On the square of x-n

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Abstract
The square of x-n is evaluated, using Raju's definition of the pointwise product of distributions, and shown to differ from x-2n by a finite multiple of the infinite (nonstandard) distribution corresponding to the square of delta (n-1). This result is used to derive identities establishing the finiteness of one-variable analogues of some propagator products in quantum electrodynamics. The one-variable identities may be applied to distributions f, g, concentrated on the null cone, lambda =0, by defining f( lambda )*g( lambda )=fg( lambda ) and regularising to include the vertex of the null cone. It is pointed out that this procedure would lead to a finite electron selfenergy without restricting the domain of the S matrix and without introducing an infinite difference between the bare and renormalised mass. It is concluded that any polynomial in delta - and its derivatives is finite and that unrestricted associativity holds for products for such polynomials.

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