Neutrino billiards: time-reversal symmetry-breaking without magnetic fields
- 8 July 1987
- journal article
- Published by The Royal Society in Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
- Vol. 412 (1842), 53-74
- https://doi.org/10.1098/rspa.1987.0080
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.
Keywords
This publication has 11 references indexed in Scilit:
- Riemann's Zeta function: A model for quantum chaos?Published by Springer Science and Business Media LLC ,2008
- Chaotic motion and random matrix theoriesPublished by Springer Science and Business Media LLC ,2005
- Statistics of energy levels without time-reversal symmetry: Aharonov-Bohm chaotic billiardsJournal of Physics A: General Physics, 1986
- Semiclassical theory of spectral rigidityProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1985
- Aspects of DegeneracyPublished by Springer Science and Business Media LLC ,1985
- Diabolical points in the spectra of trianglesProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1984
- Quantizing a classically ergodic system: Sinai's billiard and the KKR methodAnnals of Physics, 1981
- Level clustering in the regular spectrumProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 1977
- Distribution of eigenfrequencies for the wave equation in a finite domain: III. Eigenfrequency density oscillationsAnnals of Physics, 1972
- Distribution of eigenfrequencies for the wave equation in a finite domain: I. Three-dimensional problem with smooth boundary surfaceAnnals of Physics, 1970