Holomorphic Quantization on the Torus and Finite Quantum Mechanics

We construct explicitly the quantization of classical linear maps of $SL(2, R)$ on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of $SL(2, Z)$ to the subgroup $SL(2, Z)/\Gamma_l$, $\Gamma_l$ being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, $O_{l}(2)\subset SL(2,l)$, for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.Comment: 12 pages LaTeX (needs amssymb.sty). Version as will appear in J. Phys.

Analytic Representation of Finite Quantum Systems

A transform between functions in R and functions in Zd is used to define the analogue of number and coherent states in the context of finite d-dimensional quantum systems. The coherent states are used to define an analytic representation in terms of theta functions. All states are represented by entire functions with growth of order 2, which have exactly d zeros in each cell. The analytic function of a state is constructed from its zeros. Results about the completeness of finite sets of coherent states within a cell are derived

Is Ankyrin a genetic risk factor for psychiatric phenotypes?

Background Genome wide association studies reported two single nucleotide polymorphisms in ANK3 (rs9804190 and rs10994336) as independent genetic risk factors for bipolar disorder. Another SNP in ANK3 (rs10761482) was associated with schizophrenia in a large European sample. Within the debate on common susceptibility genes for schizophrenia and bipolar disorder, we tried to investigate common findings by analyzing association of ANK3 with schizophrenia, bipolar disorder and unipolar depression. Methods We genotyped three single nucleotide polymorphisms (SNPs) in ANK3 (rs9804190, rs10994336, and rs10761482) in a case-control sample of German descent including 920 patients with schizophrenia, 400 with bipolar affective disorder, 220 patients with unipolar depression according to ICD 10 and 480 healthy controls. Sample was further differentiated according to Leonhard's classification featuring disease entities with specific combination of bipolar and psychotic syndromes. Results We found no association of rs9804190 and rs10994336 with bipolar disorder, unipolar depression or schizophrenia. In contrast to previous findings rs10761482 was associated with bipolar disorder (p = 0.015) but not with schizophrenia or unipolar depression. We observed no association with disease entities according to Leonhard's classification. Conclusion Our results support a specific genetic contribution of ANK3 to bipolar disorder though we failed to replicate findings for schizophrenia. We cannot confirm ANK3 as a common risk factor for different diseases

Genome-wide study of association and interaction with maternal cytomegalovirus infection suggests new schizophrenia loci.

Genetic and environmental components as well as their interaction contribute to the risk of schizophrenia, making it highly relevant to include environmental factors in genetic studies of schizophrenia. This study comprises genome-wide association (GWA) and follow-up analyses of all individuals born in Denmark since 1981 and diagnosed with schizophrenia as well as controls from the same birth cohort. Furthermore, we present the first genome-wide interaction survey of single nucleotide polymorphisms (SNPs) and maternal cytomegalovirus (CMV) infection. The GWA analysis included 888 cases and 882 controls, and the follow-up investigation of the top GWA results was performed in independent Danish (1396 cases and 1803 controls) and German-Dutch (1169 cases, 3714 controls) samples. The SNPs most strongly associated in the single-marker analysis of the combined Danish samples were rs4757144 in ARNTL (P=3.78 × 10(-6)) and rs8057927 in CDH13 (P=1.39 × 10(-5)). Both genes have previously been linked to schizophrenia or other psychiatric disorders. The strongest associated SNP in the combined analysis, including Danish and German-Dutch samples, was rs12922317 in RUNDC2A (P=9.04 × 10(-7)). A region-based analysis summarizing independent signals in segments of 100 kb identified a new region-based genome-wide significant locus overlapping the gene ZEB1 (P=7.0 × 10(-7)). This signal was replicated in the follow-up analysis (P=2.3 × 10(-2)). Significant interaction with maternal CMV infection was found for rs7902091 (P(SNP × CMV)=7.3 × 10(-7)) in CTNNA3, a gene not previously implicated in schizophrenia, stressing the importance of including environmental factors in genetic studies

GBA and APOE ε4 associate with sporadic dementia with Lewy bodies in European genome wide association study

Dementia with Lewy Bodies (DLB) is a common neurodegenerative disorder with poor prognosis and mainly unknown pathophysiology. Heritability estimates exceed 30% but few genetic risk variants have been identified. Here we investigated common genetic variants associated with DLB in a large European multisite sample. We performed a genome wide association study in Norwegian and European cohorts of 720 DLB cases and 6490 controls and included 19 top-associated single-nucleotide polymorphisms in an additional cohort of 108 DLB cases and 75545 controls from Iceland. Overall the study included 828 DLB cases and 82035 controls. Variants in the ASH1L/GBA (Chr1q22) and APOE ε4 (Chr19) loci were associated with DLB surpassing the genome-wide significance threshold (p < 5 × 10). One additional genetic locus previously linked to psychosis in Alzheimer's disease, ZFPM1 (Chr16q24.2), showed suggestive association with DLB at p-value < 1 × 10. We report two susceptibility loci for DLB at genome-wide significance, providing insight into etiological factors. These findings highlight the complex relationship between the genetic architecture of DLB and other neurodegenerative disorders