Related papers: Finite-Difference Equations in Relativistic Quantu…
Let $G$ be a countable discrete group with an orthogonal representation $\alpha$ on a real Hilbert space $H$. We prove $L_p$ Poincar\'e inequalities for the group measure space $L_\infty(\Omega_H,\gamma)\rtimes G$, where both the group…
The class of accelerated and rotating reference frames has been studied on the basis of generalized Fermi-Walker coordinates. We obtain the infinitesimal transformations connecting any two of these frames and also their commutation…
A great number of problems of relativistic position in quantum mechanics are due to the use of coordinates which are not inherent objects of spacetime, cause unnecessary complications and can lead to misconceptions. We apply a…
We introduce two information-theoretical invariants for the projective unitary group acting on a finite-dimensional complex Hilbert space: PVM- and POVM-dynamical (quantum) entropies. They quantify the randomness of the successive quantum…
General relativity has two fundamental problems that render it unsuitable for tackling the gravitational field's quantization. The first problem is the lack of a genuine gravitational variable representing gravitation only, inertial forces…
A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A corresponding framework has been developed by Woronowicz, more…
We study several classes of indecomposable representations of quivers on infinite-dimensional Hilbert spaces and their relation. Many examples are constructed using strongly irreducible operators. Some problems in operator theory are…
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a…
Quantum-mechanical concepts can be formulated in constructive finite terms without loss of their empirical content if we replace a general unitary group by a unitary representation of a finite group. Any linear representation of a finite…
The extensive analysis of the dynamics of relativistic spinning particles is presented. Using the coadjoint orbits method the Hamiltonian dynamics is explicitly described. The main technical tool is the factorization of general Lorentz…
Because of the importance of unitarity in quantum physics, work on the representations of the de Sitter group has focussed on the unitary case, which necessarily means infinite dimensional matrices for this non-compact group. Here we…
The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combinining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained…
We investigate a quantum geometric space in the context of what could be considered an emerging effective theory from Quantum Gravity. Specifically we consider a two-parameter class of twisted Poincar\'e algebras, from which Lie-algebraic…
We present in the article the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg-Weyl symmetry with position and momentum operators transforming as Minkowski…
The uncertainty relation of three quantities in quantum mechanics is estimated in terms of commutators. The Pauli matrices are used to find a contribution of third-order commutators. The resulting inequality refines the Heisenberg…
The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is…
We propose a method for deforming an extended Galilei algebra that leads to a nonstandard realization of the Poincar\'e group with the Fock-Lorentz linear fractional transformations. The invariant parameter in these transformations has the…
Born proposed a unification of special relativity and quantum mechanics that placed position, time, energy and momentum on equal footing through a reciprocity principle and extended the usual position-time and energy-momentum line elements…
Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary…
In this paper the relativistic quantum mechanics is considered in the framework of the nonstandard synchronization scheme for clocks. Such a synchronization preserves Poincar{\'e} covariance but (at least formally) distinguishes an inertial…