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

M. V. Berry, R. J. Mondragon

## Abstract

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, $H = -i\hbar c\sigma\cdot\nabla + V(r)\sigma_z$, where $\sigma = (\sigma_x, \sigma_y)$ and $\sigma_z$ 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 $\pi$ 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$^2$/4$\pi$, 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 $J_l(k_n) = J_{l+1}(k_n)$, 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.