Ambiguities have recently been found in the definition of the partial derivative (in the case of presence of both explicit and implicit dependencies of the function subjected to differentiation). We investigate the possible influence of this subject on quantum mechanics and the classical/quantum field theory. Surprisingly, some commutators of operators of space-time 4-coordinates and those of 4-momenta are not equal to zero.
Notes added in the Abstract: Two iterated limits are not equal each other, in general. Thus, we present an example when the massless limit of the function of E, p, m does not exist in some calculations within quantum field theory.
This article concludes our critical analysis on the role of non-commutativity in quantum theory. After a brief introduction of the necessary notions on point processes, we re-analyse model B proposed in "On non-commutativity in quantum theory (II): toy models for non-commutative kinematics", using the point process theory. This viewpoint allows to generalize and modify the space process of model B in a simple manner, obtaining a new model (model C). This new model allows the recovery of non-relativistic quantum mechanics in a suitable limit.
In this article, we continue our investigation on the role of non-commutativity in quantum theory. Using the method explained in "On non-commutativity in quantum theory (I): from classical to quantum probability", we analyze two toy models which exhibit non-commutativity between the corresponding position and velocity random variables. In particular, using ordinary probability theory, we study the kinematics of a point-like particle jumping at random over a discrete random space. We show that, after the removal of the random space from the model, the position and velocity of the particle do no…
Some ambiguities have recently been found in the definition of the partial derivative (in the case of presence of both explicit and implicit dependencies of the function subjected to differentiation). We investigate the possible influence of this subject on quantum mechanics and the classical/quantum field theory. Surprisingly, some commutators of operators of space-time 4-coordinates and those of 4-momenta are {\it not} equal to zero. We postulate the non-commutativity of 4-momenta and we derive mass splitting in the Dirac equation. Moreover, two iterated limits may not commute each other, in…
Given a finite quiver, its double may be viewed as its non-commutative "cotangent" space, and hence is a non-commutative symplectic space. Crawley-Boevey, Etingof and Ginzburg constructed the non-commutative reduction of this space while Schedler constructed its quantization. We show that the non-commutative quantization and reduction commute with each other. Via the quantum and classical trace maps, such a commutativity induces the commutativity of the quantization and reduction on the space of quiver representations.
A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. In this article, we present a critical analysis on the role of non-commutativity in quantum theory, focusing on its consequences in the probabilistic description. Typically, a random phenomenon is described using the measure-theoretic formulation of probability theory. Such a description can also be done using algebraic methods, which are capable to deal with non-commutative random variables (like in quantum mechanics). Here we propose a method to construct a non-commutative prob…
We consider commutativity equations $F_i F_j =F_j F_i$ for a function $F(x^1, \dots, x^N),$ where $F_i$ is a matrix of the third order derivatives $F_{ikl}$. We show that under certain non-degeneracy conditions a solution $F$ satisfies the WDVV equations. Equivalently, the corresponding family of Frobenius algebras has the identity field $e$. We also study trigonometric solutions $F$ determined by a finite collection of vectors with multiplicities, and we give an explicit formula for $e$ for all the known such solutions. The corresponding collections of vectors are given by non-simply laced ro…
We examine the non-inertial effects of a rotating frame on a Dirac oscillator in a cosmic string space-time with non-commutative geometry in phase space. We observe that the approximate bound-state solutions are related to the biconfluent Heun polynomials. The related energies cannot be obtained in a closed form for all the bound states. We find the energy of the fundamental state analytically by taking into account the hard-wall confining condition. We describe how the ground-state energy scales with the new non-commutative term as well as with the other physical parameters of the system.
