  It is natural to consider a quantum system in the continuum limit of
space-time configuration. Incorporating also, Einstein's special relativity,
leads to the quantum theory of fields. Non-relativistic quantum mechanics and
classical mechanics are special cases. By studying vacuum expectation values
(Wightman functions W(n; z) where z denotes the set of n complex variables) of
products of quantum field operators in a separable Hilbert space, one is led to
computation of holomorphy domains for these functions over the space of several
complex variables, C^n. Quantum fields were reconstructed from these functions
by Wightman. Computer automation has been accomplished as deterministic exact
analog computation (computation over "cells" in the continuum of C^n) for
obtaining primitive extended tube domains of holomorphy. This is done in a one
dimensional space plus one dimensional time model. By considering boundary
related semi-algebraic sets, some analytic extensions of these domains are
obtained by non-deterministic methods. The novel methods of computation raise
interesting issues of computability and complexity. Moreover, the computation
is independent of any particular form of Lagrangian or dynamics, and is uniform
in n, qualifying for a universal quantum machine over C^infinity.
