Relativistic quantum theory is an entirely different subject from non-relativistic quantum mechanics (NRQM). Fundamentally, the theory only exists (in non-trivial form) as a field theory. NRQM can of course be reformulated as a field theory, but it is a very different (and simpler) affair in comparison to a relativistic theory, if for no other reason that in the non-relativsitic case one has a mass superselection rule, and indeed the automatic conservation of particle number.
It is otherwise with relativistic theory, where characteristically charge , rather than particle number, is conserved. What explains this difference?
A good deal of my early research (see my Ph.D. thesis) was concerned with understanding this question: why, as soon as one goes over to the Lorentz group, is the situation so completely transformed?
My answer is that there are two notions of complex number present in relativistic theory, which coincide in the non-relativistic case: and that a number of other characteristic features of RQT, including the absence of local number densities, and a position operator, can be traced to the same source. My conclusions were published in two related papers, dealing with fermions and bosons respectively:
But it is one thing to diagnose the problem, and in particular the origins of the localization problem in RQFT; another to solve or dissolve it. What, after all, do the localized entities we see in the laboratory correspond to? That is the question answered in:
Of the many other questions in the foundations of RQFT, here I will mention only two. One concerns the meaning of the zero-point energy, and the value of the cosmological constant. This in turn feeds through into the so-called fine-tuning problem. Download my Is the Zero-Point Energy Real? for an idea of the seriousness of the problem, and a possible solution. The other concerns the meaning of scale in physics, and specifically, scale as it is determined by the renormalization group. I have only worked on this at the margins, but for a review of some of the questions in this field, and the relationship of the renormalization group in RQFT to scaling in condensed matter physics, see my Critical Notice of Cao's book on field theory.
Copyright Simon Saunders 2001. Last updated: 7 October 2004.