Related papers: On the group generated by $\mathbf C$, $\mathbf{P}…
We consider discrete transformations ($C, P, T$ and $CP$) of baryon- and lepton-nonconserving processes. It has long been thought that values ($\pm 1, \pm i$) form the correct set of fermionic arbitrary phases for discrete transformations.…
In a remarkable development Bender and coworkers have shown that it is possible to formulate quantum mechanics consistently even if the Hamiltonian and other observables are not Hermitian. Their formulation, dubbed PT quantum mechanics,…
In the present paper we show that it is possible to obtain the well known Pauli group $P=\langle X,Y,Z \ | \ X^2=Y^2=Z^2=1, (YZ)^4=(ZX)^4=(XY)^4=1 \rangle $ of order $16$ as an appropriate quotient group of two distinct spaces of orbits of…
This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition $H^\dagger=H$ on the Hamiltonian, where $\dagger$ represents the mathematical operation of complex conjugation and matrix…
We study certain polynomial trace identities in the group $SL(2,\IC)$ and their application in the theory of discrete groups. We obtain canonical representations for two generator groups in \S 4 and then in \S 5 we give a new proof for…
Charge conjugation (C), mirror reflection (R), time reversal (T), and fermion parity $(-1)^{\rm F}$ are basic discrete spacetime and internal symmetries of the Dirac fermions. In this article, we determine the group, called the C-R-T…
The notion of formal duality in finite Abelian groups appeared recently in relation to spherical designs, tight sphere packings, and energy minimizing configurations in Euclidean spaces. For finite cyclic groups it is conjectured that there…
In conventional relativistic quantum field theory, the discrete operators $\textbf{C}$, $\textbf{P}$, and $\textbf{T}$ are matrix operators with no renormalization scale dependence. However, in a Lorentz-violating theory with a fermion…
Let $S$ be a semigroup, $\Lambda$ a non-empty set and $P$ a mapping of $\Lambda$ into $S$. The set $S\times \Lambda$ together with the operation $\circ _P$ defined by $(s, \lambda)\circ _P(t, \mu )=(sP(\lambda)t, \mu )$ form a semigroup…
We extend the $CPT$ theorem to quantum field theories with non-Hermitian Hamiltonians and unstable states. Our derivation is a quite minimal one as it requires only the time independent evolution of scalar products, invariance under complex…
Heisenberg's uncertainty relation can be written in terms of the step-up and step-down operators in the harmonic oscillator representation. It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the…
The parity transformation law of the fermion field $\psi(x)$ is usually defined by the "$\gamma^{0}$-parity" $\psi^{p}(t,-\vec{x}) = \gamma^{0}\psi(t,-\vec{x})$ with eigenvalues $\pm 1$, while the "$i\gamma^{0}$-parity"…
$CPT$ groups for spinor fields in de Sitter and anti-de Sitter spaces are defined in the framework of automorphism groups of Clifford algebras. It is shown that de Sitter spaces with mutually opposite signatures correspond to Clifford…
The smallness of the 1-3 lepton mixing angle $\theta_{13}$ and of the neutrino mass-squared-difference ratio $\Delta m^2_{12}/\Delta m^2_{23}$ can be understood as the departure from a common limit where they both vanish. We discuss in…
Totally positive (TP) and totally nonnegative (TN) matrices connect to analysis, mechanics, and to dual canonical bases in reductive groups, by well-known works of Schoenberg, Gantmacher-Krein, Lusztig, and others. TP matrices form a…
In the literature the $CPT$ theorem has only been established for Hamiltonians that are Hermitian. Here we extend the $CPT$ theorem to quantum field theories with non-Hermitian Hamiltonians. Our derivation is a quite minimal one as it…
It is shown that the specific "charge conjugation" transformation used to define the Majorana fermions in the conventional seesaw mechanism, namely $(\nu_{R})^{C}=C\bar{\nu_{R}}^{T}$ for a chiral fermion $\nu_{R}$ (and similarly for…
The recent failure of the scenario which uses only $\mathbf{10,\oot}$ Higgs multiplets to yield large enough neutrino masses in the $\mathbf{SO(10) }$ \textbf{MSGUT} motivates an alternative scenario\cite{blmdoom} where the $\mathbf{120}$…
We analyze several non-Hermitian Hamiltonians with antiunitary symmetry from the point of view of their point-group symmetry. It enables us to predict the degeneracy of the energy levels and to reduce the dimension of the matrices necessary…
A group structure of the discrete transformations (parity, time reversal and charge conjugation) for spinor field in de Sitter space are studied in terms of extraspecial finite groups. Two $CPT$ groups are introduced, the first group from…