Generalized Electromagnetic Fields Associated with the Hydrogen-Like Atom Problem

It is known that the principle of minimal coupling in quantum mechanics determines a unique interaction form for a charged particle. By properly redefining the canonical commutation relation between (canonical) conjugate components of position and momentum of the particle, e.g. an electron, we restate the Dirac equation for the hydrogen-like atom problem incorporating a generalized minimal electromagnetic coupling. The corresponding interaction keeps the $1/\left|\mathbf{q}\right|$ dependence in both the scalar potential $V\left({\left|\mathbf{q}\right|}\right)$ and the vector potential $\mathbf{A}\left(\mathbf{q}\right)$ ($\left|{\mathbf{A}\left(\mathbf{q}\right)}\right|\sim 1/\left|\mathbf{q}\right|$). This problem turns out to be exactly solvable; moreover, the eigenstates and eigenvalues can be obtained in an elementary fashion. Some feasible models within this approach are discussed. Then we make a few remarks about the breaking of supersymmetry. Finally, we briefly comment on the possible Lie algebra (dynamical symmetry algebra) of these relativistic quantum systems.

Green’s functions for translation invariant star products

We calculate the Green’s functions for a scalar field theory with quartic interactions for which the fields are multiplied with a generic translation invariant star product. Our analysis involves both non-commutative products, for which there is the canonical commutation relation among coordinates, and nonlocal commutative products. We give explicit expressions for the one-loop corrections to the two- and four-point functions. We find that the phenomenon of ultraviolet/infrared mixing is always a consequence of the presence of non-commuting variables. The commutative part of the product does not have the mixing.

Deformation of second and third quantization

In this paper, we will deform the second and third quantized theories by deforming the canonical commutation relations in such a way that they become consistent with the generalized uncertainty principle. Thus, we will first deform the second quantized commutator and obtain a deformed version of the Wheeler–DeWitt equation. Then we will further deform the third quantized theory by deforming the third quantized canonical commutation relation. This way we will obtain a deformed version of the third quantized theory for the multiverse.

Deformation of the Wheeler–DeWitt equation

In this paper, we will analyze the consequences of deforming the canonical commutation relations consistent with the existence of a minimum length and a maximum momentum. We first generalize the deformation of first quantized canonical commutation relation to second quantized canonical commutation relation. Thus, we arrive at a modified version of second quantization. A modified Wheeler–DeWitt equation will be constructed by using this deformed second quantized canonical commutation relation. Finally, we demonstrate that in this modified theory the big bang singularity gets naturally avoided.

DE VAUCOULEURS–IKEUCHI DIAGRAM AND COMMUTATION RELATIONS AMONG PHASE SPACE COORDINATES

We consider the relations between de Vaucouleurs–Ikeuchi diagram and generalized commutation relations among the coordinates and momenta. All physical objects in the Universe ranging from elementary particles to super-cluster of galaxies are confined within the Triangle of the de Vaucouleurs–Ikeuchi diagram on the matter density versus scale length plane. These three boundaries are characterized by the quantum uncertainty principle, gravitational event horizon, and cosmological constant. These are specified by the nonzero commutation relations [xμ, pν], [xμ, xν] (strictly [xi, t]) and [pμ, pν], respectively. The canonical commutation relation [xi, pj] are slightly modified, which preserves the self consistency as a whole.