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.
I recently sorted out the relation between the two-spinors of Penrose and Rindler and the usual QFT gamma-matrix/four-spinor formalism. The Penrose and Rindler formalism is entirely self-consistent and does not use gamma-matrices, although they do give a conversion in the notes. However, this conversion does not mesh with the usual QFT conventions; it is better to use another one.
For explicit formulas, see the appendix of http://arxiv.org/abs/1408.2729.
Peter,
1. If p^2=-m^2 does not look awkward to you then fine. It looks awkward however to many other people.
2. Weinberg’s equation 1.1.19 is missing a factor of “-i” next too kinetic term(compared to other textbooks). That factor was incorporated into the definition in 1.1.20 so that the metric in 1.1.21 is as you prefer. However standard textbooks like Landau, Bjorken-Drell or Itzykson-Zuber, leave the -i factor next to the derivative and so the Dirac operator is
D= -i\gamma^{\mu}\partial_{\mu} + m (see e.g. eqn 2-9 in I-Z)
while in Weinberg it is
D= \gamma^{\mu}\partial_{\mu} + m
because the gammas are redefined.
3. Come on Peter, it’s just a mathematical trick from complex analysis. That no-one so far has found a better way of defining QFT, well it’s still just a trick.
Moreover the Wick rotation plays a role only in QFT. It doesn’t play a role in QM or classical theory. Unlike the p^2=-m^2 which appears everywhere where particle dynamics is considered. To me it looks awkward as well as the Dirac equation (without the -i factor).
nicola,
Changing the space-time signature changes the definition of the gammas. As I tried to point out, one virtue of the conventions I advocate is that you can choose the gammas real. As for the Dirac equation, I think you’ve just provided another argument for my conventions: Weinberg’s Dirac equation with no gammas is simpler than one where you have to use i (and allows you to describe Majorana fields more simply).
Sorry, but I don’t think analytic continuation in QFT is just a “trick”. I’d claim it’s a fundamental insight into the nature of time (and the existence of anti-particles). Yes, you only need this in relativistic quantum field theory, but that’s the fundamental theory.
Am I the only one who thinks that the topic of this post qualifies as Not Even Wrong? Much more so than string theory, which by any reasonable metric is just wrong.
Sign conventions are like interpretations of QM – if no experiment can tell them apart, they are all equally right or wrong. But of course it is a bad idea to use several conventions in the same calculation, as the people who claimed to have found a muon anomaly 15 years ago did.
Peter,
Arguments that you find in favor of many+ convention I find actually against it. Well, I guess that’s why we call it a convention.
Many people would prefer to leave that -i factor because together with the derivative it gives the momentum operator and so the Dirac operator admits a very nice form D=gamma p + m. Using Weinberg’s approach we have D=i gamma p +m with an unnecessary factor of i only because you want p^2=-m^2.
By the way, another argument in favor of many- convention is the form of the action of the point particle. In many-, the Lagrangian is a square-root of (4-velocity)^2, something manifestly positive. In many+ you have a square-root of -(4-velocity)^2, with an awkward minus.
Just as a curiosity,
there is a famous writer of hard science fiction (the one which relies heavily on the science) called Greg Egan. He is a mathematician and is famous because his writings are the hardest science fiction ever.
In a trilogy of books (Orthogonal) he especulates with a universe with a Riemmanian metric instead of the Lorentzian of our universe. This creates very strange things, such as in that universe the speed of light varies with the frequency and that the “twin paradox” is upside down (time goes by faster with the twin that remains at home), among many many other things. I have not read the book, this is what I have discovered investigating about it.
This a detailed discussion in Egan’s web page, in case you are interested http://gregegan.customer.netspace.net.au/ORTHOGONAL/ORTHOGONAL.html#CC
I’m sorry to come late to this topic, but I learned only recently
of (IMHO profound) joint work of Graeme Segal and Maxim Kontsevich
on Wick rotation, discussed in a talk
https://www.youtube.com/watch?v=vTvXHL6ZJik
by GS at a birthday conference for MK last June. The key technical
point occurs around 25:00 in the video; in particular it seems
to me to provide convincing evidence that the East Coast convention
is NATURAL in the technical sense of the term.
Jack,
Thanks a lot. I agree that Segal’s work on Wick rotation (which as far as I know he has lectured about, but not really written up), is the deepest thinking about the question around. As you note, for what he’s saying to make sense, you need the East Coast metric (he actually very quickly but explicitly says that having 3 minus signs won’t work: another way of saying that analytically continuing three directions of space instead of one of time is a bad idea).
Hi Peter and thanks for a great article!
I’m having an argument with a friend over what Zee did and I’m hoping you can clarify it. Did Zee complexify the SO(3,1) in the book such that his statement is correct contextually, or is the excerpt you are showing simply wrong mathematically and not deducible from the introduction? Thank you very much!
Alma,
What Zee says would be correct if you complexify the two Lie algebras: what is true is that the complexified Lie algebra of SO(3,1) is the same as the complexified Lie algebra of the product of two copies of SU(2). This is not true for real Lie algebras, which is what the notation he is using refers to (if you ignore the fact that he’s using the notation for Lie groups to refer to Lie algebras…) My point was just that for him and Schwartz (and a lot of physicists), there is no difference between a real Lie algebra and its complexification.
The complexification of su(2) is sl(2, C), and of so(3,1) is so(4,C). So, what the “mathematically sophisticated” would really say is
so(4,C)=sl(2,C) + sl(2,C)
When you complexify a real Lie algebra so(p,q), you get something that doesn’t depend on the signature (so(q+p,C)), since once you have complexified you can just appropriately put in factors of i to make your inner product the standard one.
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