The one example of Lorentz group

It is shown that a subgroup of $SL(2,{\mathbb H})$, denoted $Spin(2,{\mathbb H})$ in this paper, which is defined by two conditions in addition to unit quaternionic determinant, is locally isomorphic to the restricted Lorentz group, $L_+^\uparrow$. On the basis of the Dirac theory using the spinor group $Spin(2,{\mathbb H})$, in which the charge conjugation transformation becomes linear in the quaternionic Dirac spinor, it is shown that the Hermiticity requirement of the Dira...

In this dissertation, we study the implications generated by the Lorentz breaking symmetry in quantum electrodynamics. We analyze fermions interacting with an electromagnetic field in the contexts of quantum mechanics and make radiative corrections. In quantum mechanics, the terms of the Lorentz breaking symmetry were treated as perturbations to the Dirac equation, and their expected values were obtained in a vacuum. In the radiative corrections, the Lorentz breaking symmetry...

Special relativity, the symmetry breakdown in the electroweak standard model, and the dichotomy of the spacetime related transformations with the Lorentz group, on the one side, and the chargelike transformations with the hypercharge and isospin group, on the other side, are discussed under the common concept of "relativity". A relativity is defined by classes $G/H$ of a "little" group in a "general" group of operations.Relativities are representable as linear transformations...

The paper discusses the following topics: spinor coverings for the full Lorentz group, intrinsic parity of fermions, Majorana fermions, spinor structure of space models, two types of spacial spinors, parametrization of spinor spaces by curvilinear coordinates, manifestation of spinor space structure in classifying solutions of the quantum-mechanical equations and in the matrix elements for physical quantities.

We generate non-linear representations of the Lorentz Group by unitary transformation over the Lorentz generators. To do that we use deformed scale transformations by introducing momentum-depending parameters. The momentum operator transformation is found to be equivalent to a particle momentum transformation. The configuration space transformation is found to depend on the old momentum operator and we show that this transformation generates models with two scales, one for th...

I discuss the indefinite metrical structure of the time-space translations as realized in the indefinite inner products for relativistic quantum fields, familiar in the example of quantum gauge fields. The arising indefinite unitary nondiagonalizable representations of the translations suggest as the positive unitarity condition for the probability interpretable positive definite asymptotic particle state space the requirement of a vanishing nilpotent part in the time-space t...

This is the first chapter of a new and unconventional textbook on quantum mechanics and quantum field theory. The chapter introduces standard quantum mechanics by means of a symmetry principle, without reference to classical mechanics. The mathematical foundation of this approach comes from a recent paper of Naudts and Kuna on covariance systems. The standard representation of quantum mechanics is derived. Next, spin and mass of a quantum particle are explained as labels of p...

An unconventional outlook on relationship between the quantum mechanics and special relativity is proposed. We show that the two fundamental postulates of quantum mechanics of Planck and de Broglie combined with the idea of comparison scale (explained in the paper), are enough to introduce relativistic description. We argue that Lorentz group is the symmetry group of quantum, preferred frame description. We indicate that the departure from the orthodox relativity postulate al...

The Lorentz transformation is derived without assuming the existence of Maxwell's equations, or that the speed of light is a constant, or even that light exists. This leads us logically to sonsider the existence of a primal field called stuff which at every space-time point is traveling in every posssible direction with the speed of light. All physical quantities are to be derived from operations on the stuff field. To familiarize ourselves with the implications of the new pa...

The problems connected with a choice of the spinorial basis in the $(j,0)\oplus (0,j)$ representation space are discussed. It is shown to have profound significance in relativistic quantum theory. From the methodological viewpoint this fact is related with the important dynamical role played by space-time symmetries for all kind of interactions.