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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 tex2html_wrap_inline624 , 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 tex2html_wrap_inline628 . 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. gif It follows now that G is positive, Hermitian with respect to the indefinite metric scalar product, and the same holds for tex2html_wrap_inline642 . 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