The use of exp(iS[x]) in the sum over histories

The use of $\sum \exp(iS[x])$ as the generic form for a sum over histories in configuration space is discussed critically and placed in its proper context. The standard derivation of the sum over paths by discretizing the paths is reviewed, and it is shown that the form $\sum \exp(iS[x])$ is justified only for Schrodinger-type systems which are at most second order in the momenta. Extending this derivation to the relativistic free particle, the causal Green's function is expressed as a sum over timelike paths, and the Feynman Green's function is expressed both as a sum over paths which only go one way in time and as a sum over paths which move forward and backward in time. The weighting of the paths is shown not to be $\exp(iS[x])$ in any of these cases. The role of the inner product and the operator ordering of the wave equation in defining the sum over histories is discussed.Comment: 22 pages, Latex, Imperial-TP-92-93-4

A Large-Scale Genetic Analysis Reveals a Strong Contribution of the HLA Class II Region to Giant Cell Arteritis Susceptibility

We conducted a large-scale genetic analysis on giant cell arteritis (GCA), a polygenic immune-mediated vasculitis. A case-control cohort, comprising 1,651 case subjects with GCA and 15,306 unrelated control subjects from six different countries of European ancestry, was genotyped by the Immunochip array. We also imputed HLA data with a previously validated imputation method to perform a more comprehensive analysis of this genomic region. The strongest association signals were observed in the HLA region, with rs477515 representing the highest peak (p = 4.05 × 10−40, OR = 1.73). A multivariate model including class II amino acids of HLA-DRβ1 and HLA-DQα1 and one class I amino acid of HLA-B explained most of the HLA association with GCA, consistent with previously reported associations of classical HLA alleles like HLA-DRB1∗04. An omnibus test on polymorphic amino acid positions highlighted DRβ1 13 (p = 4.08 × 10−43) and HLA-DQα1 47 (p = 4.02 × 10−46), 56, and 76 (both p = 1.84 × 10−45) as relevant positions for disease susceptibility. Outside the HLA region, the most significant loci included PTPN22 (rs2476601, p = 1.73 × 10−6, OR = 1.38), LRRC32 (rs10160518, p = 4.39 × 10−6, OR = 1.20), and REL (rs115674477, p = 1.10 × 10−5, OR = 1.63). Our study provides evidence of a strong contribution of HLA class I and II molecules to susceptibility to GCA. In the non-HLA region, we confirmed a key role for the functional PTPN22 rs2476601 variant and proposed other putative risk loci for GCA involved in Th1, Th17, and Treg cell function