VECTOR FIELDS
AND THE UNITY OFMATHEMATICS AND PHYSICS
by
Daniel HenryGottlieb
Department of Mathematics
Purdue University
West Lafayette, Indiana 47906
Abstract
We give an argument that magnetic monopoles should not exist. It is based o*
*nthe
concept of the index of a vector field. The thrust of the argument is that indi*
*ces of vector
fields are invariants of space-time orientation and of coordinate changes,and t*
*hus physical
vector fields should preserve indices. The index is defined inductively bymean*
*s of an
equation called the Law of Vector Fields. We give extended philosophical argume*
*nts that
this Law of Vector Fields should play an important role in mathematics, andwe b*
*ack up
this contention by using itin a mechanical way to greatly generalize the Gauss-*
*Bonnet
theorem and the Brouwer fixed point theorem and get new proofs of many other th*
*eorems.
1. Introduction
We will give an argument that magnetic monopoles donot exist. Magnetic monop*
*oles
were predicted by Dirac [F]based on an alteration of Maxwell's equations which *
*made them
more symmetric. Despite Dirac's ideas, magnetic monopoles have not been found i*
*n nature.
This is the case even though Dirac used similar considerations to predict antip*
*articles and
other phenomena.
Our argument goes as follows: First we state a general principle which we wi*
*ll assume
holds for every "physical vector field."
Principle of Invariance of Index. The index of any "physical" vector field is
invariant under changes of coordinates and orientation of space-time. It must b*
*e physically
significant. If the index is undefined, it signals either radiation or unrealis*
*tic physical
hypotheses.
Consequence. Every "physical" pseudo-vector field has index zero or the inde*
*x is
undefined.
Now the magnetic vector field !B is a pseudo-vector field. That means if we *
*change
the orientation of space!B changes to !B. Now Ind(V ) = (1)nInd(V ) where n is *
*the
dimension of the manifold on which V is defined. Thus Ind(!B ) = (1)3Ind(!B ). *
*So
either Ind(!B ) is not defined or Ind(!B) = 0.
Now a magnetic monopole will give rise to a !Bwith index 1. As this is incon*
*sistent
with the Invariance of Index Principle, we predict that magnetic monopoles do n*
*ot exist.
We have some arguments which tend to explain why the principle of invariance*
* of index
is reasonable. These arguments concern things mathematical rather than physical*
*. They
are not theorems, rather they are predictions and explanations of some things w*
*hich will
happen or have happened in mathematics.
These considerations lead us to the prediction that a certain equationdue to*
* Marston
Morse, [M], will play avery active role in mathematics, and by extension physic*
*s. This
equation,which we call the Law of Vector Fields was discovered in 1929 and has *
*not played