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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.*

*Converted to HTML by Robert Coquereaux *

Tue Dec 29 10:54:13 WET 1998