DeWitt-Schwinger expansion for projective and Grassmann spaces
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.
scientific article; zbMATH DE number 4209172