"[Renormalization is] just a stop-gap procedure. There must be some fundamental change in our ideas, probably a change just as fundamental as the passage from Bohr's orbit theory to quantum mechanics. When you get a number turning out to be infinite which ought to be finite, you should admit that there is something wrong with your equations, and not hope that you can get a good theory just by doctoring up that number."
Quantum Field Theory purports to be the most fundamental of sciences in that it concerns the ultimate constituents of matter. The term "quantum field theory" is used interchangeably with "particle physics" and "high energy physics" on the grounds that the experimental support for this theory comes from expensive experiments involving high-energy beams of particles. Although such multi-billion-dollar experiments are needed to push the boundaries, the theories of course claim to be universal, and should apply equally to the familiar and everyday world.
"The shell game that we play ... is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate."
- Richard Feynman, Nobel laureate 1965
Current practitioners in the field will no doubt bemoan the fact that taxpayers of the world are increasingly less willing to find the money to pay for this esoteric study. Do they really care whether there are Higgs particles or heavier flavours of quarks? Probably not. Why not? Because it really makes no difference to their own lives. Their message is clear: study theology, philosophy or "useless" branches of science if you will, but if the cost is more than a few academic salaries, blackboards and chalk, then don't expect us to pay.
It was not always thus. In the late seventies and early eighties, many in the general public followed the unfolding drama of the fundamental particle world with keen interest, and did not seem to mind about the cost of the experiments. It was, after all, a prestige project, like the Apollo program.
A lot of it had to do with the names: "quarks" with their different "flavours" and "colours", with qualities like "strangeness" and "charm". The "gluons" that bound them together. The enormous and mind-blowing scale of the experiments at CERN or Fermilab or SLAC. The enthusiastic (if largely uncomprehended) explanations by academics in the field, with their diagrams and manic gesticulations. At one time, it seemed that every other popular science program on TV was about particle physics. Indeed, I remember a marathon one on BBC2 around 1976, whose climactic moment was completing of the pattern of SU(3) Hyperons by the Omega Minus. A particle predicted by theory and subsequently discovered by experiment. What could be more satisfying? I remember in particular interviews with Abdus Salam and Richard Feynman. Salam did his plug for Indian culture by talking about the domes of the Taj Mahal and how their symmetry made them beautiful. His point was that this principle of symmetry could be applied to physics as well. The other thing I remember was Feynman saying that he was not entirely comfortable with "gauge" theories, but then he was an old timer, and what did he know? (Looking back on it now, that was rich, coming from him, since he won a Nobel prize for the original gauge theory - quantum electrodynamics).
It was hard not to get swept up in this. Oxford's contribution at the time (1980) was a series of public lectures at Wolfson College, the most memorable of which was given by Murray Gell-Mann, one of the leading lights in the field.
Both Tim Spiller, my tutorial partner, and I wanted to do research in the field, and both of us succeeded. He did a Ph.D. at Durham University and I a D.Phil. at Oxford, following a one-year course at Cambridge to study the relevant mathematics. Tim and I were the bane of our tutors as undergraduates because of the way we would never accept "hand-waving" (unrigorous) explanations. I like to think that the good side of this fussiness was that the theses we eventually produced (in totally different branches of field theory) were of higher quality than average.
Apparently the thing about gauge theories was that they were "renormalizable". Just what renormalization and gauge theories were I did not discover until I went to Cambridge.
This renormalization failed the "hand-waving" test dismally.
This is how it works. In the way that quantum field theory is done - even to this day - you get infinite answers for most physical quantities. Are we really saying that particle beams will interact infinitely strongly, producing an infinite number of secondary particles? Apparently not. We just apply some mathematical butchery to the integrals until we get the answer we want. As long as this butchery is systematic and consistent, whatever that means, then we can calculate regardless, and what do you know, we get fantastic agreement between theory and experiment for important measurable numbers (the anomalous magnetic moment of leptons and the Lamb shift in the Hydrogen atom), as well as all the simpler scattering amplitudes.
"You may have eleven significant figures of agreement, but you cheated to get it, and so it does not count," I say.
"What does it matter," they say. "This can't be a coincidence. What we have here has got to be the best theory ever."
"It's not a theory," I say. "It's just rubbish."
