Spinor Solutions Research Articles

Fermions and gauge fields in the Einstein static universe

Exact solutions of the massive Dirac equation are obtained in an SU(2) gauge field background in the Einstein static universe. The static, finite-energy gauge field used as background is the one obtained by continuing the meron-antimeron solution to this base space. The regular spinor solutions lead to a quantization condition.

Exact scalar and spinor solutions in some rotating universes

In this paper exact solutions of the Klein–Gordon and Weyl equations in some rotating universes are presented.

A note on spatial behaviour of solutions of the one-dimensional dirac equation

It is shown that, out of the functions determining the spinor solutions to the one-dimensional Dirac equation, we can construct a combination which is spatially constant. The implications and applicability of this result as well as some other allied ones, have been indicated.

Spinors in periodic self-dual gauge field backgrounds

Normalizable zero modes of the Dirac operator are constructed for a class of self-dual, periodic SU(2) gauge field backgrounds characterized by two independent integer invariants. The integers are (ST/8π2), where ST is the action over one period T, and the asymptotic winding number (q) in R3, the solutions reducing to static ‘‘monopoles’’ for large spatial distances independently of the time. The spinor solutions are obtained for the simplest class of the hierarchy presented in Chakrabarti [Phys. Rev. D 35, 696 (1987)], corresponding to q=1 and ST=8π2⋅2n (n=1,2,3,...). The full number of zero modes for such backgrounds is ((ST/8π2)−q)=(2n−1). They are all constructed explicitly. It is shown how these results can be obtained through a simple scaling limit by starting with special classes of instantons with finite action over R4. A derivation of ST is also given.

Separation of Dirac and Kähler equations in spherically symmetric space-times

The Dirac and Kähler equations are separated in a class of spherically symmetric space-times by using new techniques of Clifford analysis. New operators, which raise and lower the ‘‘spin weight’’ of spin-weighted functions by (1)/(2), are also introduced. The separated solutions enable us to analyze the distinction between spinor solutions to the Dirac separation equation and Kähler solutions.

Spinor solutions to Maxwell's equations in free space

Using the Hankel transform, we obtain, in their most general form, the spinor solutions of Maxwell's equations in free space for a light beam propagating along 0z with cylindrical symmetry around 0z. Then we compute a paraxial approximation of these solutions and we discuss the case of Gaussian beams. A comparison of the spinor solutions with the scalar ones supplied by the parabolic equation is also given.

Quantization of spinor fields. IV. Joint Bose–Fermi spectral problems

In this continued study of the connection between classical c-number spinor models and their quantized Fermi partners, we elaborate further necessary consequences of the bosonization. The genuine (c-number) path integral representation of tr exp(−iHt) is derived for the Fermi oscillator and simple lattice Fermi models. We find that the underlying Hamiltonian of the Fermi system HF can be equivalently written as PHBP, where HB is the related Bose Hamiltonian, P is an appropriate projection in the state space of the Bose system, and [P, HB]−=0, HF=PHBP. Grassmann algebras are not used. We prove further that both for the massive Thirring model (MT) and the chiral invariant Gross–Neveu model (CN), the Bethe ansatz eigenstates for the Fermi Hamiltonians are exact eigenstates of the Bose MT and CGN Hamiltonians, so introduced that HF=PHBP, [P, HB]−=0. As a consequence, through studying the c-number path integral representation for tr exp(−iHFt), we establish a class of classical (c-number) spinor solutions of the underlying field equations, which at the same time stationary both the c-number Bose and c-number Fermi actions.

Chiral invariant Gross-Neveu model: Classical versus quantum

We present a unification of different and independently investigated aspects of the chiral invariant Gross-Neveu model. Special emphasis is placed on the relevance of classical ( c-number, non Grassmann) spinor solutions of the G-N field equations for the construction, and thus understanding of the respective quantized Fermi model. To get an insight into the “quantum meaning of classical field theory” if specialized to the G-N case, we perform the path integral quantization procedure which first leads to the Fermi oscillator problem, and then, after appropriate generalizations, to the quantum Fermi G-N model. Path integrals are carried out with respect to c-number spinor paths only, and in fact no reference is necessary to the Grassmann algebra methods, which are conventionally used to integrate out fermions.

Conformal relativity

The free-particle equations implied by conformal symmetry are developed in general and solved for the basic case of the «spinor» irreducible unitary representations. Isospin labels and symmetry are shown to arise necessarily, as a «dynamical symmetry», out of conformal symmetry, a pure space-time construct. It is speculated that the correct «internal symmetry»,SU 3 orSU 4, etc., may originate this way in conformal symmetry. The massive spinor solutions describe discrete resonance families of the typeT=1/2,J=1/2, such as the nucleon, u and d quarks, etc. The massless spinor solutions fit the known leptons.

Spinor solutions to the Schwarzschild metric

The class of solutions represented by Hermitian two-by-two matrices associated to time-independent metrics which correspond to a symmetry Euclidian Killing vector is derived. The elements of this class are related by means of point-dependent unitary unimodular two-by-two matrices. It is shown that these spinor fields associated to the Schwarzschild metric are solutions of the gravitational spinor equations proposed by Sachs. The properties of covariance of the formalism under the transformations of the groupSU2 are analysed.

Quasi-Classical Theory of the Spinning Electron

We derive a modified Hamilton-Jacobi theory for a classical spinning dipole and show that this classical theory can be put in a form almost identical with the Pauli spin theory. We quantize this theory by requiring that a classical spinor "wave function" be continuous and single-valued, and that it satisfy the usual energy eigenvalue equation. In this manner we deduce the correct energy levels for a hydrogen atom in an external magnetic field. We derive the integral kernel for the Pauli equation from this classical model and discuss the properties of our spinor solutions under canonical transformation. We exhibit the charge-conjugate solution to the classical spin equation.