AN APPLICATION OF NEUTRIX CALCULUS TO QUANTUM FIELD THEORY

Neutrices are additive groups of negligible functions that do not contain any constants except 0. Their calculus was developed by van der Corput and Hadamard in connection with asymptotic series and divergent integrals. We apply neutrix calculus to quantum field theory, obtaining finite renormalizations in the loop calculations. For renormalizable quantum field theories, we recover all the usual physically observable results. One possible advantage of the neutrix framework is that effective field theories can be accommodated. Quantum gravity theories will then appear to be more manageable.

The neutrix convolution product inZ′(m)and the exchange formula

One of the problems in distribution theory is the lack of definition for convolutions and products of distribution in general. In quantum theory and physics (see e.g. [1] and [2]), one finds that some convolutions and products such as1x⋅δare in use. In [3], a definition for product of distributions and some results of products are given using a specific delta sequenceδn(x)=Cmnmρ(n2r2)in anm-dimensional space. In this paper, we use the Fourier transform onD′(m)and the exchange formula to define convolutions of ultradistributions inZ′(m)in terms of products of distributions inD′(m). We prove a theorem which states that for arbitrary elementsf˜andg˜inZ′(m), the neutrix convolutionf˜⊗g˜exists inZ′(m)if and only if the productf∘gexists inD′(m). Some results of convolutions are obtained by employing the neutrix calculus given by van der Corput [4].