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.