Wick Rotation

There’s a quite interesting discussion going on about Wick rotation over at Lubos Motl’s weblog.

In flat space-time, the situation is well-understood: if your Hamiltonian has good positivity properties you can analytically continue to imaginary values of time, and when you do this you end up with “Euclidean” path integrals, which actually make sense, unlike QFT path integrals expressed on Minkowski space, which don’t. You can see the problem even in free field theory: the propagator is given by an integral that goes through two poles, so is ill-defined. The correct way to define it to get causal propagation for a theory with positive energies is to go above one pole, below the other, which is equivalent to “Wick rotating” the integration contour 90 degrees to lie on the imaginary time axis.

In a curved space time, things are much trickier. And in a path integral approach to quantum gravity it is very tricky. Do you integrate over all metrics with Lorentz signature (ignoring the fact that the path integral doesn’t really make sense for a single one), or do you integrate over Euclidean signature metrics (Euclidean Quantum Gravity)? There are arguments against either choice, not to mention the non-renormalizability problems that both may have. For some of the arguments, see the debate in Lubos’s comment section, which gives some idea of how confused the state of this question is. Another good reference is the article by Gary Gibbons in the Hawking 60th birthday celebration volume. It doesn’t seem to be on-line, but his talk at the workshop is.

I’ve always thought this whole confusion is an important clue that there is something about the relation of QFT and geometry that we don’t understand. Things are even more confusing than just worrying about Minkowski vs. Euclidean metrics. To define spinors, we need not just a metric, but a spin connection. In Minkowski space this is a connection on a Spin(3,1)=SL(2,C) bundle, in Euclidean space on a Spin(4)=SU(2)xSU(2) bundle, and these are quite different things, with associated spinor fields with quite different properties. So the whole “Wick Rotation” question is very confusing even in flat space-time when one is dealing with spinors.

Over the years I’ve tried to sell the outrageous idea that one should define QFT in Euclidean space time, with one of the two SU(2)s in Spin(4) being Spin(3), the spatial rotations, the other being the SU(2) of the electroweak gauge group. I’ve never been able to get anyone to take this seriously, partly because I’ve never come up with a well-defined way of writing down path integrals which implement this idea.

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61 Responses to Wick Rotation

  1. Quantum_Ranger says:

    Is there any chance of ‘Peter’, actually giving a glimpse of :Nucl Phys. B paper?..a direct link would be useful.

  2. Anonymous says:

    Fixing the causal structure of spacetime will not work. A simple thought experiment will show this. A universe with no black holes initially which forms a black hole as a clump of matter collapses has a very different causal structure from a universe where no black hole ever forms. But this is the crucial point; whether or not some matter collapses to form a black hole depends upon the dynamics of matter and gravity. Clearly, this means making the causal structure independent of the dynamics will lead to a contradiction.

  3. Anonymous says:

    Dear Lubos,

    I must say that I find your posts, always, very entertaining. But, please, remember that you are no longer in your home satellite. In the US, the terms of discourse are somewhat different. Your commissar-like behaviour of trying to shout down critics with insults and denunciations will not work. You merely appear ridiculous. You may keep your stalinist committments (if you must), but you need to change your methods (hint: learn from Distler). I think there is a book called “Getting power in the US while keeping your stalinist committments intact”; will let you know the Amazon link.

    Peter: I don’t think Distler is “better” because of english. He just understands the subtle power-politics here much better than Lubos, who is still trying out his stalinist vilifications techniques.

  4. Anonymous says:

    Incidentally, \int dx e^{ikx}=2 \pi \delta(k).

    Therefore, we have a clear example of a case where Wick rotation doesn’t hold. πŸ™‚

  5. Anonymous says:

    formally, we can still perform calculations with unbounded path integrals. Let me give you an example of what I mean.

