Date: Apr 1, 2013 2:18 PM Author: Rock Brentwood Subject: Solution to the Mass Gap Problem in submission to Annalen der Physik A long-standing open problem in axiomatic quantum field theory, it is

widely believed that the intractibility of the Mass Gap Problem

reflects a gap in the knowledge or understanding of foundational

principles in physics; and has attracted the attention of the Clay

Math foundation, which placed a bounty on the solution of the problem,

expressing the hopes and expectation that its resolution will involve

the formulation of some kind of newer foundation to quantum field

theory, itself.

The following is the extended abstract of a paper awaiting submission

to Annalen der Physik, which contains such a far-reaching

reformulation that involves a simultaneous resolution to the Coleman-

Mandula Theorem, Raifeartaigh Theorem, Leutwyler ("No Interaction")

Theorem and Haag Theorem; and entails, as a consequence of the

resolution of the latter theorem, a reformulation of Quantum Field

Theory that contains a solution to the Mass Gap Problem.

Reference:

M.W. Hopkins, "Unification of Galilei, Poincare and Euclidean

Symmetry",

Quantum Topology / Hopf Algebra Seminar, UI-Chicago, 2008 October 7,

http://www.math.uic.edu/seminars/view_seminar?id=998

On the Relativistic Dynamics of Moving Bodies

Extended Abstract

In the 20th century we came to a revised understanding of just what

classical physics actually was, over three centuries after its initial

formulation by Newton. Unfortunately, it came a few decades after the

older paradigm had already been superseded by the theory of

relativity.

More unfortunately, still, is that this newer understanding was never

properly reintegrated into the newer paradigm, thereby unwittingly

cutting the moorings loose from any simple formulation of the

Correspondence Limit. Had our revised view originally been present in

the formative 1905-1908 period, it would have also entailed an altered

course in the development of relativity, leading to an outcome that,

itself, represents an upward revision of what we presently understand

it to be.

Consequently, the development of the theory of relativity is not yet

complete.

When the revision is made (restoring a more cohesive form of the

Correspondence Limit in the process), the result is that the paradoxes

comprising the Haag Theorem, Coleman-Mandula Theorem, Raifeartaigh

Theorem and Leutwyler (?No Interaction?) Theorem can all be resolved,

thereby leading the way to a bona fide relativistic dynamics of

interacting bodies ? both in the classical and quantum settings. In

the latter case, this entails non-trivially interacting quantum

systems composed of a finite number of degrees of freedom per unit

volume and leads directly to a reformulation of a quantum field theory

free of infinities, in which the mass gap problem may be resolved in

the affirmative.

This modification represents the integration of Newton's Third Law

into relativity, where it has been conspicuously absent up to now;

with the integration containing the kernel of the A-R approach to the

Self-Force problem, as a consequence.

It leads to a unified framework that not only encompasses relativistic

and non-relativistic dynamics, but also the (similarly revised)

descriptions of free and interacting systems that conform to the

(anti-)De Sitter, Newton-Hooke, Para-Poincare', Para-Galilei, Carroll,

Static, Euclidean, Spherical and Hyperbolic groups, in the process

also providing two separate routes of connection and transformation

between the 3+1 Lorentzian and 4+0 Euclidean forms of dynamics.

In this paper we will spell out the details of the modification, the

revised formulation of dynamics and, within the new unified framework,

shall construct counter-examples to each of the four No Go theorems

cited. The most important of these counter-examples is of the Coleman-

Mandula theorem and involves a reconstruction of the electroweak Higgs

model that contains a predicted value for the Higgs mass that falls

within the currently established experimental bounds. We will also lay

out a reformulation of quantum field theory and prove, within this

reformulation, the existence of a mass gap for gauge theories

constructed from a large family of Langrangians that includes the Yang-

Mills Lagrangans.