The West Coast Metric is the Wrong One

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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61 Responses to The West Coast Metric is the Wrong One

  1. auxsvr says:

    One very common mistake physicists make is that the ladder operators taught in quantum mechanics are elements of su(2), but they’re elements of its complexification, as su(2) is a real algebra. This distinction is essential if one needs to study group theory in depth for quantum gravity etc.

  2. Peter Woit says:

    auxsvr,
    Yes, that’s the same issue I’m talking about. Raising and lowering operators occur in representations of the complexification sl(2,C), not of su(2).

  3. Peter Morgan says:

    The invariant distinction for 4-vectors (in flat Minkowski space) is between space-like definite, light-like, and time-like definite (or semi-definite or indefinite, and with similar distinctions for bivectors and for spinors), so I suppose it’s better to formulate geometrical argument in these terms rather than in terms of +0- (or -0+). For the algebra, this is just one of very many sign and factor conventions (anyone for a five-minute argument about 2\pi?); we have no choice but to be clear which conventions we are using and to keep perfect track of them all. But I think signs and factors are the least of my worries when trying to understand other people’s notations and formalisms (as you note above, people are not as clear as could be about whether they are working over the real or complex field).

  4. Sudip Paul says:

    Mark Srednicki also uses the mostly plus metric in his QFT textbook. Also I was always confused by this East Coast/West Coast nomenclature. Why can’t people simply say mostly plus/mostly minus?

  5. Peter Woit says:

    Sudip Paul,
    Thanks, Srednicki’s book is one of the ones I don’t have, I suppose I should get a copy. Glad to know that there’s at least one recent example of the right convention.

    Instead of East/West, or mostly plus/mostly minus, how about right/wrong?

  6. Alex says:

    While I agree with your conclusion, I think I should point you to this preprint indicating the Pin groups may cause physically-meaningful phenomena…depending on which signature you use…well, neutrinoless beta decay can only be described in the East coast metric.

  7. Gandalf says:

    Someone has to defend the “West coast metric” here, it seems. For in that metric, energies p^2 are positive, as they should be! Ok, you have to accept that spatial distances x^2 are negative .. you can’t have it all. But who wants to impose a negative energy condition? I rather travel -1000 miles 😉

  8. Al says:

    Interestingly I’ve never heard of this “West/East Coast” nomenclature. Instead when I studied it, we had “the one that particle physicists use” (mostly minus) and “the one that general relativists use” (mostly plus). If I remember correctly, the mostly plus one stems from Minkowski’s geometric formulation of special relativity (introducing imaginary time coordinate), so it’s the one in use by relativists.

  9. Al says:

    By the way, that might explain Michael Dine’s choice. When string theorists want to emphasize string theory’s potential as TOE and connect it to gravity, they use mostly plus (“East Coast”) one.

  10. John Baez says:

    I agree that it’s better to take spacelike vectors to have v · v > 0, because this convention makes spacetime geometry backwards-compatible with the usual way of describing the geometry of space. This is how I do things in my book Gauge Fields, Knots and Gravity.

    However, the convention for Clifford algebras requires another decision, as you note. And this decision is lot more interesting to me. Should a vector with v · v = 1 give a square root of 1 in the Clifford algebra, or a square root of -1? Thinking about the relation between Clifford algebras and division algebras, I decided that square roots of -1 were more natural here. You can see the thought process hiding in here: starting from the idea of a normed division algebra, you can quickly see that an n-dimensional normed division algebra is the same thing as an n-dimensional representation of a Clifford algebra generated by n-1 anticommuting square roots of -1. Another way to put it is that a skew-adjoint orthogonal matrix – that is, a matrix lying both in the Lie group O(n) and the Lie algebra o(n) – must be a square root of -1.

    By contrast, your argument that 2×2 matrices of quaternions and purely imaginary gamma matrices are “confusing” seems based on taste rather than theorems.

    (My argument that spacelike vectors should have v · v > 0 is also based on taste – or rather, history: I’m saying this is nice because it’s the convention we’re used to from ordinary Euclidean and Riemannian geometry. But my argument that square roots of -1 are better than square roots of 1 is based on theorems. The simplest of these theorems, of course, is that taking the real numbers and adjoining a square root of -1 gives a field, the complex numbers, while adjoining a square root of 1 does not.)

