I’m having a blast teaching the introductory quantum field theory class, as you may have gathered from several previous posts. It has been taking a lot of time, but I’ve been doing detailed computations with the students (and hence taking up a lot of preparation time) to make sure they really get how to compute in a quantum field theory (such as Quantum Electrodynamics (QED)) and see how it connects to the real world. Having spent time on the electron anomalous magnetic 
moment computation (I told you about that spectacular feature of QED earlier) we went back to basics to discuss in more detail the nature of the physics that is to be extracted from what are called one-loop diagrams – diagrams of the sort I drew before that have a closed loop in them.
This is where I get to try to clear up a huge conceptual issue that still afflicts most people’s popular-level knowledge of quantum field theory due to poor writing – the nonsense often spouted about there being “hidden infinities” and so on and so forth. Covering the poorly-named “renormalization” procedure and treating it in a more physical way to see that the whole “infinity” business is dreck can be fun, since it allows for an emphasis on a lot of the key issues I care about in the science that we do, such as the idea of effective field theory, the importance of separation of scales in your physics, and most importantly the reminder: Let’s not confuse the tools we use to describe Nature with Nature itself.
I like to use that phrase, and it comes back again and again, whether it is to do with 
basic concepts in quantum mechanics (no, that photon is neither a wave nor a particle… and that’s just fine, actually…), or quantum field theory (there’s nothing profound about that parameter in your model coming out infinite – measurable quantities are entirely finite just when they should be), or in current research (see my post “but is it real?“).
Anyway, such issues aside, as another excuse to play with this new pen nib I got, on the sides are some diagrams from QED we’ve been computing, involving procedures called things like “regularization” and “renormalization”. The three on the top left are (in a sense) more fundamental than the rest. This all will lead us to the important concept of the Renormalization Group next week…(In case you’re interested, the wiggly lines represent a photon moving along, mediating the electromagnetic interaction, while the straight lines represent electrons or positrons, depending upon which direction you read the arrow in. Pick a direction on the page and think of time ticking along as you go along that direction… the diagram tells a story of what can happen as these particles interact.)
-cvj
Euler: exp(iS) = cos S + i*sin S. Just drop the i*sin S term.
I agree that QFT rocks. Have you any thoughts about Haag’s theorem and problem of proving self-consistency of renormalization for Dirac’s interaction picture in Hilbert space? Someone naughty thinks cos S can be used to replace exp(iS) to overcome Haag’s problem in the path integral, because the direction of the resultant phase arrow (on an argand diagram) for exp(iS) is always parallel to the real axis (interferences only affect the magnitude – length – of the arrow, not its direction). Otherwise, the sum over histories or path integral would give cross-sections with complex numbers, which is unphysical. It turns out that Weyl first came up with the exp(iS) factor in 1918 to scale the metric of general relativity, and Schroedinger played with it from 1922-6 before coming up with the time-dependent Schroedinger equation, which Sirac solved in 1933 to give the interaction picture, exp(-iHt) ~ exp(iS). So it’s not really solid physics, just a guess from Weyl and Schroedinger that Bohr’s discrete energy levels can be modelled by the periodic real solutions to such an equation, like exp 0 = 1. (This is kind of crazy because the time-independent Schroedinger isn’t complex, and the switch-over between equations gives the wavefunction collapse in 1st quantization, as Thomas Love of CSU points out.) The short story is, forget exp(iS). Replace it with cos(iS). End of Haag’s problem, everybody is happy.
Thanks. Good choice. So much better than what I had to use (Bjorken and Drell) when I took QFT.
Hi,
Yes I did… and no I don’t.
-cvj
Hi Clifford,
Did you already mention the textbook that you’re using? Also, are you aware of any QFT-specific software against which one can check one’s calculations?
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Anonymous Snowboarder- sounds interesting. I have not read it yet…. Thanks!
-cvj
Well, I think science _does_ have reality…. Don’t get me wrong on that. I just think it is interesting and important to be constantly mindful of where our tools end and where the things we are describing them with begin…
Best,
-cvj
I am especially fond of these kind of drawings for some reason. It’s the abstraction, the utility, and the fact they’re handmade.
I also really enjoyed your post on ‘Is it real?’ My school here is interdisciplinary and part of our program is a seminar where we’ve been reading ‘Where Mathematics Comes From’, a book about embodied mathematics and the cognitive aspects of engineering new math tools. Funny how many people took the system for granted or had romantic notions about it. I’m doing some study on the period where math separated from Philosophy (according to Etymology Online science is first academically described as ‘non-arts’ about 1670). Your comment on the calculus is interesting because Newton went to college in 1661, and this issue of science supposing to have reality is the result of the separation of sciencific ideas from art ideas. I’ve really come to appreciate Newton as a master of cognitive engineering (making tools out of ideas).
C – have you read “Drawing Theories Apart” about the early years of Feynman diagrams and the different ways they were adopted for use in QED and later applied to completely different problems? While no direct application to your class per se, it might interest some students.