DeWitt-Schwinger expansion for projective and Grassmann spaces
type: Item
instance of: scholarly article
title: DeWitt-Schwinger expansion for projective and Grassmann spaces
zbMATH Open document ID: 0731.58064
DOI: 10.1007/BF00401641
author: N. N. Shtykov
author: Vladimir Lyakhovsky
author: D. V. Vassilevich
published in: Letters in Mathematical Physics
publication date: 1991-01-01T00:00:00Z
review text: Calculation of the DeWitt-Schwinger coefficients (DWSC) for the internal manifold give important information on the quantum corrections on the Kaluza-Klein models. The authors propose a method of calculating these coefficients on d-dimensional coset-spaces G/H from the asymptotics of the series \[ (1)\quad (x,e^{z\square}x)_{IJ}=\frac{1}{(4\pi)^{- d/2}}z^{-d/2}\sum^{\infty}_{k=0}a_ k(x,x)_{IJ}z^ k \] where \(\square\) is the Laplace-Beltrami operator for scalar fields and the squared Dirac operator for spinors and I, J denote spinor indices in the latter case while x is a local coordinate on G/H. The authors reexpress the left-hand side of (1) in the form (2) \((\cdot)_{IJ}=\delta_{IJ}\sum_{m,\xi}d_ m(Vd)^{-1}e^{-M^ 2_ m(\xi)t}\), where \(\xi\) labels multiple components D(H) in the branching \(D^{(m)}\downarrow H\), \(d_ m=\dim D\), \(d=\dim D(H)\), \(V=vol(G/H)\) and \(M^ 2_ m(\xi)\) are eigenvalues of \(\square\) restricted to \(D^{(m)}\). Further they make detailed harmonic expansion calculations for symmetric spaces of dimension \(d\leq 8\) using the negative Young tableaux method. Finally the DWSC's are evaluated in the case of symmetric spaces with \(d\leq 8\) by an asymptotic analysis of (1) and (2) as \(t\to 0\) using a method of analytic continuation of the associated \(\zeta\)-function.
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 83E15
Mathematics Subject Classification ID: 81R99
zbMATH DE Number: 4209172
zbMATH Keywords: quantum corrections
zbMATH Keywords: heat kernel
zbMATH Keywords: DeWitt-Schwinger coefficients
zbMATH Keywords: symmetric spaces
zbMATH Keywords: \(\zeta \) -function
MaRDI profile type: Publication
Wikidata QID: Q124840382
cites work: Algebraic approach to Kaluza-Klein models
cites work: Q4095018
full work available at URL: https://doi.org/10.1007/bf00401641
OpenAlex ID: W2060876503
description: scientific article; zbMATH DE number 4209172