tag:blogger.com,1999:blog-8666091.post110952206275983914..comments2020-03-13T03:28:45.250+01:00Comments on The Reference Frame: Wick rotationLuboš Motlhttp://www.blogger.com/profile/17487263983247488359noreply@blogger.comBlogger69125tag:blogger.com,1999:blog-8666091.post-11693303130846298002011-03-03T20:01:53.749+01:002011-03-03T20:01:53.749+01:00Yes, Fulig! The cut I was talking about has to be ...Yes, Fulig! The cut I was talking about has to be purely on the physical real axis and the discontinuity is fully determined from unitarity by decay rates etc.<br /><br />There can't be such irregular singularities in the bulk of the complex plane because they would correspond to physical states or resonances that are however "qualitatively different" and they would produce non-local physics or other pathological things.Luboš Motlhttps://www.blogger.com/profile/17487263983247488359noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-41377726959223887032011-03-03T14:42:13.087+01:002011-03-03T14:42:13.087+01:00Lubos, thanks for your very instructive remarks.
...Lubos, thanks for your very instructive remarks.<br /><br />As you wrote, branch cuts make the amplitude discontinuous across the cut. It is true that the function can be made unique, but in general the cut is arbitrary, in the sense that it can be moved around, as in the log example provided by the link in my previous message. So the value of the amplitude depends on how you position the cut. Is the cut determined by some physical condition? Does the discontinuity in the amplitude correspond to a concrete physical phenomenon?Fulighttps://www.blogger.com/profile/01302279334452728279noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-75780060200451416572011-03-02T13:18:23.709+01:002011-03-02T13:18:23.709+01:00Dear fulig, a good question. It's a topic for ...Dear fulig, a good question. It's a topic for a much longer article: singularities in the complex plane.<br /><br />First, it is not true that a pole makes the continuation ambibuous. A pole is a first-order singularity, like in 1/z, and the function continues to be single-valued.<br /><br />Such poles have the interpretation of physical states - resonances etc. But one can go on.<br /><br />However, there may also be other singularities such as branch cuts. In particular, they make the amplitudes (their imaginary part) discontinuous along the physical real axis.<br /><br />Nevertheless, in some proper sense, the singularities that may occur in the complex plane may be considered as "sequences of poles", at least when it comes to the consequences, so the function is still unique. That remains the case in the quadrant of the complex plane between the physical value of k0 and i-times this value.<br /><br />No, there is certainly no restriction that k0 has to be small. After all, small relatively to what? k0 is a dimensionful quantity, an energy, and by causality, it's always bigger (for physical particles) than the absolute value of the other components of k.<br /><br />You seem to think about some spectacular breakdown of the analytic continuation in physics. No breakdown of this sort exists. If you reach as brutal conclusions as the claim that k0 has to be tiny, you make much more progress if you start to think, on the contrary, that there's never any ambiguity about the analytic continuation because the ambiguities that may occur are really much more subtle than what you suggest.Luboš Motlhttps://www.blogger.com/profile/17487263983247488359noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-59343153979306829192011-03-02T09:40:49.722+01:002011-03-02T09:40:49.722+01:00Lubos.
Thanks for the nice introduction.
