Abstract
The fundamental space ζ is defined as the set of entire analytic functions [test functions φ(z)], which are rapidly decreasing on the real axis. The variable z corresponds to the complex energy plane. The conjugate or dual space ζ′ is the set of continuous linear functionals (distributions) on ζ. Among those distributions are the propagators, determined by the poles implied by the equations of motion and the contour of integration implied by the boundary conditions. All propagators can be represented as linear combinations of elementary (one pole) functionals. The algebra of convolution products is also determined. The Fourier transformed space ζ̃ contains test functions φ̃(x). These functions are extra-rapidly decreasing, so that the exponentially increasing solutions of higher-order equations are distributions on ζ̃.
Highlights
The usual quantum field theories can be considered to be mathematically sound when they are constructed from expectation values of products of field operators
The fundamental space 5 is defined as the set of entire analytic functions [test functions &o(z)], which are rapidly decreasing on the real axis
We consider the fundamental space 5 as the set of all entire analytic functions cp(z) which are rapidly decreasing on the real axis, i.e., ~~(z>l,=~= p(x) is a function belonging to the space Y of Schwartz test functions
Summary
The usual quantum field theories can be considered to be mathematically sound when they are constructed from expectation values of products of field operators. Real exponential certainly lies outside the space Y ’of tempered distributions To enlarge this space in the sense we want, we have to take test functions that go to zero faster than any exponential of the type ePalxl (a>O). Such is the case for the example of the spaces yi, (O
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