XXII WORKSHOP ON GEOMETRIC METHODS IN PHYSICS
29 JUNE - 5 JULY 2003, BIAŁOWIEŻA, POLAND
Dmitri Vassiliev -
Do we really live in a Riemannian world?
The notions of metric and affine connection play a fundamental
role in classical differential geometry. The first one allows us
to measure distances, whereas the second allows us to define the
concept of parallelism (more precisely, parallel transport of
vectors).
Riemannian geometry is based on the assumption that the way we
measure distances determines the way we define the concept of
parallelism. Namely, the connection coefficients are assumed to be
expressed via the components of the metric tensor in accordance
with a certain explicit formula. Such a connection is called the
Levi-Civita connection, and such connection coefficients are
called Christoffel symbols. The Levi-Civita connection can be
invariantly characterised as the (unique) metric compatible
connection with zero torsion.
Non-Riemannian geometry is based on the assumption that the way we
measure distances is unrelated to the way we define the concept of
parallelism. Namely, there is no a priori relationship between the
connection and the metric. In particular, the connection may not
be metric compatible and (or) may have nonzero torsion.
We suggest a field theory [1] based on non-Riemannian geometry. We
choose an arbitrary Lorentz-invariant quadratic action and vary it
independently with respect to the metric and the connection. We
show that the solutions of the resulting field equations are
perfectly sensible. Einstein spaces turn out to be solutions, so
the theory reproduces the main effects of General Relativity.
Another solution is an explicitly constructed wave of torsion in
Minkowski space; this solution may be interpreted as a model for
an elementary particle.
[1] D.Vassiliev, Pseudoinstantons in Metric-Affine Field Theory,
"General Relativity and Gravitation" 2002 vol. 34 p. 1239-1265.