Narrow Results

In this paper, we have constructed the Green function of the Feshbach–Villars (FV) spinless particle in a noncommutative (NC) phase-space coordinates, where the Pauli matrices describing the charge symmetry are replaced by the Grassmannian odd variables. Subsequently, for the perform calculations, we diagonalize the Hamiltonian governing the dynamics of the system via the Foldy–Wouthuysen (FW) canonical transformation. The exact calculations have been done in the cases of free particle and magnetic field interaction. In both cases, the energy eigenvalues and their corresponding eigenfunctions are deduced.

The finite embeddability property (FEP) for knotted extensions of residuated lattices holds under the assumption of commutativity, but fails in the general case. We identify weaker forms of the commutativity identity which ensure that the FEP holds. The results have applications outside of order algebra to non-classical logic, establishing the strong finite model property (SFMP) and the decidability for deductions, as well as to mathematical linguistics and automata theory, providing new conditions for recognizability of languages. Our proofs make use of residuated frames, developed in the context of algebraic proof theory.

As a historical remark the similarities are pointed out between a form of chiral action introduced by Schwinger and the formalism used in the noncommutative extensions of electromagnetism.