    \int_{-\infty}^{\infty} dx e^{kx} looks divergent and not well-defined, but let’s pretend its value is f(k), which is of course infinite. Then, f'(k)=\int dx x e^{kx}. Ignore for a moment the fact that this integral diverges badly and pretend that integration by parts works. Then, f'(k)=-\int dx 1/k e^{kx}=-f(k)/k. Solving this differential equation, we obtain f(k)=f(1)/k. But recall f(k) is infinite. So, we normalize everything by dividing infinity by infinity to get a finite value.

    \int dx e^{kx}/\int dx e^{x}=1/k

    I know this is vulgar speech which wouldn’t be appropriate in the polite company of mathematicians, but we’re all physicists, right? You won’t get offended. Peter may be a “mathematician”, but when it comes to physics, he has the mindset of a physicist.

  6. Anonymous says:

    Is it possible to add higher order terms to the Einstein-Hilbert action so that the Euclidean action is bounded from below?

  7. D R Lunsford says:

    Lubos,

    The thing that makes invariant integrals is sqrt(det(g)), not -det g whatever that is. So the issue is, do I take sqrt(det(-g)) as do the relativists, or, as demanded by tensor analysis, sqrt(det(g)) ipse, which, if g has 1, 3, 5.. – signs, imaginary (and branched)?

    -drl

  8. Anonymous says:

    Lubos, how can we have both signature changing metrics AND a global gauge fixing where -|g| = 1 everywhere?

  9. Anonymous says:

    Rest assured, Lubos will never have to distort his text to look less smart.

  10. Frank says:

    On the Topic of Lorentzian vs Euclidean, I’m reminded on stuff by Renate Loll in 2 Dimensions a while back:
    a couple of slides: http://cgpg.gravity.psu.edu/online/Html/Seminars/Spring1999/Loll/Slides/s01.html

    Insisting on causal paths in the path integral the theory can be defined in the continuum limit and differs from what you get in Euclidean theory.
    Something analogue to the Wick rotation is still going on in that an imaginary cosmological constant is required to ensure the existence of the continuum limit.

    More recently numerical investigations suggest that a smooth 4D (Hausdorff Dimension) macroscopic geometry emerges from these causal path integrals.

    http://arxiv.org/abs/hep-th/0404156

  11. Lubos Motl says:

    Dear “”, diffeomorphisms don’t change the value of the action which is exactly the reason why the modes induced by diffeomorphisms are pure gauge. Locally, one can always choose a gauge where (-det g)=+1.

    Sometimes I wonder, “”, that “” is really Jacques Distler who just distorts his text a bit to look less smart, by a few orders of magnitude, so that there is an “independent” debater who supports his viewpoints. πŸ˜‰

  12. Peter says:

    I think it would be best if people discuss Lubos’s mistakes on his blog, mine on this one….

  13. Anonymous says:

    Dear Lubos, what you say sounds as an error that would prevent a first-year grad student of general relativity from passing her exam.

    Of course that your confusion about diffeomorphisms and the conformal factor shows a complete lack of understanding of general relativity.

    That’s not so hard to see. Diffeomorphisms would not change the value of the action.

    One can’t be confused about elementary general relativity theory if she wants to seriously work in quantum gravity. πŸ˜‰

    By your own admission, Distler’s objections are perfectly valid. So now, you’re completely unable to describe what the mistake in Distler’s reasoning was supposed to be, are you?

  14. D R Lunsford says:

    Well that joke went over like a plaster of paris bagel…

    -drl

  15. Lubos Motl says:

    Peter, you can always learn new things here. For example, Dr. Lunsford now teaches you that the determinant of the metric tensor (i.e. the product of -1 times 1 times 1 times 1) equals “i”. Enjoy your learning! πŸ™‚

  16. D R Lunsford says:

    Um, -det g = -i πŸ™‚

    Funny, this is also the y5 issue.

    -drl

  17. Anonymous says:

    In response to Lubos:

    The gravitational action is arbitrarily large if the conformal factor varies fast enough.
    You thought this behavior can be “gauged away” because one can “always set (-det g) = 1.

