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. Peter says:

    I saw that too, and suppose he is tired of Lubos’s rants and the nonsense they generate. Funny that he should have more trouble with Lubos than me. I guess it’s because I don’t engage in political commentary here, so Lubos and I have only one topic to disagree about.

  2. Anonymous says:

    just read on preposterous universe that sean banned lubos from his weblog. is he serious?

  3. Peter says:

    The main thing that is weird about this idea is that you don’t have the full Lorentz symmetry. You have in some sense picked a time direction, which determines an SU(2)=Spin(3) subgroup of Spin(4), which will be the spatial rotations, which are not spontaneously broken. The weak SU(2) acts non-trivially on this choice of time direction, which behaves somewhat like a Higgs field, perhaps spontaneously breaking the weak SU(2). (But still, I haven’t written down dynamics that does this).

  4. Anonymous says:

    One obvious problem would seem to be that the electroweak gauge symmetry is spontaneously broken, while Lorentz symmetry is not. How would you get around this?

  5. Peter says:

    Gravitons aren’t quanta of the spin connection. With the standard Lagrangian, the field equation for the spin connection determines it in terms of the vierbeins. It’s the vierbein fields, or equivalently the metric, whose quanta are the gravitons.

    I wrote a paper about this “Euclideanized boosts=weak SU(2)” idea many years ago

    Nucl. Phys. B303, pg. 329, 1988

    but I certainly know a lot more now than I did then, and should write an updated version someday.
    For one thing that paper wasn’t even written in the context of QFT, just of a single-particle model.

  6. Anonymous says:

    If one of the Wick rotated SU(2)’s happens to be the electroweak SU(2), this would explain why only left handed fermions interact with the weak interaction, but but wouldn’t it also mean the W and Z bosons are gravitons?

  7. D R Lunsford says:

    I’ve never seen any evidence that LM has a clue about anything at all.


  8. D R Lunsford says:

    Peter – well said again, I agree in detail that the real problem in physics is the relation of QFT to actual geometry.

    Have a look Finkelstein/Jauch, “Quaternion Quantum Mechnics”, which may be related to what you are “selling”.


  9. Peter says:

    Distler is no slouch himself when it comes to “straw man arguments, willful misreading, and insults”, although he is capable of more subtle insults than Lubos since English is his native language. Check out his contributions to some of the early postings of this weblog.

    Pretty funny to see him and Lubos in action. Do you think it’s statistically significant that the two most prominent string theorists with weblogs are both incredibly arrogant and incapable of admitting that anyone who disagrees with them might have a point?

  10. Fabio says:

    That LM discussion is pretty interesting, especially the part where Jacques Distler tries to make a point, runs up against the trademark Lubos Motl mixture of straw man arguments, willful misreading, and insults, then ultimately decides it’s not worth the bother.

  11. Alejandro Rivero says:

    Wow! Welcome to marginality, Peter!

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