Related papers: Separation of noncommutative differential calculus…
The central structure in various versions of noncommutative geometry is a differential calculus on an associative algebra. This is an analogue of the calculus of differential forms on a manifold. In this short review we collect examples of…
It is shown that quantum mechanics on noncommutative spaces (NQM) can be obtained by the canonical quantization of some underlying second class constrained system formulated in extended configuration space. It leads, in particular, to an…
We develop a construction of the unitary type anti-involution for the quantized differential calculus over $GL_q(n)$ in the case $|q|=1$. To this end, we consider a joint associative algebra of quantized functions, differential forms and…
A method has been recently proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example a version of quantized space-time is considered here. It is found that there is a…
We introduce a construction of the differential calculus on the quantum supergroup GL$_{p,q}(1| 1)$. We obtain two differential calculi, respectively, associated with the left and right Cartan-Maurer one-forms. We also obtain the quantum…
There ought to exist a reformulation of quantum mechanics which does not refer to an external classical spacetime manifold. Such a reformulation can be achieved using the language of noncommutative differential geometry. A consequence which…
An exterior derivative, inner derivation, and Lie derivative are introduced on the quantum group $GL_{q}(N)$. $SL_{q}(N)$ is then found by constructing matrices with determinant unity, and the induced calculus is found.
We propose a natural family of higher-order partial differential equations generalizing the second-order Klein-Gordon equation. We characterize the associated model by means of a generalized action for a scalar field, containing…
In our previous publications we have introduced analogs of partial derivatives on the algebras U(gl(N)). In the present paper we compare two methods of introducing these analogs: via the so-called quantum doubles and by means of a…
The multiplicative (or geometric) calculus is a non-Newtonian calculus derived from an arithmetic in which the operations of addition/subtraction/multiplication are replaced by multiplication/division/exponentiation. A major difference…
Recent studies show that deformations in quantum mechanics are inevitable. In this contribution, we consider a relativistic quantum mechanical differential equation in the presence of Dunkl operator-based deformation and we investigate…
We propose two families of differential algebras of classical dimension on kappa-Minkowski space. The algebras are constructed using realizations of the generators as formal power series in a Weyl super-algebra. We also propose a novel…
This paper shows how the forms of gauge theory, Hamiltonian mechanics and quantum mechanics arise from a non-commutative framework for calculus and differential geometry. Discrete calculus is seen to fit into this pattern by reformulating…
We study covariant differential calculus on the quantum spheres S_q^2N-1. Two classification results for covariant first order differential calculi are proved. As an important step towards a description of the noncommutative geometry of the…
Explicit construction of the second order left differential calculi on the quantum group and its subgroups are obtained with the property of the natural reduction: the differential calculus on the quantum group $GL_q(2,C)$ has to contain…
We define a noncommutative differential calculus constructed from the inner derivation, then several relevant examples are showed. It is of interest to note that for certain $C^*$-algebra, this calculus is closely related to the classical…
The non-commutative differential calculus on quantum groups can be extended by introducing, in analogy with the classical case, inner product operators and Lie derivatives. For the case of $\GL$ we show how this extended calculus induces by…
Quantum field theories on noncommutative Minkowski space are studied in a model-independent setting by treating the noncommutativity as a deformation of quantum field theories on commutative space. Starting from an arbitrary Wightman…
We express the defining relations of the $q$-deformed Minkowski space algebra as well as that of the corresponding derivatives and differentials in the form of reflection equations. This formulation encompasses the covariance properties…
QR factorisation plays an important role in matrix computations. Within the context of optimisation and of automatic differentiation of such computations, we need to compute the derivative of this factorisation. For tall matrices, however,…