As long as I have known about it I have argued the case against renormalization. On the other hand I did want to get my degree, so I just chose research that avoided confronting the issue. I tried to get to grips with some of the things that one has to deal with before getting to renormalization: issues related to non-interacting fields, such as the spin-statistics theorem and Lagrangians for particles of higher spin. There were a few interesting things to explore, mostly in clarifying the connection between work in the area and the underlying principles, which is what I did for my doctoral thesis. This was done by March, 1984, which left me with a few months in hand, so I started looking at renormalization again to see if I could make any more sense of it second time round.
I then discovered something very nice. If the field is written as a power series in the coupling constant, then the field equations enable a simple reduction of an interacting field in terms of the free field and any amplitude can be calculated just by inspection. I wrote this up in a preprint here . The idea was so simple that I found it hard to believe that I was the first to see it. Well, there is nothing new under the sun, and sure enough - as I discovered in late 2005/early 2006 - what I had was an old idea that had just withered on the vine. Stueckelberg1 thought of it first, in 1934, but Källén2 (who seems not to have been aware of Stueckelberg's work), also thought of it in 1949.
The idea has two consequences that ought to have given the founders of quantum mechanics a lot of grief. First, the local field equations they would expect to be able to use give nonsensical, infinite answers (a feature, incidentally, of every other treatment of quantum field theory). Secondly, properties such as orthonormality of a basis of particle states at constant time no longer apply. For the former, I think that the answer is to not use rigidly local field equations. The only reason for choosing one field equation over another is to get agreement with experiment (although normally in such a way as to incorporate experimentally-founded beliefs in invariance principles such as special relativity). If local field equations give infinite answers then obviously they are not agreeing with experiment. However, it is possible to make an adjustment - known as "normal-ordering" - which eliminates the problem, at least in the Källén-Stueckelberg approach. The latter problem is Haag's theorem: in the presence of interactions, it is always assumed that the Hamiltonian can be split into a "free" part and an "interaction". The "free" part is used to define an orthonormal basis of states to which the interaction applies. But Haag's theorem says that this is not possible, or to put it another way, it is not possible to construct a Hamiltonian operator that treats an interacting field like a free one. Haag's theorem forbids us from applying the perturbation theory we learned in quantum mechanics to quantum field theory, a circumstance that very few are prepared to consider. Even now, the text-books on quantum field theory gleefully violate Haag's theorem on the grounds that they dare not contemplate the consequences of accepting it.
However, in my view, acceptance of Haag's theorem is a very good place to start. The next paper I wrote, in 1986, follows this up. It takes my 1984 paper and adds two things: first, a direct solving of the equal-time commutators, and second, a physical interpretation wherein the interaction picture is rediscovered as an approximation.
With regard to the first thing, I doubt if this has been done before in the way I have done it3, but the conclusion is something that some may claim is obvious: namely that local field equations are a necessary result of fields commuting for spacelike intervals. Some call this causality, arguing that if fields did not behave in this way, then the order in which things happen would depend on one's (relativistic) frame of reference. It is certainly not too difficult to see the corollary: namely that if we start with local field equations, then the equal-time commutators are not inconsistent, whereas non-local field equations could well be. This seems fine, and the spin-statistics theorem is a useful consequence of the principle. But in fact this was not the answer I really wanted as local field equations lead to infinite amplitudes. It could be that local field equations with the terms put into normal order - which avoid these infinities - also solve the commutators, but if they do then there is probably a better argument to be found than the one I give in this paper. Substituting Haag expansions (arbitrary sums of normal-ordered tensor products of free fields) directly into the commutators is an obvious thing to try here. I did make fumbling attempts around October 2001, without making much progress, but I think that with a little more ingenuity, a solution could be found.
With regard to the second thing, the matrix elements consist of transients plus contributions which survive for large time displacements. The latter turns out to be exactly that which would be obtained by Feynman graph analysis. I now know that - to some extent - I was just revisiting ground already explored by Källén and Stueckelberg4 .
My third paper applies all of this to the specific case of quantum electrodynamics, replicating all scattering amplitudes up to tree level. As for reproducing the "successes" of traditional QED, namely the Lamb shift and the anomalous magnetic moment of leptons, I do not know. I would want to be confident that I had an understanding of bound states before I attempted a Lamb shift calculation and would want to be sure I understood the classical limit of the photon field before I tried to calculate the anomalous magnetic moment of leptons. Finding time to do it is the problem.
