Beyond the Heisenberg time: Semiclassical treatment of spectral correlations in chaotic systems with spin 1/2

The two-point correlation function of chaotic systems with spin 1/2 is evaluated using periodic orbits. The spectral form factor for all times thus becomes accessible. Equivalence with the predictions of random matrix theory for the Gaussian symplectic ensemble is demonstrated. A duality between the underlying generating functions of the orthogonal and symplectic symmetry classes is semiclassically established

Two-point correlations of the Gaussian symplectic ensemble from periodic orbits

We determine the asymptotics of the two-point correlation function for quantum systems with half-integer spin which show chaotic behaviour in the classical limit using a method introduced by Bogomolny and Keating [Phys. Rev. Lett. 77 (1996) 1472-1475]. For time-reversal invariant systems we obtain the leading terms of the two-point correlation function of the Gaussian symplectic ensemble. Special attention has to be paid to the role of Kramers' degeneracy.Comment: 7 pages, no figure

Periodic-Orbit Theory of Universality in Quantum Chaos

We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory, that full classical chaos is paralleled by quantum energy spectra with universal spectral statistics, in agreement with random-matrix theory. For dynamics from all three Wigner-Dyson symmetry classes, we calculate the small-time spectral form factor $K(\tau)$ as power series in the time $\tau$. Each term $\tau^n$ of that series is provided by specific families of pairs of periodic orbits. The contributing pairs are classified in terms of close self-encounters in phase space. The frequency of occurrence of self-encounters is calculated by invoking ergodicity. Combinatorial rules for building pairs involve non-trivial properties of permutations. We show our series to be equivalent to perturbative implementations of the non-linear sigma models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of orbit pairs are one-to-one with Feynman diagrams known from the sigma model.Comment: 31 pages, 17 figure

Spectral Statistics for the Dirac Operator on Graphs

We determine conditions for the quantisation of graphs using the Dirac operator for both two and four component spinors. According to the Bohigas-Giannoni-Schmit conjecture for such systems with time-reversal symmetry the energy level statistics are expected, in the semiclassical limit, to correspond to those of random matrices from the Gaussian symplectic ensemble. This is confirmed by numerical investigation. The scattering matrix used to formulate the quantisation condition is found to be independent of the type of spinor. We derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation

Semiclassical form factor for chaotic systems with spin 1/2

We study the properties of the two-point spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the so-called diagonal approximation can be recovered. The incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics. Provided this skew product is mixing, we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory.Comment: 20 pages, no figure

Quantum cat maps with spin 1/2

We derive a semiclassical trace formula for quantized chaotic transformations of the torus coupled to a two-spinor precessing in a magnetic field. The trace formula is applied to semiclassical correlation densities of the quantum map, which, according to the conjecture of Bohigas, Giannoni and Schmit, are expected to converge to those of the circular symplectic ensemble (CSE) of random matrices. In particular, we show that the diagonal approximation of the spectral form factor for small arguments agrees with the CSE prediction. The results are confirmed by numerical investigations.Comment: 26 pages, 3 figure

Effects of common haplotypes of the ileal sodium dependent bile acid transporter gene on the development of sporadic and familial colorectal cancer: A case control study

<p>Abstract</p> <p>Background</p> <p>The genetics of sporadic and non-syndromic familial colorectal cancer (CRC) is not well defined. However, genetic factors that promote the development of precursor lesions, i.e. adenomas, might also predispose to CRC. Recently, an association of colorectal adenoma with two variants (c.507C>T;p.L169L and c.511G>T;p.A171S) of the ileal sodium dependent bile acid transporter gene (<it>SLC10A2</it>) has been reported. Here, we reconstructed haplotypes of the <it>SLC10A2 </it>gene locus and tested for association with non-syndromic familial and sporadic CRC compared to 'hyper-normal' controls who displayed no colorectal polyps on screening colonoscopy.</p> <p>Methods</p> <p>We included 150 patients with sporadic CRC, 93 patients with familial CRC but exclusion of familial adenomatous polyposis and Lynch's syndrome, and 204 'hyper-normal' controls. Haplotype-tagging <it>SLC10A2 </it>gene variants were identified in the Hapmap database and genotyped using PCR-based 5' exonuclease assays with fluorescent dye-labelled probes. Haplotypes were reconstructed using the PHASE algorithm. Association testing was performed with both SNPs and reconstructed haplotypes.</p> <p>Results</p> <p>Minor allele frequencies of all <it>SLC10A2 </it>polymorphisms are within previously reported ranges, and no deviations from Hardy-Weinberg equilibrium are observed. However, we found no association with any of the <it>SLC10A2 </it>haplotypes with sporadic or familial CRC in our samples (all P values > 0.05).</p> <p>Conclusion</p> <p>Common variants of the <it>SLC10A2 </it>gene are not associated with sporadic or familial CRC. Hence, albeit this gene might be associated with early stages of colorectal neoplasia, it appears not to represent a major risk factor for progression to CRC.</p