Abstract
This paper presents a spectral calculus for computing the spectra of causal Lorentz invariant Borel complex measures on Minkowski space, thereby enabling one to compute their densities with respect to Lebesque measure. The spectra of certain elementary convolutions involving Feynman propagators of scalar particles are computed. It is proved that the convolution of arbitrary causal Lorentz invariant Borel complex measures exists and the product of such measures exists in a wide class of cases. Techniques for their computation in terms of their spectral representation are presented.
Highlights
Let B(R4 ) denote the Borel algebra of R4 [1] and letB0 (R4 ) = {Γ ∈ B(R4 ) : Γ is relatively compact}. (1) + Hm= { p ∈ R4 : p 2 = m 2, p 0 > 0 }, (2) −= { p ∈ R4 : p 2 = m 2, p 0 < 0 }, (3)Let and be the mass shells corresponding to mass m > 0 (m = 0) and let
We have defined a spectral calculus that enables one to compute the spectrum of any causal
Lorentz invariant Borel complex measure on Minkowski space whose spectrum is a continuous function. This calculus can be used in many applications in quantum field theory (QFT) and leads to a method called spectral regularization [21]
Summary
Let B(R4 ) denote the Borel algebra of R4 (with respect to the Euclidean topology) [1] and let. As mentioned above one of the main results of the present paper is a presentation of a spectral calculus that enables one to compute the spectral function of a causal Lorentz invariant Borel complex measure on Minkowski space This spectral calculus is quite easy to use in practice but it is somewhat tedious to prove rigorously its validity.
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