Neutrino Billiards: Time-Reversal Symmetry-Breaking Without Magnetic Fields

Details Neutrino Billiards: Time-Reversal Symmetry-Breaking Without Magnetic Fields Neutrino Billiards: Time-Reversal Symmetry-Breaking Without Magnetic Fields

Jul 1987

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1987rspsa.412...53b&link_type=abstract

Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Volume 412, Issue 1842, pp. 53-74

Statistics

101

Scientific paper

A Dirac hamiltonian describing massless spin-half particles (`neutrinos') moving in the plane r = (x, y) under the action of a 4-scalar (not electric) potential V(r) is, in position representation, H = -ihbar cσ\cdotnabla + V(r)σ_z, where σ = (σ_x, σ_y) and σ_z are the Pauli matrices; hat{H} acts on two-component column spinor wavefunctions psi(r) = (psi_1,psi_2) and has eigenvalues hbar ck_n. H does not possess time-reversal symmetry (T). If V(r) describes a hard wall bounding a finite domain D (`billiards'), this is equivalent to a novel boundary condition for psi_2/psi_1. T-breaking is interpreted semiclassically as a difference of π between the phases accumulated by waves travelling in opposite senses round closed geodesics in D with odd numbers of reflections. The semiclassical (large-k) asymptotics of the eigenvalue counting function (spectral staircase) N(k) are shown to have the `Weyl' leading term Ak^2/4π, where A is the area of D, but zero perimeter correction. The Dirac equation is transformed to an integral equation round the boundary of D, and forms the basis of a numerical method for computing the k_n. When D is the unit disc, geodesics are integrable and the eigenvalues, which satisfy J_l(k_n) = Jl+1(k_n), are (locally) Poisson-distributed. When D is an `Africa' shape (cubic conformal map of the unit disc), the eigenvalues are (locally) distributed according to the statistics of the gaussian unitary ensemble of random-matrix theory, as predicted on the basis of T-breaking and lack of geometric symmetry.

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