  11. Noboru Nakanishi says:

    The use of the words, “East Coast Metric”and “West Coast Metric”, is unwelcome; high-energy physicists are not necessarily Americans. I once used the words, “Space-favored Metric” and “Time-favored Metric”. The former is more convenient than the latter except in the momentum-space consideration. But it must be emphasized that this is the matter of convenience but not the matter of the right/wrong business.

    As was noted by Schwinger (and independently by Nakano), the x of a field operator has no intrinsic metric signature in quantum field theory. People just introduce the Minkowski metric by hand. Similarly, in general relativity, the Einstein equation contains no information about metric signature. One just introduces the Lorenzian metric signature by hand as the boundary condition. From these facts, I proposed that the quantity x should be something living in the space which has no metric; concretely, the space of x should be an affine space. Time should be embedded into this space. I formulated quantum Einstein gravity on this standpoint. One of the consequences of this formulation is that SUSY is a wrong symmetry!

  12. Sudip Paul says:

    Just remembered, Polchinski’s two volumes of String Theory are completely in the mostly plus convention.

    Btw, you might wanna change the title of this post to “West Coast Metric Considered Harmful.” 🙂

  13. Magnema says:

    Personally, I prefer the -+++ convention. However, I do see the following advantage in the +— convention: obviously, when it makes squared spacial intervals negative, it makes squared time intervals positive. Practically speaking, the only paths one cares (that I’ve seen in my studies of classical GR) are the geodesics of particles, which are timelike curves, thus why would would prefer timelike curves to have positive length.

  14. Peter Woit says:

    John,
    I agree that the question of the convention for Clifford algebras is a much more interesting one. I’ve used both, when I first was teaching representation theory I used the negative one, see
    http://www.math.columbia.edu/~woit/notes17.pdf
    for pretty much the same reason you mention (that Clifford algebras are generalizations of C, or H, where the generators have square -1). More recently I changed to the positive convention, I forget exactly why. Bourbaki does use the positive convention, and this is the kind of thing they usually put a lot of thought into.

    I do think it’s not just a matter of taste that if you want to work over the reals with an algebra that is isomorphic to an algebra of real matrices, you should use real matrices, and not introduce an “i” into your calculations (e.g. when thinking about Majorana fermions using a Clifford algebra isomorphic to M(4,R)). It is a very interesting question though whether the right Clifford algebra to think about for Minkowski space is M(4, R) or M(2,H) (which is equivalent to the question of which sign to use), they are two different real Clifford algebras (and as Alex points out, the Pin groups are different). My understanding of Schwinger’s argument was that he had a “Euclidean principle”, that what you see in Minkowski space is only things that analytically continue from Euclidean space, and this explains why you don’t see Majorana fermions (a real spinor representation). I don’t know if his argument would change if he was using the opposite convention and the Clifford algebra was M(2,H).

  15. Robert Delbourgo says:

    I guess the metric you choose depends on how you were brought up, which then gets ingrained in your brain. I was brought up as a particle person so use the +— choice. It really does not matter all that much although I do appreciate Peter’s arguments for the other choice.

  16. John Fredsted says:

    To mind comes the following passage from the monograph Clifford Algebras and Spinors, by Pertti Lounesto:

    Physicists might want to observe that the Clifford algebras Cl(3,1) = Mat(4,R) and Cl(1,3) = Mat(2,H) are not isomorphic as associative algebras, even though both of them have the same complexification M(4,C) …

  17. Ralph says:

    Much as I’m tempted to agree with you, the only really convincing point there is Weinberg.

  18. Mike Duff says:

    Type IIB supergravity is an example of a “bi-coastal’’ theory: its equations of motion are the same in either east or west conventions. This is because it exhibits a metric reversal symmetry g_ab goes to minus g_ab.

    hep-th/0605273
    hep-th/0605274

  19. manfred Requardt says:

    I think, the dividing line runs typically between high energy physicists and general relativists. In QFT one usually likes to have timelike orbits of particles to have positive length, while in GR one likes to have the spatial metric to be positive.

