# Metric tensor transformed into momentum space?

by Jay R. Yablon
Tags: metric, momentum, space, tensor, transformed
 P: n/a Metric tensor transformed into momentum space? wrote in message news:eg6pj2$4bl$1@skeeter.ucdavis.edu... > Jay R. Yablon wrote: >> The metric tensor g_uv is typically understood to be a function >> of >> spacetime coordinates, that is, g_uv(x^u). > >> Has anyone ever considered a metric tensor Fourier transformed into >> momentum space, > >> g_uv(p^u) = int {d^4x exp (-2 pi i p x) g_uv(x^u)}? > > In what coordinate system? The quantity in your exponent is not > well-behaved under coordinate transformations. Neither is your > integral -- only the integral of a scalar density (or equivalently > a four-form, in four dimensions) is independent of the choice of > coordinates. > > Steve Carlip Yes, agreed. Please permit me to modify this and be more specific. Think of, say, scattering experiments where one has fermion with an initial momentum p, final p' and a photon momentum q. Suppose we have a metric tensor which is a function of all three, g_uv(p',q,p) (similarly, say, to the Gamma_u(p',q,p) which are used to represent perturbations at the fermion/photon vertex -- I am thinking specifically of an anticommutator defined by g_uv = .5{Gamma_u Gamma_v} and trying to use this as a metric tensor because this does reduce to the Minkowski metric n_uv for q-->0, that is, g_uv(p,0,p)=n_uv). Suppose we specify a "momentum differential element" dp^u, (u=0,1,2,3) which may be represented in some coordinate system of our choosing. Is there any reason we cannot define an "invariant mass interval" dm in momentum space, such that: dm(p',q,p)^2 = g_uv(p',q,p) dp^u dp^v (1) where in momentum space, the usual spacetime elements are replaced by ds(x^u)-->dm(p',q,p); dx^u-->dp^u; g_uv(x^u) --> g_uv(p',q,p), and the Fourier transform, in keeping with what you said above about using a well-behaved four-form operating on a scalar density, is: dm(p',q,p)^ = int {d^4x exp (-2 pi i p x) ds(x^u)^2}? (2) Thus, in the simple case, if we were to choose Cartesian coordinates, the q-->0; g_uv(p,0,p)-->n_uv expression for (1) would be: dm(p,0,p)^2 = dE^2 - dp_x^2 - dp_y^2 - dp_z^2 (3) Thanks. Jay R. Yablon _____________________________ Jay R. Yablon Email: jyablon@nycap.rr.com