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Renormalization, and the log jam in modern quantum field theory

Renormalization (NB: also spelled "Renormalisation") is a mathematical inconsistency in quantum field theory that is so well established that one is forced to either accept it without question or to use it as an excuse for avoiding the study of interacting quantum field theory altogether. Renormalization can be summarised as follows: developing quantum field theory from first principles involves applying a process known as "quantization" to classical field theory. This prescription, suitably adapted, gives a full dynamical theory which is to classical field theory what quantum mechanics is to classical mechanics, but it does not quite work. Things look fine on the surface, but the more questions one asks the more the cracks start to appear. Perturbation theory, which works so well in ordinary quantum mechanics, throws up some higher-order terms which are infinite, and cannot be made to go away. This was known about as early as 1928, and was the reason why Paul Dirac, who (along with Wolfgang Pauli) was the first to seriously investigate quantum electrodynamics, almost gave up on field theory. The problem remains unsolved to this day. Perturbation theory is done slightly differently, using an approach based on the pioneering work of Richard Feynman, but, other than that, nothing has changed. One seductive fact is that by pretending that infinite terms are not there, which is what renormalization is, the agreement with experiment is good. However: (i) there always is a tendency for the results of experiments to converge on the required theoretical answer and (ii) since renormalization is based on making infinity minus infinity equal anything you like, you can adjust inconvenient theoretical numbers as well. Renormalization, in short, consists of the turning of a blind eye to the mathematical inconsistencies of interacting field theory. The inconsistency, more than anything else, was what made me devote three years of my life (1984-1987) to finding a better way. My work obviously touched a raw nerve: I could not get anything other than the lowliest academic posts, other than by fluke, as was the case for Sussex, and very few institutions in the U.K. were even prepared to hear me speak. I only got into print because Steve Barnett, who was in a different but related field, God bless him, pulled some strings for me.

I did make some progress in my work, although at the end of it all, it looked as though I had slain the wrong monster. Before one gets to perturbation theory one has to use something called the Interaction Picture. A lesser-known fact is that it has been proved that this does not exist, a fact which is known as Haag's theorem (the text book developments of quantum field theory do not so much violate Haag's theorem as rape, pillage and burn it). My work, which built the theory up in a different way, replicated the results for scattering amplitudes, but did not use the interaction picture, hence the title of my paper: Quantum Electrodynamics without the Interaction Picture. However the new methods, although in almost all respects better than the text book ones, are still not wholly free of infinities. This seems to be something to do with the field equations being local. Quasi-local field equations, which I will define here as local field equations that have had the negative energy parts commuted to the right work alright (i.e. no infinities), but I would like to see a tidy proof that they satisfy the commutators (an untidy one is given in my paper).

I believe that our failure to really get on top of quantum field theory is the reason for the depressing lack of progress in fundamental physics theory. There have been no successes of theory that are not just applications of 1950's quantum electrodynamics or simple extensions of it. There is no proper field theoretic understanding of bound states (elementary quantum mechanics does far better in this regard than supposedly more advanced quantum field theory). There is no quantum theory of gravity. In regard to that which can be termed "quantum field theory", our sketchy and inconsistent understanding of scattering processes is all we really have to hold up to posterity. I believe that this will all change once we have learned to build a consistent interacting field theory from first principles. As for the drug trips: superstrings, higher dimensions, etc. I look on these as buying a ticket for the National Lottery. You dream of winning, but cannot seriously expect it. The future, I am sure, belongs to those who attack head on. To quote Lord Rutherford: never try to solve a difficult problem, which you may read here as never try to take more than one step at a time.

I might also add that the way that whole academic system is set up is not conducive to the production of interesting and original research. And this applies to all fields, not just theoretical physics. The tone is set by burned-out old men who have long since lost any real interest in research and seem to do very little other than teaching and politicking. Since they make the decisions about who succeeds them, the few of the next generation who are prepared to abase themselves to the required degree to stay in the system have to stick to mainstream, normally uninteresting research to attain the necessary credentials. To some it comes naturally, I have to admit. But to the ones who actually have something interesting to say, it definitely does not. Nor should it. Questioning is very often an upsetting, anarchic thing to do. This mismanagement probably reflects our society's lack of interest. You will note, of course, that whenever society, or rather the governments that represent it really wants results, something much more akin to a meritocracy emerges: witness the wartime research efforts to develop radar or atomic weapons. No reference was made here to status or sycophantic tendencies. The brightest and best were employed simply because they were the brightest and best. Would that our society would feel the same about peacetime research. In case there are any academics reading this: you are all guilty. Either by things that you have done, or things that you have failed to do. I can hardly think of a single situation of my own or any of my colleagues where boldness and originality have been rewarded.

Finally, let me relate a story told by one of the senior lecturers at coffee one day in the Oxford Theoretical Physics department:

A flood forces a man to stand on the roof of his house. A man in a rowing boat comes along and asks him if he can help. The man replies that no, he needs no assistance, as God is going to help him. Another man arrives in a boat, and gets the same response. Finally a helicopter arrives, a rope ladder comes down, and they tell him to climb up. He refuses, saying that God is going to help him. The man dies shortly after, and when he gets to heaven he asks God why he did not help him. An exasperated God says, "but I sent you two rowing boats and a helicopter!"

Think about it.