  20. Anon says:

    Well, on the “west coast” (mostly minus) side, I’d like to note that even the venerable Wald switches to (+,-,-,-) when discussing spinors in curved spacetime (see the discussion under equation 13.1.18).

  21. Theo Nieuwenhuizen says:

    Einstein was thinking physically rather than mathematically. His Lagrangian has a positive kinetic term with a plus in front (west coast convention, being on the east coast). It is indeed annoying that the physical choice is connected with mathematical inconvenience. But “imaginary time” is indeed imaginary, it has nothing whatsoever to do with physics, even though Penrose may discover “physics” in this mathematical game. So the right choice depends on what you are. Physicists should go to the west coast, other can do what they want. (PS I started out on the east, but realized the betray.)

  22. tt says:

    imaginary time is used in stat mech.
    http://en.wikipedia.org/wiki/Imaginary_time

  23. Radioactive says:

    Much as I hate to agree with Lubos, and I have the emotional intelligence to identify tongue-in-cheek remarks, but the choice is completely irrelevant and quibbling about sign conventions is best left to pedants who need to feel important at lunchtime talks.

  24. Peter Woit says:

    Radioactive,
    Don’t worry, I’ll soon move on from technical pedantry and get back to blogging about less childish topics like theories of everything, the multiverse, etc….

  25. vmarko says:

    I’ve also never heard of the West/East coast terminology, and I think most people in Europe haven’t either. There was always just a particle-physicist signature (+,-,-,-) and the gravity-physicist signature (-,+,+,+).

    I was also first educated in the particle-signature fashion, but after started doing research (on GR and QG), I switched to gravity-signature, and never ever looked back. It’s much more convenient, for all the reasons Peter mentioned, and then some (raising and lowering indices, discussing spatial hypersurfaces and the Cauchy problem, working with Kaluca-Klein scenarios, canonical quantization, Dirac’s analysis of constrained systems, etc).

    As for books, note also that the “GR-phonebook” (Misner, Thorne, Wheeler) uses the (-,+,+,+) signature throughout.

    As for the energy, I don’t quite understand what is the problem — energy can be (and should be chosen to be) positive in either metric, AFAIK. The overall sign of the action is what fixes the sign of energy, and this can be chosen independently of the metric signature. One chooses the ground state to be of minimum potential energy, i.e. the action to have a maximum, which is done by multiplying any given action with -1 if needed. This has nothing to do with the choice of the metric.

  26. Peter Woit says:

    Anon,
    Thanks for pointing that out. Looking into it, it seems that Wald is doing this to try and follow Penrose, who I hadn’t realized is in the West Coast camp, despite being a relativist. It’s striking to see multiple instances of writers changing convention in the middle of a book, something that normally one would very much try hard to avoid. I suspect it is partly because the sign issues get very tricky when dealing with spinors, so authors decide it’s just too hard to figure out how to change from one to the other.

  27. Peter,
    * The link to Figueroa-O’Farrill’s notes is broken. You need a www. instead of ww. in the URL.
    * I think the choice of mostly minus or mostly plus metric when not dealing with spinors, is largely an issue of convenience, nothing to do with positivity of energy or action terms.
    * For spinors, the relevant issue is the relative sign choice in the metric and the definition of the Clifford algebra. Different relative signs would give inequivalent spinor representations as already pointed out. I don’t know of a good physical argument for choosing one over the other, unless physics some how only cares about the representation of the complexified group in which case either convention is fine.
    * As for Wald’s switch in metric signature. That is done to use the 2-spinor notation where one wants to identify an abstract index “a” on the tangent space to a pair of spinor indices “AA′”. In particular, one wants the metric \eta_ab -> \epsilon_AB \epsilon_A′B′ instead of with a minus sign where \epsilon is a real symplectic 2×2 matrix. Though I am sure this again has to do with a choice of sign for the defining equation of the Clifford algebra.
    * But overall as pointed out in the notes by Figueroa-O’Farrill “There is no difference between 3+1 and 1+3—there never is at the level of Spin group” so maybe this point is moot!?

    original post

  28. Peter Woit says:

    Kartik,
    Thanks, fixed the link.

