric phases from the observable point of view. Precisely, we introduce the notion of observable-geometric phases, which is defined as a sequence of phases associated with a complete set of…
Examples are given of q-deformed systems that may be interpreted by the standard rules of quantum theory in terms of new degrees of freedom and supplementary quantum numbers.
We present an adaptive geometry in which the yardstick co-deforms with space itself, formulated on cellular spaces where length is a count: distances are shortest cell-crossing counts. No cell shape, angles, or embedding are assumed; the…
Many basis sets for electronic structure calculations evolve with varying external parameters, such as moving atoms in dynamic simulations, giving rise to extra derivative terms in the dynamical equations. Here we revisit these derivatives…
We introduce the analogue of the metric tensor in case of $q$-deformed differential calculus. We analyse the consequences of the existence of such metric, showing that this enforces severe restrictions on the parameters of the theory. We…
Fractional calculus and q-deformed Lie algebras are closely related. Both concepts expand the scope of standard Lie algebras to describe generalized symmetries. For the fractional harmonic oscillator, the corresponding q-number is derived.…
We present a simple geometric construction linking geometric to deformation quantization. Both theories depend on some apparently arbitrary parameters, most importantly a polarization and a symplectic connection, and for real polarizations…
This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.
We apply De Haro's Geometric View of Theories to one of the simplest quantum systems: a spinless particle on a line and on a circle. The classical phase space M = T*Q is taken as the base of a trivial Hilbert bundle E ~ M x H, and the…
We present a systematic introduction to the geometry of linear braided spaces. These are versions of $\R^n$ in which the coordinates $x_i$ have braid-statistics described by an R-matrix. From this starting point we survey the author's…
We present in this paper quantum real lines as quantum defomations of the real numbers $\R$.Upon deforming the Heisenberg algebra $\cL$ generated by $(a, a^\dagger)$ in terms of the Moyal $\ast$-product,we first construct q-deformed…
One of the most celebrated accomplishments of modern physics is the description of fundamental principles of nature in the language of geometry. As the motion of celestial bodies is governed by the geometry of spacetime, the motion of…
We study the quantization of geometry in the presence of a cosmological constant, using a discretiza- tion with constant-curvature simplices. Phase space turns out to be compact and the Hilbert space finite dimensional for each link. Not…
Some possible applications of deformed algebras to Quantum Physics are considered based on a rigorous approach. Jackson integrals are expressed in the context of the equipped separable Hilbert space. Jackson integrals are expressed in the…
In this work, we construct the algebra of differential forms with the cube of exterior differential equal to zero on one-dimensional space. We prove that this algebra is a graded q-differential algebra where q is a cubic root of unity.…