The effects of a strict cutoff on quantum field theory

James Fredric Sturnfield, Purdue University

Abstract

Standard Quantum Field Theory has a number of integrals which are infinite. Although these are eliminated for some cases by renormalization, this aspect of the theory is not fully satisfactory. A number of theories with fundamental lengths have been introduced as alternatives and it would be useful to be able to distinguish between them. In particular, we look at the effects that a strict cutoff would have on Quantum Field Theory. It is noted that care must be taken in the method used to apply a strict cutoff. We are lead to considering a theory where the cutoffs are defined by restricting each internal line. This theory is only piecewise analytic. The resulting scattering matrix is frame dependent, yet the theory still satisfies the special relativity view that all frames are subjectively identical. The renormalization of this theory is finite. The change in mass from the electron self-energy will be a spinor operator. The main distinctions of this theory from standard theory will occur at super high energies. We will find new poles and resonances which arise from new endpoint singularities. The locations of these singularities will be frame dependent. Some of these singularities will correspond to creations or interactions of the normal particles with tachyons. We will show that for the one loop diagram, the form of the cutoff singularities are closely related to the standard singularities. When there is more than one loop, there can appear some new type of behavior. In particular, we will find a cube root type of behavior in the two loop self-energy diagram. We also look at the asymptotic behavior of the ladder diagram. We find that in the center of mass frame the behavior is only slightly modified from standard Regge theory. In the lab frame, the behavior will be much more complicated.

Degree

Ph.D.

Advisors

Thurber, Purdue University.

Subject Area

Mathematics

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