    Ignoring spinors, for standard flat space calculations, clearly you can use either convention if you want to. I still think though that working with a negative definite metric (for space or analytically continued space-time) is always going to be at best confusing, if not worse, and that’s what the West Coast convention makes you do. The issue about analytic continuation in QFT seems to me a real one, there’s far too little clear discussion of the issue (non-perturbatively), and maybe none in the West Coast convention.

    I would have thought you could do two component spinor notation with either convention, and a bit of googling shows that it has been done. See
    http://www.niu.edu/spmartin/spinors/
    and
    http://www.niu.edu/spmartin/TASI11/
    for details. These papers evidently include a tex file where you can change a single variable, and get the paper in either convention (I’ll add this info to the posting).

    Yes, for the spin groups, doesn’t matter. The physical significance of the two different Clifford algebras and different Pin groups is still obscure to me.

    To say that “physics doesn’t depend on the real structure, just the complexification” I’d interpret as a claim that the physical picture for the physics of spinor fields in different real structures is related by analytic continuation. This may or may not be true, it’s exactly what I’d like to understand better, but have been finding very little in the literature that addresses this.

  29. Peter,
    Being from a GR background, I agree that it is very convenient to work in the mostly plus signature and being in Chicago I’ll skip the East/West dichotomy! 🙂

    * I am not sure what significance one should attach to “Wick rotation” or analytic continuation procedures emplyed in textbook QFT. Firstly, those work only in analytic and static/stationary spacetimes, which again from a GR perspective is highly restrictive.

    * Yes, one can do 2-component spinor notation in either signature if one wants to carry around the “sigma matrices”. Penrose’s notation gets rid of the sigma matrices entirely ( See: Spinors and Spacetime Vol1 — Penrose & Rindler ) and one can write equations like g_ab = \epsilon_AB \epsilon_A′B′ which works only in the mostly minus signature. If using a mostly plus signature this would have a minus sign which is very inconvenient. So this notation by Penrose sacrifices the goodness of the mostly plus metric for the goodness of skipping sigma matrices entirely, which is also the convention Wald uses. I have also seen Bob Geroch use this notation in his lectures. In the http://www.niu.edu/spmartin/spinors/ link that you cite this corresponds to their Eq.2.48 which does have a sign change when changing the metric signature.

    * I still don’t understand the physical significance (if any) of pinors. As far as I have seen, physics theories only use spinors and the Spin group in which case this sign choice is irrelevant.

    * Again, I am pretty skeptical of the importance of analytic continuation to physics in general.

    original post

  30. Peter Woit says:

    Kartik,
    Thanks for the comments on the two-component formalism.

    About analytic continuation, I realize this doesn’t work in arbitrary backgrounds, but I do think it’s a deep issue in QFT. As far as I’ve ever been able to tell, there really is no such thing as a rigorous, non-perturbative formulation of QFT that does not use the analytic continuation. Even in the simplest considerations of the free field propagator, you have to add some extra piece of information (how do you integrate through the poles) that is very nicely expressed by the analytic continuation requirement.

    Anyway, analytic continuation issues are somewhat of a different topic, but the point here is just that they seem to not have been sorted out for spinors, with the general assumption being one of “only depends on complexification”, in the sense of assuming the analytic continuation can be done. Your comment led me to take a look at Penrose (at least to the extent of picking up “Road to Reality”), and I was intrigued to learn that he does use the West Coast metric, but that he also clearly thinks of the question in terms of complexified Minkowski space and analytic continuation. Quite possibly the right way to think about these questions is in terms of twistors.

  31. * I also think that twistors are in some sense capturing the some part of this issue. I would highly recommend both volumes of Spinors and Spacetime — Penrose & Rindler. They present the formalism in a very clear style.