Here is the correspondence I had with the journals. It seems that my greatest adversaries were the so-called "axiomatic field theorists", who not content just to disagree, appeared to be determined to ensure that nothing I wrote ever got into print. Maybe if I had acknowledged and referenced their own work things might have been different. However there was, and is no real reason to do so. I am amazed that so many supposedly intelligent people can work for so many years and have so little of interest to show for it. In particular, results obtained for spacetimes other that those with three space and one time dimension are not relevant. Do it in 3+1 dimensions, or not at all. I am also surprised that none of them seemed to be aware of the work of Källén or Stueckelberg, or at least to have recognised the connection between these papers and my own. Was this laziness on their parts, or did they genuinely not know? The process of selecting papers for publication in scientific journals seems mostly to be about enforcing conformity. But, to borrow Tony Blair's phrase, I do not wear unpopularity as a badge of honour. If I could have conformed, I would have. I have not gone off on this tangent to make some kind of point, but simply because it was the best way I could see at the time of solving the relevant problems.
In a nutshell, my proposal is this: write the interacting fields as sums of tensor products of free fields. Use coefficients in the expansion that are almost those which follow from the usual local equations of motion. I say "almost" because the terms must appear in normal order. Then use the known properties of free fields to evaluate the matrix elements directly. Comparison of these expressions with time-dependent perturbation theory (from ordinary quantum mechanics) shows that these consist of transients plus the tree-level Feynman graph amplitudes. As no re-definition of the mass, coupling or field operators is required (infinite or otherwise), there is no renormalization.
Unfortunately for me, though, most practitioners in the field appear not be be bothered about the inconsistencies in quantum field theory, and regard my solitary campaign against infinite subtractions at best as a humdrum tidying-up exercise and at worst a direct and personal threat to their livelihood. I admit to being taken aback by some of the reactions I have had. In the vast majority of cases, the issue is not even up for discussion.
The explanation for this opposition is perhaps to be found on the physics Nobel prize web site. The five prizes awarded for quantum field theory are all for work that is heavily dependent on renormalization. These are as follows:
Although by these awards the Swedish Academy is in my opinion endorsing shoddy science, I would say that, if anything, particle physicists have grown to accept renormalization more rather than less as the years have gone by. Not that they have solved the problem: it is just that they have given up trying. Some even seem to be proud of the fact, lauding the virtues of makeshift "effective" field theories that can be inserted into the infinitely-wide gap defined by infinity minus infinity. Nonetheless, almost all concede that things could be better, it is just that they consider that trying to improve the situation is ridiculously high-minded and idealistic. None that I have talked to expect to see a solution in their lifetimes. They think it possible that the universe might have 10, 11 or 26 dimensions (according to Edward Witten's mood that day), but they absolutely do not believe that calculations (in four dimensions) that they studied when they were graduate students can be done without mathematical sleight of hand. Neither do any appear to be interested in investigating the possibility. As with a lot of things, Feynman had a nice way of putting it: "Renormalization is like having a toilet in your house. Everyone knows it's there, but you don't talk about it." But personally, I do not see how fundamental physics can move on until the problem is solved. Before it can be solved, it needs to be addressed, and before it can be addressed, it needs to be acknowledged. But nowadays, even to get the problem acknowledged is hard, so we have a situation where if you asked a theoretical particle physicist, "Would you like to be able to calculate the Lamb Shift without any renormalization?" you would get the answer, "Of course!" But if you then asked, "Are you prepared to sponsor a project that attempts this?" the answer would always be "No". Personally, I believe that in taking this attitude, they only harm themselves. Botching may be an unavoidable part of many practical endeavours, where deadlines have to be met and customers have to be satisfied, but on the research frontier there can be no justification.
For those of you who are swayed by arguments from authority (I like to think that I am not one of them, by the way), one could almost make the case against renormalization on these grounds. Backing the view of one Nobel prizewinner (Dirac) against the twelve listed above could be justified by saying that excluding Feynman - who in any case had plenty of doubts about renormalization - Dirac made more impact on physics than the others put together.
Physicists are first and foremost scientists. They are not primarily mathematicians and they are not religious zealots (at least not in regard to work). The extent to which they are permitted to believe their explanations is the extent to which they are verified in experiments. They therefore are entitled to strong faith in quantum mechanics and special relativity, both of which seem to pass all of the multifarious experimental tests thrown at them. They are also entitled to believe in vector particles mediating the weak interaction. They are entitled to believe in quarks. The following however are less certain: general relativity as the theory of gravity and quantum chromodynamics as the theory of the "strong" nuclear force.
