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At present, there exists no physically plausible example of a quantum fieldtheory for which the existence of solutions has been proven mathematically. TheClay Mathematics Institute has offered a prize for proving existence for aclass of Yang-Mills theories defined by Jaffe and Witten. This paper proposes amulti-stage strategy for proving existence for a quantum version of the"'tHooft Polyakov" (tP) field theory, and argues that this theory, while notrenormalizable or physically plausible as it stands, opens up a clear path to aphysically plausible well-defined theory. The key initial challenge is to firstprove stability for a classical version of this theory, in the spirit of WalterStrauss. The stability results of Bogomolnyi for classical PDE systems arewidely cited as a foundation of string theory, but they leave key questionsunaddressed, and may even call for small modifications of the tP model itself.This paper calls attention to the questions which require more attention, andproposes some partial possible answers. It also specifies a new numericalmethod to check for stability for the usual "hedgehog" types of solution, amethod which may have broader potential uses for any ODE problems whereaccuracy in derivatives is as important as accuracy in function values.

Stability Theory and the Foundations of Physics: A Challenge to Real Analysis and Numerical Calculation