    * Another related annoyance wrt complex vs real forms of groups is that we physicists do not distinguish these in Yang-Mills theory. I have had the hardest time finding a good reference on the issues of group structure, complex vs real forms, Cartan subalgebras… as it relates to Yang-Mills theories.

    original post

  32. John Baez says:

    Kartik wrote: “I still don’t understand the physical significance (if any) of pinors.”

    One place they can be important is when you’ve got a theory where spacetime is non-orientable. Then you need to know how your “spinors” transform under orientation-reversing symmetries – so what you really need is pinors, which are a representation of a double cover of O(p,q) rather than merely SO(p,q).

    Here I will not enter into the fact that O(p,q) has up to 8 different double covers! After choosing a sign convention for your Clifford algebras, the Clifford algebra will contain a group Pin(p,q) which is a specific double cover of O(p,q). We can hope that nature will be kind enough to use this one.

    I saw Cécile DeWitt-Morette give a talk about this stuff. Once she and her husband Bryce DeWitt both did computations for a free fermion field on a nonorientable spacetime. They got physically different answers! And it turned out the reason was that one was using a pinor representation of Pin(1,n), while the other was using a pinor representation of Pin(n,1).

    They wrote a paper about this.

  33. John,

    Thanks for pointing that out and for the link to the DeWitts’ paper. I was completely ignoring orientability issues.

    In this case, wouldn’t one want the universal cover of O(p,q) rather than merely a double cover? If I am remembering correctly for d > 3 the Spin(p,q) group is also the universal cover of the oriented part of SO(p.q). For instance in d=3, I thought the universal cover gave rise to representations like anyons.

    original post

  34. Marc Nardmann says:

    More information related to John Baez’s remark about double covers of O(p,q) can be found in the following two articles:

    1.) Trautman, Andrzej: Double covers of pseudo-orthogonal groups. Clifford analysis and its applications, NATO Science Series 25 (2001), 377-388. (http://link.springer.com/chapter/10.1007%2F978-94-010-0862-4_32)

    For p,q>0, eight inequivalent double covers of O(p,q) are constructed in that article; see §7.2. Trautman says there are precisely 16 inequivalent double covers of O(3,1), but he does not construct explicitly the other 8.

    2.) Trautman, Andrzej: On eight kinds of spinors. Acta Physica Polonica B 36 (2005), 121-130. (go to http://www.actaphys.uj.edu.pl/_cur/pl/acta_physica_polonica_b, then browse)

    This article ends with the following conclusion: “Physicists have been wise to restrict their attention to the groups Pin(1,3) and Pin(3,1) as these are the only double covers of [the full Lorentz group] that admit faithful irreducible spinorial representations and allow the construction of real invariants and covariant currents.”

  35. Lowell says:

    Well, my book “Quantum Field Theory” (Cambridge Univ. Press) has the proper
    (-,+,+,+) metric.

  36. Lowell Brown says:

    Might as well give my full name.

  37. Tony Smith says:

    John Baez said in his week156 web page (17 Sep 2000):
    “… any self-dual irreducible unitary group representation H must admit an antiunitary intertwiner J: H → H with either J^2 = 1 or J^2 = -1.
    In the first case H comes from a real representation;
    in the second case it comes from a quaternionic representation

    in Volume 1 of Weinberg’s “Quantum Field Theory” …
    the CPT operator on the Hilbert space of a spin-j representation of the Poincare group is an antiunitary operator with (CPT)^2 = -1^(2j).
    So indeed we do have (CPT)^2 = 1 in the bosonic case, making these representations real,
    and (CPT)^2 = -1 in the fermionic case, making these representations quaternionic. …”.

    Tony Smith

  38. John Baez says:

    Kartik wrote:

    In this case, wouldn’t one want the universal cover of O(p,q) rather than merely a double cover?

    You’re right that we need to carefully think about all these issues rather than merely grabbing a double cover and using that. But part of the problem is that the concept of “universal cover” applies to connected Lie groups, and O(p,q) typically has 4 connected components. So, for example, O(3,1) has 4 connected components that contain the elements 1, P, T, and PT respectively. If we restrict attention to the connected component containing the identity, its fundamental group is Z/2 so its universal cover is its double cover, isomorphic to SL(2,C). But things get more complicated when we consider all of O(3,1). It has different double covers. All of them become isomorphic when we restrict them to obtain covers of the connected component containing the identity. That is, all restrict to give universal covers of this connected component! I think what’s happening is that they are different central extensions of SL(2,C) by the group {1,P,T,PT} = Z/2 × Z/2.