    Is this explanation good enough for you?
    If not I suggest you read the available literature.

  18. Lubos Motl says:

    Dear “”, it was not subtle, except that you’re completely unable to describe what the mistake was supposed to be, are you? πŸ˜‰

  19. Anonymous says:

    In response to Lubos Motl:

    I did not write (-det g) = 1 and your mistake was not subtle at all.

  20. D R Lunsford says:

    Peter,

    Yes, something “funny” is going on and it amounts to using the wrong “i” in the context of spinors, which need spacetime algebraic invariants. In a representation of the Dirac algebra in which y5 is diagonal (Weyl rep) the spacetime “I” (-iy5) reduces (almost) to the usual “i”. This issue comes up again and again and leads me to believe the “via regia” to geometry in the context of field theory is y5.

    Here are references to the papers I mentioned before:

    D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Foundations of quaternion quantum mechanics, Journal of Mathematical Physics 3, 207 (1962)

    D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Some physical consequences of general Q-covariance, Helvetica Physica Acta 35, 328-329 (1962)

    D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Principle of general Q-covariance, Journal of Mathematical Physics 4, 788-796 (1963)

    D. Finkelstein, J.M. Jauch, S. Schiminovich and D. Speiser, Quaternionic representations of compact groups, Journal of Mathematical Physics 4, 136-140 (1963)

    You’ll find, essentially, WSG electroweak theory, long before gauge theory was cool.

    -drl

  21. Lubos Motl says:

    Who are the others you learn from, Peter?

    The anonymous guy below who believes that one can’t set (-det g)=1 by gauge transformations (diffeomorphisms)?

    Of course that you can always find a simple coordinate redefinition such that (-det g) is whatever function or constant you want.

    I retracted a statement that all problems related to (-det g) are pure gauge problems (it’s a very subtle question), and an anonymous reader immediately celebrates that I retracted something, even though she or he misunderstands what exactly was retracted.

  22. Peter says:

    Actually one ot the best aspects for me of this weblog is that I often learn new things from people who write in with comments. Sometimes even from Lubos….

  23. Anonymous says:

    It is not such a big deal as long as you can admit them quickly WITHOUT insulting everybody.

  24. Anonymous says:

    If Lubos celebrates the fact that Peter stumbled a bit in this case, then I would like to point out that he made a much worse blunder 2 days ago on his
    Wick rotation thread.
    He was of the opinion that the conformal factor in higher-dimensional gravity can be set to 1 by gauge transformation.
    People make mistakes. It is not such a big deal as long as you can admit them quickly with insulting everybody.

  25. Peter says:

    Unlike some people I’m not claiming to have a wonderful theory here. All I’m pointing out is that there seems to me to be something funny going on when you try and think about Wick-rotating spinors. The standard point of view is that this is a purely technical problem, but I suspect it might be a clue to something much more interesting and mentioned some reasons for thinking this. There are others in my 1988 paper. But this is a vague idea about where to look for a new theory, not a new theory, and I haven’t claimed otherwise.

  26. Anonymous says:

    Theatre critics should never act. It’s an important rule. Probably best to just keep knocking everyone else’s work Peter. Much easier.

  27. Anonymous says:

    The kinematics doesn’t seem to work. It may work for fermions; the left handed ones are also the ones charged under electroweak SU(2), while the right handed ones are uncharged. The photon, however, transforms under both SU(2)s of the Lorentz group, while it is uncharged under the electroweak symmetry. This might already invalidate your proposed identification.

  28. Peter says:

    Hmm, it’s too late at night for me to straighten this out but I think you’re right I was saying something incorrect. But all I was trying to say was SO(4)/SO(3)=S^3, which as a manifold can be identified with SU(2), although you’re right it doesn’t get its group law from the SO(4) one.

