I’m trying to get back to blogging about quantum mechanics slogans, this one is about relativistic quantum mechanics. I’m hoping it will stir up more trouble than my last East vs. West one.
If you’re doing calculations in relativistic quantum field theory, you typically handle the Minkowski nature of space-time by introducing an indefinite signature metric, and there are two possible choices:
- Mostly minus signs (for the spatial components), positive sign for the time direction. This is commonly known as the “West Coast metric”, I’m guessing because Feynman used it.
- Mostly plus signs (for the spatial components), negative sign for the time direction. This is commonly known as the “East Coast metric”, I’m guessing because Schwinger used it.
While I was educated on the East Coast, most courses I took and textbooks I used favored the West Coast metric, and that’s very much true of more recent textbooks (I don’t know of a recent one using the East Coast convention). When I started on this project I used the West Coast convention. After a while though, I finally found this conceptually more and more confusing, and switched to the East Coast convention. As time has gone on, I’ve become more and more convinced that this is the right convention to choose, that the West Coast convention is just a mistake, and a source of conceptual confusion in the subject.
Here are some reasons:
- With the East Coast convention, the treatment of spatial coordinates is just like in the non-relativistic case. In the West Coast convention, as far as space goes, you have decided to work with a negative definite metric, which is a quite misguided thing to do for obvious reasons.
- With the East Coast convention, if you do what you always do to make a QFT well-defined, analytically continue to imaginary time, you end up working with the standard Euclidean metric in four dimensions. In the West Coast version, you end up with a negative definite metric, again a bad idea (thanks to Peter Orland for emphasizing this to me). You could instead do your analytic continuation by analytically continuing all three of the space coordinates, also a really bad idea.
- With the East Coast convention, the Clifford algebra Cliff(3,1) is the algebra of real four by four matrices. In other words, you can choose your gamma-matrices to be real matrices and work with a real spinor representation (the Majorana representation). Going with the West Coast, Cliff(1,3) is the algebra of two by two quaternionic matrices, a confusing thing to work with. Ignoring that, what physicists end up doing is working with gamma matrices that are pure imaginary, which is highly confusing (odd powers of gamma matrices are pure imaginary, even ones real). According to Figueroa-O’Farrill, in this case you are working with:
pseudo-Majorana spinors – a nebulous concept best kept undisturbed.
To be fair, for this problem you can do what he does, and just change the sign in your definition of the Clifford algebra.
- Weinberg’s quantum field theory textbook uses the East Coast convention.
One reason this issue came to mind is that I’ve been trying to understand (not very successfully…) Schwinger’s old papers on Euclidean quantum field theory, where he makes some quite interesting claims. Schwinger used the East Coast convention for this, and as explained above, it’s only with this convention that you get something sensible after analytic continuation in time. There is very little literature following up on Schwinger’s arguments, I’m suspecting partly because in the West Coast convention following Schwinger’s arguments becomes virtually impossible.
The problem here is part of a more general problem, that I think most physicists don’t appreciate the mathematical concept of “real structure” or complexification. Given any formulas, they’re happy to just start using complex numbers, even when the quantities involved are real, with the idea that only at the end, when you get observable numbers, do you need to impose some reality condition. For an example of this, one often finds in qft texts claims that sound very strange to mathematicians, I have in mind especially things like
The mathematically sophisticated say that the algebra SO(3,1) is isomorphic to SU(2) X SU(2) (Tony Zee, QFT in a nutshell, page 113 first edition).
One problem here is that Zee is using the standard math notation for the Lie group to denote the Lie algebra, an unfortunately common practice. But the real problem is that the two Lie algebras are only isomorphic if you complexify, as real Lie algebras they are quite different. If you have always from the beginning complexified everything, this distinction doesn’t make sense to you, but it is often an important one if you want to really understand what is going on in some calculation involving spinors. For another example, there’s
the 3D rotation algebra, which has multiple names so(3)=sl(2,R)=so(1,1)=su(2), due to multiple Lie groups having the same algebra. So we have shown that so(3,1)=su(2) + su(2) (Matthew Schwartz, QFT and the Standard Model, Page 162).
Schwartz properly distinguishes by notation between the group and the Lie algebra, so is in better shape than Zee, but, again, so(3,1)=su(2) + su(2) is only true for complexified Lie algebras. The statement “so(3)=sl(2,R)=so(1,1)=su(2)” suffers from a typo (so(1,1) should be so(2,1)), but again an identification is being made that only makes sense if you have complexified everything. What’s really true here is that you have SL(2,R) a double cover of SO(2,1), and sl(2,R)=so(2,1), as well as SU(2) a double cover of SO(3), with su(2)=so(3). But SU(2) is a very different Lie group than SL(2,R) (although they share the complexification SL(2,C)).
So, my modest proposal is that the HEP community should just admit that the West Coast convention was a mistake, and rewrite all the textbooks (Weinberg doesn’t have to…).
Update: A commenter tells me there is at least one recent textbook with the right convention, Srednicki’s.
Checking some books, I remembered one other intriguing recent choice, that of Michael Dine, who wrote the first half of his book (the QFT part) in the West Coast metric, but the second half (the string theory part) in the East Coast metric.
Update: For those interested in how to translate back and forth between Coasts in the two-spinor notation, I noticed that Dreiner, Haber and Martin have written review papers, with a line in the tex that lets you choose which Coast. See here and here.
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