To take the first, the "proofs" of General Relativity are light bending, the precession of the perihelion of Mercury and gravitational red-shift. All of these are tiny effects, and whilst the results do not contradict G.R., they do not mean that it is the only possible explanation either. General Relativity is like quantum mechanics in that it is not so much a theory as a whole way of thinking, and it can be very hard to fit something as grandiose as this with other frameworks, quantum mechanics in particular. If there is a conflict between quantum mechanics and G.R. then the scientist (if not the mathematician) is forced to choose quantum mechanics. With gravity the experimental data, or at least, data that cannot be explained by Newtonian gravity are incredibly sparse compared to the results that support quantum mechanics. What we would like are experiments that test gravity at the microscopic level, in the same way that Quantum Optics tests electromagnetism at the quantum level, but will we ever get these? The inability to get an experimental handle on quantum gravity makes me wonder whether it even exists at all in its own right. Might gravity be just some kind of residual of other forces, like the van der Waals attraction in chemistry? Assuming that this notion is wrong, what about strong gravitational fields? The fact is that we know nothing about the world of strong gravitational fields, a fact which has not stopped Astrophysicists have giving names to objects that are supposed to have such, such as neutron stars and black holes. Unfortunately, an observatory is not a laboratory. It is very hard to understand or even demonstrate the existence of such objects unless you have a degree of control over them.
The other area of uncertainty is, to my mind, the "strong" nuclear force. The quark model works well as a classification tool. It also explains deep inelastic lepton-hadron scattering. The notion of quark "colour" further provides a possible explanation, inter alia, of the tendency for quarks to bunch together in groups of three, or in quark-antiquark pairs. It is clear that the force has to be strong to overcome electrostatic effects. Beyond that, it is less of an exact science. Quantum chromodynamics, the gauge theory of quark colour is the candidate theory of the binding force, but we are limited by the fact that bound states cannot be done satisfactorily with quantum field theory. The analogy of calculating atomic energy levels with quantum electrodynamics would be to calculate hadron masses with quantum chromodynamics, but the only technique available for doing this - lattice gauge theory - despite decades of work by many talented people and truly phenomenal amounts of computer power being thrown at the problem, seems not to be there yet, and even if it was, many, including myself, would be asking whether we have gained much insight through cracking this particular nut with such a heavy hammer.
This article would not be complete
without saying more about Superstring theory. If this was just a quiet
mathematical backwater, as it was twenty years ago, I would not object. But it
now arrogantly claims not only to be physics, but to be about to deliver a
"theory of everything", despite the overwhelming evidence to the contrary. This
makes me glad that I am not in the academic mainstream these days5.
Brave souls like Peter Woit point
out the utter lack of any scientific value of this ridiculous phenomenon, but
unfortunately, they are merely voices crying in the wilderness. I have heard
the argument that runs: if enough people believe it then it must be true,
and to that I give the answer, five million lemmings can't be wrong. There
are plenty of historical precedents of the majority getting it wrong, but when
a person calling him/herself a scientist comes out with statements like, it's
just so beautiful that it must be true, or Superstrings are the
language in which God wrote the world, then you just know that such
persons must be wrong. They are not scientists, they are merely religious
zealots. Unfortunately, science is the opposite of religion. A scientist is a
professional sceptic, and as such is obliged to discard theories that cannot be
substantiated by experiment. In the absence of some unforseeable dramatic
breakthrough, this has to include Superstrings. There is in any case the
obvious philosophical objection to choosing complicated explanations when the
simple ones have not been exhausted (Occam's razor).
Although I never believed that the Superstring program had much hope of success, the thing that really clinched it for me was a behavioural change in my fellow young researchers around 1985. I was used to being able to have completely frank discussions about physics with my peers. An outsider might have perceived these discussions as unnecessarily adversarial, but for us it was merely part of the fun of doing physics. The sea change around 1985 was that the fight suddenly went out of them. Rather than try to answer my criticisms, they would plead that Superstring theory was just so wonderful that one had to just go with the flow. There was little satisfaction to be derived from baiting converts to this new religion, especially as they were becoming so influential, so I soon gave up. But acquiescing to the fashion did not necessarily guarantee academic careers, as within a short time most, like me, were out of academia anyway. An exception here was Brian Greene, the nearest thing to a rock star that theoretical physics can currently provide. Although I obviously disagree with him about Superstrings, I concur with Peter Woit in observing that if there was anyone who could talk me into it, it would be Brian6.
