DIFFERENTIAL GEOMETRY AND ELEMENTARY PARTICLES

P.K.Smrz

Key words: Elementary particles, manifolds, connection,
        spacetime.

1991 Math. Subject Classification: 83D05, 53B50.

Introduction

Applications of the differential geometry to theoretical physics may be
broadly divided into two classes.

In the first class the differential geometry plays a role of a mathematical
tool that helps to formulate and solve problems in a basically finished
physical theory. Classical newtonian mechanics is the best example. The
theory itself is complete, but it contains number of unsolved problems
which become more transparent when the geometrical concepts like symplectic
manifolds and associated structures are introduced.

The second class of applications is more concrete in the sense that the
geometrical structures are closely related to the structure of the physical
world. The main example is Einstein's theory of gravitation. We do not live
in symplectic manifolds of the classical mechanics, they just happen to fit
the mathematical structure of the theory, but in certain sense we do live
in the space-time of the theory of relativity. It is this type of applications
of the differential geometry which I want to address in the present
contribution.

The remarkable success of Einstein's theory and its ties with the
differential geometry generated a strong feeling among scientists that
Physics as such is in fact a branch of applied differential geometry. It has
been known since the beginning of the twentieth century that the world is
fundamentally composed from fields and particles. For the sake of unified
description many physicists tried to think of particles as of special or
singular field configuration. Einstein himself tried to represent a particle by
Schwarzschild solution of his equations. It was already known that particles
behave according to the strange rules of quantum theory. Einstein hoped
that the quantum behaviour could somehow be derived from the gravitational
field equations which, due to their nonlinear character, could in principle
lead to unexpected results.${ }^{[1]}$ His attempts remained unsuccessful,
and today it is clear why. I want to list two main reasons.

Firstly, a singularity of a classical field like in the Schwarzschild
solution will always have a definite position in space, which precludes
explanation of the double-slit experiment performed with individual particles.
In Einstein's time such experiments were always performed with particle
beams where the interference was understood to be of one particle with another,
while in the present experiments the single particle must somehow pass
through both slits in order to interfere with itself.

Secondly, it is difficult to see how the spin of fermions could be faithfully
represented using only the real four-dimensional space-time geometry (by what
I do not want to say that people never tried it). In order to describe
fermions under spatial rotations one has to introduce complex vectors which
do not form a natural part of the original geometry.

If one maintains that the whole physical world including both fields and
particles is to be described by geometrical properties of some fundamental
continuum, then one has to search for such a continuum beyond the space-time
manifold. In fact, one should expect that the existence of the space-time
manifold itself is only a manifestation of the properties of the fundamental
continuum, a manifestation that depends on the method of observation. In the
present contribution I want to sketch a few main points of such a search,
concentrating on the role of the differential geometry.