    I got into this mixup by trying to oversimplify things by just counting degrees of freedom. A more accurate way of saying what is in my 1988 paper is that I look at the twistor space of orthogonal complex structures over a 4d Riemannian manifold, and try and identify the electroweak U(2) as the subgroup of SO(4) at each point in twistor space that commutes with the complex structure.

    I haven’t thought about this much in a while and don’t have time to try and write out more details, especially not here where it is hard to write math. If people want more details now, they should take a look at the paper.

  29. Anonymous says:

    But the set of elements of the form (q_1,q_1^{-1}) doesn’t form a group! SU(2) isn’t Abelian. And we know the electroweak SU(2) forms a gauge group. And if you wish to identify the electroweak SU(2) with torsion, since you’re not identifying it with gravitation, that can’t work. The electroweak coupling has the form \bar{\psi} A_\mu \gamma^\mu \psi, not \bar{\psi} \Gamma_{\mu\nu} \sigma^{\mu\nu} \psi. It’s a vector (well, V-A) coupling, not a tensor coupling.

  30. Peter says:

    Here’s one way of thinking about what I meant:

    Pick an identification of R^4=H (H is the quaternions). Then elements of SO(4) are given by pairs (q_1,q_2) of unit quaternions, and they act on an element q of H=R^4 by

    q goes to q_1qq_2^{-1}

    Note that SU(2)=group of unit quaternions, SO(4)=SU(2)xSU(2) modulo a Z_2, since (q_1,q_2) and (-q_1,-q_2) give the same action. Spin(4)=SU(2)xSU(2), but if you only act on vectors you just see SO(4).

    With the standard choice of identification q=x_0+x_1i+x_2j+x_3k, the diagonal subgroup of elements of the form (q_1,q_1) leaves x_0 invariant, acts as SO(3) on the other 3 variables. By “anti-diagonal” SU(2) I was thinking of elements of the form (q_1,q_1^{-1}).

  31. Anonymous says:

    My apologies, that’s only true for Spin(n) for even n. “Inversions” act trivially upon Spin(n).

  32. Anonymous says:

    Sorry, I accidentally hit the Post button.

    SU(2) × SU(2) has infinitely many diagonal subgroups, one for each automorphism of SU(2). Basically, if f is an automorphism of SU(2), the subgroup of elements ( g,f(g) ) forms a diagonal subgroup. The outer automorphism group of SU(2) is the two element group Z2, i.e. correspoding to a “reflection”. This, I’m pretty sure is what Peter meant by the anti-diagonal subgroup.

  33. Anonymous says:

    SU(2) × SU(2) has infinitely many diagonal subgroups, one for each automorphism of SU(2). Basically, if f is an automorphism of SU(2), the subgroup of elements ( g,f(g) ) forms a diagonal subgroup. The outer automorphism group of SU(2) is the two element group Z2

  34. Lubos Motl says:

    Dear Peter,

    your comment about the trademark Lubos Motl (R) is very illuminating and entertaining. Nevertheless you may want to focus on the essence which is the SU(2) x SU(2) group in this particular case. Once you see that there are only two truly inequivalent ways how to embed SU(2) into SO(4) and only one of them has another SU(2) left, you may decide whether you want to follow my advise or the advise of the other straw man.

    Good luck
    Lubos

  35. Alejandro Rivero says:

    A glimpse of the role of the electroweak group:

    a_\mu=
    {m_\mu \over m_Z}+ {m_e \over m_W}+
    \frac12 {m_\mu^2-m_e m_\tau \over m_W^2}=.001165825

    The conventional perturbative calculation (without hadronic corrections) of muonic (g-2)/2 gives .0011658487, about a 0.003% respect to the value given by the above empirical formula.

  36. Peter says:

    Hi Lubos,

    I can’t imagine why someone here was referring to the “trademark Lubos Motl mixture of straw man arguments, willful misreading, and insults”….

  37. Lubos Motl says:

    Dear Peter, what you say sounds as an error that would prevent a first-year undergrad student of linear algebra from passing her exam.