    You’re right that O(2,1) is very different: here the fundamental group of the connected component containing the identity is Z, not Z/2, so we get anyons.

  39. Mesmar Djehha says:

    I always imagined funnily (I am an autodidact in physics and mathematics from north Africa) that the terminology West Coast/East Coast came from the fact that when we look down at the metric matrix in front of us, we see the + (representing sunrise) coming from the west wtr to the matrix in West-Coast convention, whereas we see the sunrise coming from the east in the Est-Coast convention.
    Thanks a lot Peter for this posting. We really learn a lot about mathematics and physics from this blog.

  40. Martin says:

    Srednicki kindly provides a free draft copy plus errata:

    http://web.physics.ucsb.edu/~mark/qft.html

  41. Here is a discussion at mathoverflow about the related issue of the convention used in defining Clifford algebras. Note that I don’t claim to understand the first (and meta) answer there…

  42. Justin says:

    Love this blog post. Do not worry about the detractors. I do feel that this blog is not quite as good as it could be because it focuses way too much on multiverse nonsense. I’d like to see more postings about trying to understand quantum field theory better, and this post is right on target.

  43. Geoff says:

    I have only skimmed the comments so I apologize if I’m repeating something already mentioned more succinctly above. I was under the impression that the most obvious reason for using the mostly plus convention was that at some point one is forced to do violence to General Relativity and cast it into Hamiltonian form thru the ADM decomposition. Although Ted Jacobsen has been voicing valid reasons for not quantizing the metric, it seems that the vast majority of successful quantization schemes have passed thru the Hamiltonian formulation.

    A wonderful resource that I’ve just come across that doesn’t appear to be on the ArXiv is an article about spinors by Geroch in the Springer Handbook of Spacetime where he uses the mostly minus convention. I’ve heard a lecture in which he expressed bewilderment that spinors which were originally used to describe electrons make the proof the Positive Energy Theorem in GR substantially easier. And if every tensor and tensor operation has a counterpart in the spinor formulation, it certainly seems worth learning. It does seem from my quite limited viewpoint that something profound is happening when spinors show up in such an unexpected manner.

    Professor Baez – I had always skimmed the descriptions of non-orientable spacetimes as it seemed that they lead to violations in causality. Are there non-orientable spacetimes which preserve a global causal structure or are they merely interesting from a theoretical standpoint?

    Professor Lowell Brown – I’ve quite enjoyed having your textbook as welcome addition to my shelf and was hoping you’d get around to someday writing the second half of the book.

    I have always been fascinated by stochastic quantization as it too seems entirely unexpected. For those who know, can you point me in the direction of a reference that might explain its physical relevance or is it simply a trick which also involves complexification?

    Does anyone happen to know if Schwinger changed conventions when he left Harvard for UCLA?

    Thanks in advance for help anyone can provide.

  44. Lowell Brown says:

    Nope, Schwinger stayed East Coast when he moved to the West Coast.

  45. John Baez says:

    I wrote:

    I think what’s happening is that they are different central extensions of SL(2,C) by the group {1,P,T,PT} = Z/2 × Z/2.

    Sorry, on second thought that doesn’t sound right: even in O(3,1) the transformations T and P don’t lie in the center, only 1 and PT are. So, I think we should delete the word “central” here.

  46. Peter Woit says:

    Justin,
    Don’t worry, I’m quite good at ignoring detractors who I don’t think have a case. And I do know when a topic is substantive, attracts an intelligent discussion, and I learn something, will do my best to try to make that happen more often. Sorry about the next posting….

