Let us consider now a relativistic Dirac particle.[31]
The main point in our approach
is to treat the relativistic case as a non-relativistic one, but replacing
time t with "proper time" parameter , and replacing Hamiltonian H
with Fock-Schwinger super-Hamiltonian. Explicitly:
We will take the standard representation of gamma matrices:
and define indefinite metric space by
where .
The Dirac matrices are Hermitian with respect to this scalar product, and so is the
Dirac operator:
Let us consider now a particle position detector which, for simplicity, is at rest with respect to the coordinate system. We associate with it the operator G defined by
where g(x) is a positive, bell-like function centered over the detector
position.
It follows now that G is positive, Hermitian with respect to
the indefinite metric scalar product, and the same holds for
. We postulate the following relativistic version of the PDP
algorithm:
The above prescription is not the only one possible. But it has one
very important property: the algorithm is independent of any local
observer. It is, in fact, a somewhat strange algorithm - it
works as if it was quite natural for Nature to be working in more than
four space-time dimensions.