    Of course that your things would violate the Coleman-Mandula theorem. But more easily, there is nothing such as “anti-diagonal” SU(2). You can divide the SU(2) x SU(2) generators to the diagonal ones, and the rest. The diagonal ones form a closed algebra, but the anti-diagonal don’t.

    That’s not so hard to see. These are generated by J14, J24, J34, and the commutators of those are again in the diagonal algebra generated by J23, J31, J12.

    One can’t be confused about elementary SU(2) group theory if she wants to seriously unify interactions. πŸ˜‰

  38. Peter says:

    Coleman-Mandula says you can’t find a larger symmetry group that includes both Poincare and internal symmetries, mixing them in a non-trivial way. I’m not trying to do that.

  39. Thomas Larsson says:

    Peter,

    If you try to unify spatial rotations with electroweak SU(2) without SUSY, don’t you run into trouble with the Coleman-Mandula theorem?

    Nobody seems to have mentioned AJL’s simulations, which distinguish between Lorentzian and Euclidean spacetimes in a different way. Thay insist on strict causality everywhere, which means that topology change is ruled out. They work with Wick-rotated Lorentzian spacetime, which is thus different from Euclidean spacetime with topology change allowed.

    I like their results for several reasons. The obvious one is that they get a smooth 4D manifold rather than a crumpled mess, in agreement with observation. Another feature is that if you couple the Ising model to Lorentzian gravity, as in hep-th/9904012, the critical exponents remain at their Onsager values, unlike Ising coupled to Euclidean gravity. This is good, because the one thing we do know that quantum gravity is that it is unimportant compared to electromagnetism; the Ising model is a condensed-matter model, and as such it is a rough model of electromagnetism. Moreover, claims that AJL violate unitarity are, AFAIU, false.

  40. Peter says:

    When you decompose Spin(4) as SU(2)xSU(2), you’re making a choice of how to do it. Standard thing is to identify R^4 with 2×2 complex matrices, identifying the time-direction as the unit matrix. Then the diagonal action leaves this invariant, but rotates the space directions. From this point of view I’m hoping to identify the anti-diagonal SU(2) with the weak SU(2). The details of this are in my old Nucl Phys. B paper.

  41. Lubos Motl says:

    Dear Fyodor,

    your criticism is of course completely valid. On the other hand, mixing up one of the SU(2)s with the diagonal SU(2) looks like an innocent idea compared to Peter’s main proposal that the other part of the Euclideanized Lorentz symmetry is actually the SU(2) electroweak symmetry. πŸ˜‰

    With such Woitian proposals, anything goes.

    Incidentally, in loop quantum gravity there is also a confusion about the self-dual vs. spatial SU(2) within SO(4). Originally the SU(2) gauge group of loop quantum gravity was derived as the self-dual SU(2), i.e. one of the factors, but eventually it’s only kinematics that works, and therefore you’re equally justified to say that the SU(2) used in the spin network constructions is actually just the spatial rotational group.

    All the best
    Lubos

  42. Fyodor Uckoff says:

    The subgroup of Spin(4) corresponding to rotations is the *diagonal* SU(2), is it not? So I’m not real sure what you mean by

    one of the two SU(2)s in Spin(4) being Spin(3), the spatial rotations

  43. Peter says:

    For a biography of Gian-Carlo Wick:

    http://books.nap.edu/books/0309066441/html/333.html

    Sorry Lubos, it just can’t be helped that you’re more entertaining than Wick Rotation.

  44. Lubos Motl says:

    That’s pretty interesting. Peter’s article would suggest that the people would discuss the Wick rotation, but on this blog it’s not such an interesting topic. So the participants discuss the first interesting topic related to the Wick rotation that happens to be Lubos Motl. πŸ˜‰

    Does someone know who Wick was or is?