  47. cedric bardot says:

    …most physicists don’t appreciate the mathematical concept of “real structure”

    Well, may be one will have to wait for the experimental discovery of a fundamental Majorana neutrino to expect better awareness about clifford algebras, pin groups and real structures (cf I. Todorov arxiv.org/abs/1106.3197)
    On the theoretical side people like A. Connes et al and John .W. Barrett may have already paved the way to improve our geometric understanding of the standard model in order to orientate us in the possibly already nontrivial dimensional structure of zeptospace (arxiv.org/abs/hep-th/0608226 and arxiv.org/abs/hep-th/0608221).

  48. nicola says:

    The many-minus convention is far better because the mass-shell constraint has the clear form p^2=m^2 (while in the other you get the awkward minus sign). Moreover the Dirac equation in the many-plus convention would get an additional factor of i – either in front of m or in front of Dirac matrices – which is even more awkward.

    That Weinberg uses many-plus convention – well, Landau uses the other one.

    Wick rotation is just some other mathematical operation used in calculations. There is no deep Physics behind it. To choose a convention in Physics because Wick rotation looks nice, is beyond me.

    Backwards-compatibility is the only reasonable argument. If it weren’t for the mass-shell constraint and the Dirac equation it would probably convince me. On the other hand the creator of Python language considers backwards-compatibility as misguided since newer ideas/solutions are better then the older ones and so why constraining yourself with the wrong conventions/habits invented before? Many new languages invented are not backwards-compatible anymore.

  49. Peter Woit says:

    nicola,
    1. I don’t really see why p^2=-m^2 is a problem.
    2. I don’t see any i’s in the Dirac equation in East coast convention, either in Weinberg (equation 1.1.19), or my current notes (section 45.1, that chapter is a mess now, but I think the signs, factors of i are correct).
    3. I don’t think Wick rotation is just “some other mathematical operation”. QFTs really only are well-defined in imaginary time, it’s a fundamental fact about them.

  50. Adam Helfer says:

    As a relativist, I’d like to make some points in favor of the mostly-minus convention. Of course, one really can use either convention — one just has to include minus signs or factors of i in different places. And for certain specific applications (e.g. the ADM formalism or some QFT arguments) it may be technically easier to use the mostly-plus convention. But as a matter of principle — of trying to make the mathematics match most directly the physical foundations — in relativity, I think the mostly-minus convention is to be preferred.

    The reason is that proper intervals between timelike-separated events are directly measurable: you just let a clock run along the trajectory whose proper time you want. On the other hand, spacelike proper intervals are by definition acausal concepts. They cannot be measured directly, and working them indirectly tends to be involved.

    To some degree, of course, one can measure space-like intervals by means of “rigid rulers,” but notice that this is not at all a simple thing: the rulers really serve to define a coordinate system and what one reads off from them are coordinate differences, not invariant intervals. Alternatively, one can infer spacelike separations from systems of timelike measurements. (Note that such measurements are needed to calibrate the system of rigid rulers anyway.)

    Also the idealizations involved — and the question of how to correct for deviations from them for real measurements — are much more severe for rulers and space-like measurements than they are for clocks and time-like ones. A clock simply has to be small enough and stable enough that its mechanism is not significantly affected by curvature over its run-time. On the other hand, if one wants to extract physics from a ruler extending over a large enough region that space-time curvature is significant,
    in general one needs to solve the coupled system of the ruler’s internal dynamics and gravity (and one cannot really treat the ruler as rigid).

    (As an example, note that GPS works primarily by making time-measurements. Some reference to rulers — Earth-based reference frames — is needed, but the main measurements are temporal. Also interferometers like LIGO essentially measure time-signals, as they compare phase differences for light of a fixed frequency along different space-time paths; the spatial separations are inferred.)

    It is true that the mostly-plus convention seems to be the friendlier, less radical departure from Euclidean space. But really the lesson of relativity is that three-space is, except in restricted circumstances, a rather indirect concept of limited physical significance. (I hasten to add that in many circumstances, especially when relativity is not important, it certainly is useful!) However, the everyday concept which most directly generalizes to relativity is not space but time. So my feeling is that the friendliness of the mostly-plus convention is meretricious (as far as the foundations of relativity go), and we get a deeper sense of the geometry of space-time by embracing the perspective implicit in the mostly-minus one.

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