Structure of the plane waves for a spin 3/2 particle, massive and massless cases, gauge symmetry

The structure of the plane waves solutions for a relativistic spin 3/2 particle described by 16-component vector-bispinor is studied. In massless case, two representations are used: Rarita – Schwinger basis, and a special second basis in which the wave equation contains the Levi-Civita tensor. In the second representation it becomes evident the existence of gauge solutions in the form of 4-gradient of an arbitrary bispinor. General solution of the massless equation consists of six independent components, it is proved in an explicit form that four of them may be identified with the gauge solutions, and therefore may be removed. This procedure is performed in the Rarita – Schwinger basis as well. For the massive case, in Rarita – Schwinger basis four independent solutions are constructed explicitly.

Dirac Equation on the Kerr–Newman Spacetime and Heun Functions

By employing a pseudoorthonormal coordinate-free approach, the Dirac equation for particles in the Kerr–Newman spacetime is separated into its radial and angular parts. In the massless case to which a special attention is given, the general Heun-type equations turn into their confluent form. We show how one recovers some results previously obtained in literature, by other means.

Eigenvalue bounds for non-selfadjoint Dirac operators

We prove that the eigenvalues of the n-dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ D 0 + V , $$n\ge 2$$ n ≥ 2 , perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ L x j 1 L x ^ j ∞ , for $$j\in \{1,\dots ,n\}$$ j ∈ { 1 , ⋯ , n } . In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ σ ( D 0 + V ) = σ ( D 0 ) = R . The main tools used are an abstract version of the Birman–Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

Schwarzschild radial perturbations in Eddington–Finkelstein and Painlevé–Gullstrand coordinates

In a previous paper, we have considered the Regge–Wheeler equation for fields of spin s = 0, 1 or 2 on the Schwarzschild spacetime in coordinates that are regular at the horizon. In particular, we have constructed in Eddington–Finkelstein (EF) coordinates exact solutions in terms of series that are regular at the horizon and converge on the entire open domain from the central singularity to infinity. Here, we extend this earlier work in two different directions. First, we consider in EF coordinates a massive scalar field that can serve as a dark matter candidate. Second, we extend the treatment of the massless case to Painlevé–Gullstrand (PG) coordinates, which are associated with radially infalling observers.

FIELD THEORY OF MASSIVE AND MASSLESS VECTOR PARTICLES IN THE DUFFIN–KEMMER–PETIAU FORMALISM

Field theory of massive and massless vector particles is considered in the first-order formalism. The Hamiltonian form of equations is obtained after the exclusion of nondynamical components. We obtain the canonical and symmetrical Belinfante energy–momentum tensors and their nonzero traces. We note that the dilatation symmetry is broken in the massive case but in the massless case the modified dilatation current is conserved. The canonical quantization is performed and the propagator of the massive fields is found in the Duffin–Kemmer–Petiau formalism.

A NONLOCAL DISCRETIZATION OF FIELDS

A nonlocal method to obtain discrete classical fields is presented. This technique relies on well-behaved matrix representations of the derivatives constructed on a nonequispaced lattice. The drawbacks of lattice theory like the fermion doubling or the breaking of chiral symmetry for the massless case are absent in this method.