We discuss diagonalization of propagator for mixing fermions system based on the eigenvalue problem. The similarity transformation converting matrix propagator into diagonal form is obtained. The suggested diagonalization has simple algebraic properties for on-shell fermions and can be used in renormalization of fermion mixing matrix.
We apply the exponential operator method to derive the propagator for a fermion immersed within a rigidly rotating environment with cylindrical geometry. Given that the rotation axis provides a preferred direction, Lorentz symmetry is lost and the general solution is not translationally invariant in the radial coordinate. However, under the approximation that the fermion is completely dragged by the vortical motion, valid for large angular velocities, translation invariance is recovered. The propagator can then be written in momentum space. The result is suited to be used applying ordinary Feynman rules for perturbative calculations in momentum space.
We study the eigenvalue equation for the Cartesian coordinates observables $x_i$ on the fully $O(2)$-covariant fuzzy circle ${S^1_Lambda}_{Lambdainmathbb{N}}$ ($i=1,2$) and on the fully $O(3)$-covariant fuzzy 2-sphere ${S^2_Lambda}_{Lambdainmathbb{N}}$ ($i=1,2,3$) introduced in [G. Fiore, F. Pisacane, J. Geom. Phys. 132 (2018), 423-451]. We show that the spectrum and eigenvectors of $x_i$ fulfill a number of properties which are expected for $x_i$ to approximate well the corresponding coordinate operator of a quantum particle forced to stay on the unit sphere.
Starting from Wigners symmetry representation theorem, we give a general account of discrete symmetries (parity P, charge conjugation C, time-reversal T), focusing on fermions in Quantum Field Theory. We provide the rules of transformation of Weyl spinors, both at the classical level (grassmanian wave functions) and quantum level (operators). Making use of Wightmans definition of invariance, we outline ambiguities linked to the notion of classical fermionic Lagrangian. We then present the general constraints cast by these transformations and their products on the propagator of the simplest among coupled fermionic system, the one made with one fermion and its antifermion. Last, we put in correspondence the propagation of C eigenstates (Majorana fermions) and the criteria cast on their propagator by C and CP invariance.
We discuss the (right) eigenvalue equation for $mathbb{H}$, $mathbb{C}$ and $mathbb{R}$ linear quaternionic operators. The possibility to introduce an isomorphism between these operators and real/complex matrices allows to translate the quaternionic problem into an {em equivalent} real or complex counterpart. Interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics.
We study the gauge covariance of the fermion propagator in Maxwell-Chern-Simons planar quantum electrodynamics (QED$_3$) considering four-component spinors with parity-even and parity-odd mass terms both for fermions and photons. Starting with its tree level expression in the Landau gauge, we derive a non perturbative expression for this propagator in an arbitrary covariant gauge by means of its Landau-Khalatnikov-Fradkin transformation (LKFT). We compare our findings in the weak coupling regime with the direct one-loop calculation of the two-point Green function and observe perfect agreement up to a gauge independent term. We also reproduce results derived in earlier works as special cases of our findings.