Lewis Ryder

Neutrino oscillations induced by spacetime torsion

The gravitational neutrino oscillation problem is studied by considering the Dirac Hamiltonian in a Riemann-Cartan spacetime and calculating the dynamical phase. Torsion contributions which depend on the spin direction of the mass eigenstates are found. These effects are of the order of Planck scales.The gravitational neutrino oscillation problem is studied by considering the Dirac Hamiltonian in a Riemann-Cartan space-time and calculating the dynamical phase. Torsion contributions which depend on the spin direction of the mass eigenstates are found. These effects are of the order of Planck scales.

Relativistic treatment of inertial spin effects

A relativistic spin operator for Dirac particles is identified and it is shown that a coupling of spin to angular velocity arises in the relativistic case, just as Mashhoon had speculated, and Hehl and Ni had demonstrated, in the non-relativistic case.

Einstein-Cartan-Dirac theory in the low-energy limit

We look for manifestations of the effects of torsion in the low-energy limit in the context of Einstein - Cartan - Dirac theory (or any theory of gravity in which the torsion tensor is purely axial). To proceed, we introduce the mathematical law governing the transport of orthonormal bases or tetrads in a spacetime with torsion. This law is applied to compute the metric and connection in a rotating and accelerating frame, or laboratory. A spin- particle is placed in this rotating and accelerating frame and the low-energy limit of the Dirac equation is taken by means of the Foldy - Wouthuysen transformation. In addition to obtaining the Bonse - Wroblewski phase shift due to acceleration, Sagnac-type effects, rotation - spin couplings of the Mashhoon type, redshift of the kinetic energy and the spin - orbit coupling term of Hehl and Ni, we also obtain several interesting and significant terms as a consequence of introducing torsion into spacetime. We give a detailed interpretation of these additional terms and discuss their observability in the light of current well-known experimental techniques.

The effect of Schwarzschild field on spin 1/2 particles compared to the effect of a uniformly accelerating frame

The Dirac Hamiltonian is calculated in the Schwarzschild space and compared to the analogous one in a uniformly accelerating Minkowski frame yielding a test of the equivalence principle. Comparing these Hamiltonians, we see that the flat-space energy-mass terms and their redshifted forms are the same in the two cases, but the coefficient of the spin-orbit coupling term is different and an additional term appears in the gravitational case.

TIME-INDEPENDENT SOLUTIONS TO THE TWO-DIMENSIONAL NONLINEAR O(3) SIGMA MODEL AND SURFACES OF CONSTANT MEAN CURVATURE

It is shown that time-independent solutions to the (2+1)-dimensional nonlinear O(3) sigma model may be placed in correspondence with surfaces of constant mean curvature in three-dimensional Euclidean space. The tools required to establish this correspondence are provided by the classical differential geometry of surfaces. A constant-mean-curvature surface induces a solution to the O(3) model through the identification of the Gauss map, or normal vector, of the surface with the field vector of the sigma model. Some explicit solutions, including the solitons and antisolitons discovered by Belavin and Polyakov, and a more general solution due to Purkait and Ray, are considered and the surfaces giving rise to them are found explicitly. It is seen, for example, that the Belavin-Polyakov solutions are induced by the Gauss maps of surfaces which are conformal to their spherical images, i.e. spheres and minimal surfaces, and that the Purkait-Ray solution corresponds to the family of constant-mean-curvature helicoids first studied by do Carmo and Dajczer in 1982. A generalization of this method to include time dependence may shed new light on the role of the Hopf invariant in this model.

Relativistic Spin Operator for Dirac Particles

It is shown that a relativistic spin operator,obeying the required SU(2) commutation relations, may bedefined in terms of the Pauli-Lubanski vectorWμ. In the case of Dirac particles, thisoperator reduces to the Foldy-Wouthuysen “mean-spin”operator for states of positive energy.

Possible effects of space-time nonmetricity on neutrino oscillations

The contribution of gravitational neutrino oscillations to the solar neutrino problem is studied by constructing a Dirac Hamiltonian and calculating the corresponding dynamical phase in the vicinity of the Sun in a non-Riemann background Kerr space-time with torsion and nonmetricity. We show that certain components of nonmetricity and the axial as well as nonaxial components of torsion may contribute to neutrino oscillations. We also note that the rotation of the Sun may cause a suppression of transitions among neutrinos. However, the observed solar neutrino deficit could not be explained by any of these effects because they are of the order of Planck scale.

DIRAC EQUATION IN SPACETIMES WITH NON-METRICITY AND TORSION

Dirac equation is written in a non-Riemannian spacetime with torsion and non-metricity by lifting the connection from the tangent bundle to the spinor bundle over spacetime. Foldy-Wouthuysen transformation of the Dirac equation in a Schwarzschild background spacetime is considered and it is shown that both the torsion and non-metricity couples to the momentum and spin of a massive, spinning particle. However, the effects are small to be observationally significant.Dirac equation is written in a non-Riemannian spacetime with torsion and non-metricity by lifting the connection from the tangent bundle to the spinor bundle over spacetime. Foldy–Wouthuysen transformation of the Dirac equation in a Schwarzschild background spacetime is considered and it is shown that both the torsion and non-metricity couples to the momentum and spin of a massive, spinning particle. However, the effects are small to be observationally significant.

General relativistic treatment of the Colella–Overhauser–Werner experiment on neutron interference in a gravitational field

In the Colella–Overhauser–Werner (COW) experiment a gravity-induced phase shift of spin 1/2 particles was detected. The experimental results were explained by using the Newtonian theory of gravity. The explanation can be easily given using general relativistic arguments and the highest order term reproduces the result of Colella, Overhauser, and Werner together with additional, lower order corrections. The derivation can be considered as an interesting exercise for students with basic knowledge of the field of general relativity.

Spin and rotation in general relativity

Spin is the ultimate gyroscope. The smallest possible amount of angular momentum is ħ/Π- that possessed by a spin 1/2 particle. When the day comes that it becomes realistic to theorise about and to measure the precession of a spin 1/2 particle it will be necessary to have to hand the relevant theoretical tools; in other words, to be able to give a description of spin one-half particles which is consistent with Special Relativity, and to generalise that description to General Relativity. In general terms, then, this is an exercise in relativistic quantum mechanics, and in the case of General Relativity, in quantum mechanics in a curved space. This latter is, of course, different from quantum gravity. Quantum gravity is a theory describing the quantum nature of the gravitational field itself, for example in terms of gravitons. For our purposes the gravitational field is treated classically, as a curved space-time. The only thing to be quantised is the spin 1/2 particle.