The CPT Group of the Dirac Field

We deduce the structure of the Dirac field on the lattice from the discrete version of differential geometry and from the representation of the integral Lorentz transformations. The analysis of the induced representations of the Poincare group on the lattice reveals that they are reducible, a result that can be considered a group theoretical approach to the problem of fermion doubling.

A recent paper by Jones-Smith and Mathur extends PT-symmetric quantum mechanics from bosonic systems (systems for which $T^2=1$) to fermionic systems (systems for which $T^2=-1$). The current paper shows how the formalism developed by Jones-Smith and Mathur can be used to construct PT-symmetric matrix representations for operator algebras of the form $\eta^2=0$, $\bar{\eta}^2=0$, $\eta\bar{\eta}+\bar {\eta} =\alpha 1$, where $\bar{eta}=\eta^{PT} =PT \eta T^{-1}P^{-1}$. It is ...

Due to the non-commutative nature of quaternions we introduce the concept of left and right action for quaternionic numbers. This gives the opportunity to manipulate appropriately the $H$-field. The standard problems arising in the definitions of transpose, determinant and trace for quaternionic matrices are overcome. We investigate the possibility to formulate a new approach to Quaternionic Group Theory. Our aim is to highlight the possibility of looking at new quaternionic ...

Wigner's little groups are the subgroups of the Lorentz group whose transformations leave the momentum of a given particle invariant. They thus define the internal space-time symmetries of relativistic particles. These symmetries take different mathematical forms for massive and for massless particles. However, it is shown possible to construct one unified representation using a graphical description. This graphical approach allows us to describe vividly parity, time reversal...

This dissertation is about The history of quaternions and their associated rotation groups as it relates to theoretical physics.

Complexified spacetime algebra is defined as the geometric (Clifford) algebra of spacetime with complex coefficients, isomorphic $\mathcal{G}_{1,4}$. By resorting to matrix representation by means of Dirac-Pauli gamma matrices, the paper demonstrates isomorphism between subgroups of CSTA and SU(3). It is shown that the symmetry group of those subgroups is indeed $U(1) \otimes SU(2) \otimes SU(3)$ and that there are 4 distinct copies of this group within CSTA.

Clifford algebras are used for constructing spin groups, and are therefore of particular importance in the theory of quantum mechanics. But the spin group is not the only subgroup of the Clifford algebra. An algebraist's perspective on these groups and algebras may suggest ways in which they might be applied more widely to describe the fundamental properties of matter. I do not claim to build a physical theory on top of the fundamental algebra, and my suggestions for possible...

A variation of Dirac equation based on SO(2,1) group is suggested for treating low dimensional systems in the three dimensional x,y,t space. Non-unitary representations are developed in an analogous way to those used in the ordinary Dirac equations and quantum field theory is developed for the present SO(2,1) relativistic equation. The theory is applied for low dimensional systems including especially holes-electrons pairs, or other two-particle states, in the low dimensional...

In a recent publication the I showed how the geometric algebra ${G}_{4,1}$, the algebra of 5-dimensional space-time, can generate relativistic dynamics from the simple principle that only null geodesics should be allowed. The same paper showed also that Dirac equation could be derived from the condition that a function should be monogenic in that algebra; this construction of the Dirac equation allows a choice for the imaginary unit and it was suggested that different imagina...

The symmetry of Nature under a Space Inversion is described by a Parity operator. Contrary to popular belief, the Parity operator is not unique. The choice of the Parity operator requires several arbitrary decisions to be made. It is shown that alternative, equally plausible, choices leads to the definition of a Parity operator that is conserved by the Weak Interactions. The operator commonly known as CP is a more appropriate choice for a Parity operator.