  45. Peter says:

    Hi Robert,

    The way in which Witten’s N=2 twisted supersymmetry trick works to turn a QFT that is relatively close to the Standard Model into a TQFT, by mixing the internal and space-time SU(2) has always fascinated me. I suspect it’s related to the idea I mentioned, but don’t understand how. My general feeling about supersymmetry is that we don’t understand its geometrical significance very well. If we did, we’d see that it was some “twisted” version of it, like Witten’s TQFT, that was what is really interesting. Unfortunately I don’t see how to use this specific TQFT to get what I’m looking for, maybe someone else will. Or maybe it requires some slightly different, but still unknown, construction.

  46. Arun says:

    If Lubos is the topic, then, in my opinion, he is best when he expounds on physics that he understands well; he is lucid and is a pleasure to read.

    -Arun

  47. A comment about possible connection of electroweak gauge group with Minkowski space that Peter is pondering. I hide the real background and just represent some observations, which I find intriguing.

    a) If you construct stringy vertex operators in Minkowski space by the standard construction you have 2 transversal polarization degrees of freedom having interpretation in terms of a Kac-Moody algebra associated with some 2-D Cartan algebra. It extends by the standard vertex operator construction to SU(3) Kac Moody algebra. As if color group were somehow inherent to 4-D Minkowski space.

    In fact, G_2 having same Cartan algebra is the maximal extension, but I have not managed to find whether the construction extends to the generators of G_2 representing its short roots (Olive et al argues that this is the case but do not give the construction in their 1984 article).

    b) On the other hand, when you divide SU(3) by U(2) you get CP(2) having U(2) as holonomy group and couplings are just those of electroweak gauge group (once you couple spinors to Kahler potential to get a proper spinor structure, this was done already by Hawking and Pope). Isometry group is of course SU(3) and CP_2 spinor connection codes for standard model symmetries.

    These observations lead you to ask what do you get if you construct SU(3)/U(2) coset theory. U(2) Kac-Moody would act precisely like electroweak gauge group in this theory. Is this theory analog of an electroweak gauge theory with symmetry broken down to U(2) and constructed using solely strings in M^4? Or could one somehow generalize the theory to get also color multiplets? In any case, spin (transverse polarizations) and “color”, “ew” qnumbers, and spin are very intimately related in this picture.

    Matti Pitkanen

  48. DMS says:

    Peter,
    I am sure I am completely wrong here, but is the idea (of the two SU(2)s in Spin(4)) proposed similar to “twisting” used by Witten to derive the new invariants for 4-manifolds based on his Seiberg-Witten on N=2 SUSY?

    PS1: Yes, the contribution to your earliest posts by two senior physicists are amusing reading πŸ™‚

    PS2: Although a recipient of a few LM insults myself, I actually feel he is being misunderstood and needs help. But in the hyper-aggressive/macho string community, such talk is likely verboten. I do hope some senior guys (whose opinion he respects) have the decency and humanity to help him out of whatever is bothering him (the example of the mathematicians’ help to Nash comes to mind).

    Often, when LM talks about QFT or strings (and not busy insulting people) he can be quite lucid (if too verbose, at times). And he does take time to answer some basic questions on physics from people, something you don’t see from similarly capable physicists (ok string theorists :)).

  49. Robert says:

    This interpretation of the the left handed part of the euclidean Lorentz group sounds a bit like a topological twist: There (in the D=4 version) you start with a N=2 theory that has a SU(2) R-symmetry and swap the roles of SU(2)_left with SU(2)_R. By this trick, one of the supercharges becomes a scalar and can be used as a BRST operator. The upshot being that the BPS-states become elements of the BRST-cohomology and thus physical states. Is there a relation (given your background in TFT)?

  50. Lubos Motle gave a nice summary about Wick rotation. An approach inspired by the conviction that the difficulties of path integral approach reflect deeper problems of principle is discussed in the mini article which I titled “How to put end to the suffering caused by path integrals?” at http://matpitka.blogspot.com/ .

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