The Dirac Sea, T and C Symmetry Breaking, and the Spinor Vacuum of the Universe

We have developed a quantum field theory of spinors based on the algebra of canonical anticommutation relations (CAR algebra) of Grassmann densities in the momentum space. We have proven the existence of two spinor vacua. Operators C and T transform the normal vacuum into an alternative one, which leads to the breaking of the C and T symmetries. The CPT is the real structure operator; it preserves the normal vacuum. We have proven that, in the theory of the Dirac Sea, the formula for the charge conjugation operator must contain an additional generalized Dirac conjugation operator.

Simple nuclear C*-algebras not isomorphic to their opposites

We show that it is consistent with Zermelo–Fraenkel set theory with the axiom of choice (ZFC) that there is a simple nuclear nonseparable C∗-algebra, which is not isomorphic to its opposite algebra. We can furthermore guarantee that this example is an inductive limit of unital copies of the Cuntz algebra O2 or of the canonical anticommutation relations (CAR) algebra.

Discussion on Lorentz invariance violation of noncommutative field theory and neutrino oscillation

Depending on deformed canonical anticommutation relations, massless neutrino oscillation based on Lorentz invariance violation in noncommutative field theory is discussed. It is found that the previous studies about massless neutrino oscillation within deformed canonical anticommutation relations should satisfy the condition of new Moyal product and new nonstandard commutation relations. Furthermore, comparing the Lorentz invariant violation parameters A in the previous studies with new Moyal product and new nonstandard commutation relations, we find that the orders of magnitude of noncommutative parameters (Lorentz invariant violation parameters A) is not self-consistent. This inconsistency means that the previous studies of Lorentz invariance violation in noncommutative field theory may not naturally explain massless neutrino oscillation. In other words, it should be impossible to explain neutrino oscillation by Lorentz invariance violation in noncommutative field theory. This conclusion is supported by the latest atmospheric neutrinos experimental results from the super-Kamiokande Collaboration, which show that no evidence of Lorentz invariance violation on atmospheric neutrinos was observed.

A NEW DESCRIPTION OF MOTION OF THE FERMIONIC SO(2N+2) TOP IN THE CLASSICAL LIMIT UNDER THE QUASI-ANTICOMMUTATION RELATION APPROXIMATION

-1 The boson images of fermion SO(2N+1) Lie operators have been given together with those of SO(2N+2) ones. The SO(2N+1) Lie operators are generators of rotation in the (2N+1)-dimensional Euclidean space (N: number of single-particle states of the fermions). The images of fermion annihilation–creation operators must satisfy the canonical anticommutation relations, when they operate on a spinor subspace. In the regular representation space we use a boson Hamiltonian with Lagrange multipliers to select out the spinor subspace. Based on these facts, a new description of a fermionic SO(2N+2) top is proposed. From the Heisenberg equations of motions for the boson operators, we get the SO(2N+1) self-consistent field (SCF) Hartree–Bogoliubov (HB) equation for the classical stationary motion of the fermion top. Decomposing an SO(2N+1) matrix into matrices describing paired and unpaired modes of fermions, we obtain a new form of the SO(2N+1) SCF equation with respect to the paired-mode amplitudes. To demonstrate the effectiveness of the new description based on the bosonization theory, the extended HB eigenvalue equation is applied to a superconducting toy-model which consists of a particle–hole plus BCS-type interaction. It is solved to reach an interesting and exciting solution which is not found in the traditional HB eigenvalue equation due to the unpaired-made effects. To complete the new description, the Lagrange multipliers must be determined in the classical limit. For this aim a quasi-anticommutation relation approximation is proposed. Only if a certain relation between an SO(2N+1) parameter z and the N is satisfied, unknown parameters K and l in the Lagrange multipliers can be determined without any inconsistency.