One might ask what I think that they should have done instead. Well, here are some tips for those entering particle physics research for the first time. They may not help you get a job, but they might at least help you to feel better about yourself:
So where next? As well as Superstrings, we should also abandon Feynman-Dyson perturbation theory, which has had its day and now needs to be consigned to the dustbin of history. Then, using the better-founded Källén-Stueckelberg covariant perturbation theory, we should try to learn how to do bound states in quantum field theory. We could then, possibly, talk about progress.
I am not aware of anyone qualified having studied my arguments in detail unless they were forced to, as with the journal referees. This may be something to do with the herding instinct (see illustration) or they may see my work as an attempt to fix something (renormalized perturbation theory) that is not broken. I do not know. Despite all of this, I still do quantum field theory from time to time and am working (slowly) on a "text book" developing everything from first principles. There are a number of things to to follow up here, the most important probably being the bound state problem. I believe that as well as giving a handle on the Lamb shift, one might also get a more satisfactory framework for understanding hadrons.
But let me be clear about renormalization: particle physicists have only got away with this because they have convinced the rest of the scientific community that they are smarter than them. If (say) an electronic engineer got a divergent integral in his calculations and then - rather than finding out where exactly he went wrong - invented an infinite "counterterm" to subtract to render the integral finite, he would be laughed at.
Given the unappetising nature of the renormalization program I have a degree of sympathy for graduate students who choose to join the Superstring bandwagon in preference. They should, however, not be blind to the fact that the right of Superstring researchers to be housed in physics departments diminishes with each year that passes and each cul-de-sac explored. Sooner or later their bluff will be called, and when it is, they will be forced to answer the question, how was it that this non-physics-based speculation was able so successfully to starve out so many alternative lines of enquiry? Indeed, the so-called anthropic principle to me seems to indicate that even now, things are getting desperate. Here the lack of predictive ability of Superstrings leads one to the conclusion that nothing can ever be predicted. Interestingly, though, despite deciding that all further scientific research is pointless, advocates of the anthropic principle still continue to write papers, give talks and to draw their salaries.
Some of these people are eminent physicists nearing the end of their careers, who seem to be trying hard to convince the media that rather than admit that Superstrings has been a failure, particle physicists ought instead to just lower their standards in regard to what gets classified as science.
This is utter foolishness, and goes a long way toward undoing the good work done by teachers of physics and popularizers such as Brian Greene in getting people interested in the subject. They may have given up, but I have not, nor have most others who would wish to be called scientists. I believe in the scientific method. I believe that it is good to be able to describe nature using a few principles written in the language of mathematics. I believe that one should never give in to the temptation to cheat. I believe that if a theory is not agreeing with experiment, or is leading to no testable results at all - as is the case for Superstrings - then it should be abandoned. I also believe that the public's support for the whole enterprise should not be taken for granted. They provide the money, and if the majority of the fundamental physics community is spending its time stoned on a hyper-dimensional drug trip then they will eventually realize, and funding will be cut. With the subject in its current state, it would be a mercy killing: it deserves no better, and it is not satisfying in any way to point out that some of us could see that the subject had taken a wrong turning more than twenty years ago. One such contrarian, though, managed, with some luck, to stay in the academic profession, and now holds a permanent but untenured job at a major American university. He has written at some length on these matters ...
A book by Peter Woit, a lecturer in
mathematics at Columbia University. Top: US edition, bottom:UK
edition. Published 2006.
The title derives from an anecdote involving Wolfgang Pauli. The version I heard runs as follows:
A student is explaining a wonderful new theory he has developed to Pauli. Pauli listens in silence, and then finally announces, "Your theory is not correct. In fact, your theory is not even wrong. Your theory makes no predictions at all."
Not even wrong (the statement) describes the current state of Superstring theory. If theoretical particle physicists seriously believed that their mission was to gain understanding of the physical world, rather than just to pursue interesting but pointless hobby horses, books like this would not need to be written.
But as things stand, with the ground now littered with the corpses of attempts at "ultimate" theories, of which Superstrings is the most visible, Dr. Woit's extensive post-mortem has become essential reading.
One
of the points that Dr. Woit makes is that the working practices of theoretical
particle physicists are wrong. There is a pronounced tendency for everyone to
work on the same thing. He contrasts this with mathematics, where people are
more apt to become experts in very specific - and different - areas. The reason
for the mathematicians being right and the physicists wrong is very simple:
research consists of venturing into unknown territory. Since one does not have
the answers in advance, a "shotgun" approach, where many different avenues are
explored at once is much more likely to lead to a positive result than the
manic pursuit of a single idea. The physicists' philosophy is so ingrained that
many, including myself, have been denied the opportunity to continue research
simply because they were not part of any bandwagon. In my case, the arguments
were not even considered! It is ironic that it should be at the time that the
subject is engaged in its most speculative endeavour since the Middle Ages that
it should be least tolerant of dissent.
