Alternative potentials for the electromagnetic field

The electromagnetic field can be expressed in terms of two complex potentials $\alpha, \beta ,$ which are related to the Debye potentials. The evolution equations for these potentials are derived, which are separable either in parabolic coordinates (leading to the radiation fields) or in radial coordinates (multipole fields). Potentials corresponding to focused wave fields as well as plane waves are discussed. A conserved radiation density can be constructed in terms of these potentials, which is positive (negative) for positive (negative) helicity radiation.Comment: 13 pages, plainTex, slightly amended version of origina

Non-dispersive wavepacket solutions of the Schrodinger equation

The free Schrodinger equation has constant velocity wavepacket solutions \psi_{\bf v} of the form \psi= f({\bf r} - {\bf v}t) e^{- i m c^2 t / 2}. These solutions are eigenvectors of a momentum operator {\bf \tilde p} which is symmetric in a positive definite scalar product space. We discuss whether these \psi_{\bf v} can act as basis states rather than the usual plane wave solutions.Comment: 12 pages, parameters amended to yield correct dimension and new section added on relativistic extensio

Hundreds of variants clustered in genomic loci and biological pathways affect human height

Most common human traits and diseases have a polygenic pattern of inheritance: DNA sequence variants at many genetic loci influence the phenotype. Genome-wide association (GWA) studies have identified more than 600 variants associated with human traits, but these typically explain small fractions of phenotypic variation, raising questions about the use of further studies. Here, using 183,727 individuals, we show that hundreds of genetic variants, in at least 180 loci, influence adult height, a highly heritable and classic polygenic trait. The large number of loci reveals patterns with important implications for genetic studies of common human diseases and traits. First, the 180 loci are not random, but instead are enriched for genes that are connected in biological pathways (P = 0.016) and that underlie skeletal growth defects (P < 0.001). Second, the likely causal gene is often located near the most strongly associated variant: in 13 of 21 loci containing a known skeletal growth gene, that gene was closest to the associated variant. Third, at least 19 loci have multiple independently associated variants, suggesting that allelic heterogeneity is a frequent feature of polygenic traits, that comprehensive explorations of already-discovered loci should discover additional variants and that an appreciable fraction of associated loci may have been identified. Fourth, associated variants are enriched for likely functional effects on genes, being over-represented among variants that alter amino-acid structure of proteins and expression levels of nearby genes. Our data explain approximately 10% of the phenotypic variation in height, and we estimate that unidentified common variants of similar effect sizes would increase this figure to approximately 16% of phenotypic variation (approximately 20% of heritable variation). Although additional approaches are needed to dissect the genetic architecture of polygenic human traits fully, our findings indicate that GWA studies can identify large numbers of loci that implicate biologically relevant genes and pathways.

An Analysis of Two Genome-wide Association Meta-analyses Identifies a New Locus for Broad Depression Phenotype

AbstractBackgroundThe genetics of depression has been explored in genome-wide association studies that focused on either major depressive disorder or depressive symptoms with mostly negative findings. A broad depression phenotype including both phenotypes has not been tested previously using a genome-wide association approach. We aimed to identify genetic polymorphisms significantly associated with a broad phenotype from depressive symptoms to major depressive disorder.MethodsWe analyzed two prior studies of 70,017 participants of European ancestry from general and clinical populations in the discovery stage. We performed a replication meta-analysis of 28,328 participants. Single nucleotide polymorphism (SNP)-based heritability and genetic correlations were calculated using linkage disequilibrium score regression. Discovery and replication analyses were performed using a p-value-based meta-analysis. Lifetime major depressive disorder and depressive symptom scores were used as the outcome measures.ResultsThe SNP-based heritability of major depressive disorder was 0.21 (SE = 0.02), the SNP-based heritability of depressive symptoms was 0.04 (SE = 0.01), and their genetic correlation was 1.001 (SE = 0.2). We found one genome-wide significant locus related to the broad depression phenotype (rs9825823, chromosome 3: 61,082,153, p = 8.2 × 10–9) located in an intron of the FHIT gene. We replicated this SNP in independent samples (p = .02) and the overall meta-analysis of the discovery and replication cohorts (1.0 × 10–9).ConclusionsThis large study identified a new locus for depression. Our results support a continuum between depressive symptoms and major depressive disorder. A phenotypically more inclusive approach may help to achieve the large sample sizes needed to detect susceptibility loci for depression