Related papers: Representation-independent manipulations with Dira…
In these notes we first review Pauli's proof of his `fundamental theorem' that states the equivalence of any two sets of Dirac matrices $\{ \gamma^\mu \}$. Due to this theorem not only all physical results in the context of the Dirac…
We present a tomographic scheme, based on spacetime symmetries, for the reconstruction of the internal degrees of freedom of a Dirac spinor. We discuss the circumstances under which the tomographic group can be taken as SU(2), and how this…
We define spin frames, with the aim of extending spin structures from the category of (pseudo-)Riemannian manifolds to the category of spin manifolds with a fixed signature on them, though with no selected metric structure. Because of this…
We define and study the probability current and the Hamiltonian operator for a fully general set of Dirac matrices in a flat spacetime with affine coordinates, by using the Bargmann-Pauli hermitizing matrix. We find that with some weak…
Gamma matrices for quantum Minkowski spaces are found. The invariance of the corresponding Dirac operator is proven. We introduce momenta for spin 1/2 particles and get (in certain cases) formal solutions of the Dirac equation.
A representation of the Lorentz group is given in terms of 4 X 4 matrices defined over a simple non-division algebra. The transformation properties of the corresponding four component spinor are studied, and shown to be equivalent to the…
We investigated action of operator analogs of Dirac gamma matrices (we called them gamma operators) on a vacuum. We derived formulas for vacuum state vector and operators of the Lorentz transformations of spinors in the superalgebraic…
The spinor representation of the Lorentz group does not accept simple generalization with the group GL(4,R) of general linear coordinate transformations. The Dirac equation may be written for an arbitrary choice of a coordinate system and a…
We present the consistent approach to finding the discrete transformations in the representation spaces of the proper Poincar\'e group. To this end we use the possibility to establish a correspondence between involutory automorphisms of the…
In this largely expository paper we give a self-contained treatment of the Dirac operator. Emphasizing the algebraic point of view we first sketch the necessary prerequisites from Clifford algebras and their representations and then define…
Transformation properties of Dirac equation correspond to Spin(3,1) representation of Lorentz group SO(3,1), but group GL(4,R) of general relativity does not accept a similar construction with Dirac spinors. On the other hand, it is…
The question of how does the Dirac equation depend on the choice of the $\gamma$ matrices has partially been addressed and explored in the literature. In this paper we focus on this question by considering a general form of $\gamma$…
A simple framework for Dirac spinors is developed that parametrizes admissible quantum dynamics and also analytically constructs electromagnetic fields, obeying Maxwell's equations, which yield a desired evolution. In particular, we show…
Graph independence (also known as $\epsilon$-independence or $\lambda$-independence) is a mixture of classical independence and free independence corresponding to graph products or groups and operator algebras. Using conjugation by certain…
Previously (A. Akhmeteli, J. Math. Phys., v. 52, p. 082303 (2011)), the Dirac equation in an arbitrary electromagnetic field was shown to be generally equivalent to a fourth-order equation for just one component of the four-component Dirac…
For a directed acyclic graph, there are two known criteria to decide whether any specific conditional independence statement is implied for all distributions factorized according to the given graph. Both criteria are based on special types…
The technique for representing spinors and the definition of the discrete symmetries is used to illustrate on a toy model properties of massless and massive spinors states, in the first and the second quantized picture. Since in this toy…
Anderson's theorem asserting, that symmetry of dynamic equations written in the relativisitically covariant form is determined by symmetry of its absolute objects, is applied to the free Dirac equation. Dirac matrices are the only absolute…
In the superalgebraic representation of spinors using Grassmann densities and derivatives with respect to them, a generalization of Dirac conjugation is introduced, which provides Lorentz-covariant transformations of conjugate spinors. It…
We study the most general Fierz identities for a pair of non-contracted Dirac matrices both in the standard basis and for chiral spinors. These identities are useful in building independent effective operators of fermions that involve…