This book will be most satisfying to those who already have a background in elementary particle physics. Having said that, the more technical portions are more often than not inessential to the central argument, and one need not feel guilty about skipping them. As is his right, Dr. Woit has used the opportunity to slip in his own views about the development of quantum mechanics, views which are notable on account of the rarity value of having been thought through fully by the author himself. I am not sure about their relevance, but I found them fascinating reading, especially as, like the author, I prefer the formal approach.
For a review by science writer Robert Matthews (supportive of Dr. Woit's point of view), click here.
1 Relativistisch invariante Störungstheorie des Diracschen Elektrons, by E.C.G. Stueckelberg, Annalen der Physik, vol. 21 (1934). My German not being everything it should be, I have relied on a pre-digested version of this paper given here: The Road to Stueckelberg's Covariant Perturbation Theory as Illustrated by Successive Treatments of Compton Scattering, by J. Lacki, H. Ruegg and V. L. Telegdi (1999). My thanks to Danny Ross Lunsford for drawing attention to the latter.
2 Mass- and charge-renormalizations in quantum electrodynamics without use of the interaction representation, Arkiv för Fysik, bd. 2, #19, p.187 (1950) and Formal integration of the equations of quantum theory in the Heisenberg representation, Arkiv för Fysik, bd. 2, #37, p.37 (1950). Both by Gunnar Källén. These are actually in the 1951 volume in the library I used (the Bodleian in Oxford).
3 Actually, this paper: On quantum field theories, Matematisk-fysiske Meddelelser, 29, #12 (1955), by Rudolf Haag, solves spacelike commutators in §5, but for the restricted case of Φ3 theory, and just to first order in the power series expansion. Unlike my analysis, Haag places no restrictions on the first time derivative of the field, and finds a slightly more general solution, where some derivative couplings are allowed.
4 A few comments, though. Stueckelberg uses the power series expansion in the coupling, and the residue of the energy conservation pole to provide the physical interpretation. Something very similar is used in my papers. However, Stueckelberg is not properly second quantized: his photons are modes in a cavity, and his electrons are wave functions rather than field operators. Fermions cannot be created or destroyed and so the only process he can treat is Compton scattering (the scattering of a single photon off a single electron). Interestingly, his subsequent papers seem to indicate that he soon abandoned the covariant approach, instead switching to, and often anticipating, the non-explicitly-covariant S-matrix methods of Dyson, Feynman and Schwinger. Could he have given up because of problems with the Interaction Picture? We may never know. Given that his 1934 methods are so much simpler, more elegant and more powerful, I am still amazed that he would willingly stop working on them. Källén's papers, sixteen years later, are properly second quantised, but his physical interpretation is less elegant as his best ambition is just to reproduce Dyson's results, which, strictly, apply only to scattering processes. Källén's papers would be easier to read if he made more use of four-dimensional momentum space. He works out a graphical representation which he claims are just Feynman diagrams, but they are not. They are more like the diagrams in my papers which have two kinds of line for each particle type. The two types of line are confusingly drawn the same in his paper, even though he then goes on to calculate the amplitudes correctly by treating them differently.
5 It may well be that I could have stayed in academia had I not followed the Sinatra Doctrine so early on. This would, however, have meant doing work that I found totally depressing. My reasoning in the end was that if I was going to do something I did not really want to do then it should at least be useful to someone, technical and reasonably well paid. This path led eventually to financial software. The academic system then, as now, was set up to leave the young person entirely at the mercy of a club of burned-out old men. They felt entirely justified in not supporting any of my job applications simply because I had not chosen to work on any of their own projects. All very well, except for this: almost all of the important steps in theoretical physics in the 20th century were taken by young people. With the increasing failure to take account of the fact that innovative thinking is primarily the prerogative of the young person, I am not in the least bit surprised at the lack of progress in theoretical physics in recent years.
6 Oh, and by the way, Brian - if you get to read this - my great-uncle Willem van Stockum was Dutch, not Scottish, as claimed in The Fabric of the Cosmos. Also, since his degree was in Mathematics, and all his subsequent research was carried out in Mathematics departments, one ought to classify him as a mathematician rather than a physicist.