So Wick...Lubos.<br /><br />Thanks for the nice introduction.<br /><br />So Wick rotation yields a LOCALLY holomorphic function, i.e. you can expand locally, but at some point you hit a pole and continuation is no longer univocal. The first pic here shows what happens:<br />http://en.wikipedia.org/wiki/Analytic_continuation<br /><br />This implies that the expansion holds only for small k_0, i.e. there is an implicit cutoff.<br />Now my question: does the fact that the continuation becomes multi-valued for larger k_0 have a physical interpretation? Is the corresponding universal cover physically relevant?Fulighttps://www.blogger.com/profile/01302279334452728279noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-44305018936206323002010-11-18T14:51:38.283+01:002010-11-18T14:51:38.283+01:00Right, crasshopper. But your comment still underst...Right, crasshopper. But your comment still understates the legitimacy of the procedure. <br /><br />Even if there were poles etc. and the procedure using the imaginary time led to experimentally verifiable results, it would be well-established physics.<br /><br />The calculations in the imaginary time and/or Euclidean spacetime are actually more well-behaved, not less behaved, than those in the Minkowski spacettime.<br /><br />So this is not a silly "detour". It is the canonical, right, straightforward way to calculate things in QFTs.Luboš Motlhttps://www.blogger.com/profile/17487263983247488359noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-60883084754797041772010-11-18T14:18:02.458+01:002010-11-18T14:18:02.458+01:00So to summarize, you turn it a bit, do the math, t...So to summarize, you turn it a bit, do the math, turn it back, and this is justified because the function is holomorphic and you don't bump into a pole?Lao Tzuhttps://www.blogger.com/profile/01010110666955169076noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-77430232103057987982008-12-16T09:26:00.000+01:002008-12-16T09:26:00.000+01:00Dear Pat, thanks for your interest.Why temperature...Dear Pat, thanks for your interest.<BR/><BR/>Why temperature is imaginary time is a question where you need *some* mathematics - because the very notion of an imaginary time *is* a pretty abstract piece of mathematics. ;-) <BR/><BR/>So I will try to reduce it as much as possible.<BR/><BR/>In quantum mechanics, the evolution by time "t" is expressed by an operator, exp(i.H.t/hbar) where hbar is h/2.pi, the small Planck's constant. Also, H is the Hamiltonian, and "t" is time.<BR/><BR/>That's how it works. To exponentiate an operator means to repeat some operation many times, for some "time" or "angle" or "distance" etc. Why? It's because<BR/><BR/>exp(X) = lim (1+X/N)^N,<BR/><BR/>where N is infinite. You make the small operation, multiplicatively changing/scaling/transforming 1 to 1+X/N, N times, where N is a large number, and you will end up with an exponential growth. Imagine N=100, which is almost infinite, and X being the interest rate. (1+X/N) is the money you have from $1 after one year, and you multiply it by itself 100 times - the money is going to grow exponentially.<BR/><BR/>And in statistical physics, recall Maxwell-Boltzmann distribution, the operators and distributions and density matrices go like exp(-H/kT), also an exponential of the Hamiltonian. I would have to explain 1/3 of statistical physics here.<BR/><BR/>It's almost the same exponential but with a real coefficient instead of imaginary. You can flip in between them if you add (or remove) i. BTW, I discussed this thing in more detail in L'Equation Bogdanov. ;-)<BR/><BR/>So the evolution by an imaginary time is equivalent to adding the thermal exponentially decreasing factors to the probability amplitudes.. Tracing makes the time coordinate periodic, and so forth.<BR/><BR/>OK, when I read the rest of your text, it seems that the (non)difference between mathematics and physics is a deeper part of your problem.<BR/><BR/>You know, the world *is* mathematics. Mathematics is just the right, careful way to construct logical arguments and predict things in science or elsewhere. If you want to do things right, you need mathematics.<BR/><BR/>If you want a more famous guy who is saying the very same thing, that maths is needed (for the laws of gravity, in this case), see this <A HREF="http://www.youtube.com/watch?v=SeQ3M1JVodk&feature=PlayList&p=037BEC9B7C19BA57&index=1" REL="nofollow">Feynman lecture</A>. ;-)<BR/><BR/>So in this context, again, I can tell you: to describe the thermal properties and evolution of quantum systems, you need very accurate functions. It just happens that these functions have parameters like time and temperature, but the remaining structure is identical.<BR/><BR/>So it is a mathematical fact - which also means a physical fact - that by substituting imaginary number as an unphysical value of the parameter, you can get to the other situation. The imaginary value of time or temperature is unphysical, you can't see "i" on clocks or thermometers. <BR/><BR/>But you can calculate with it, and it is often more convenient, easily definable, and more controllable to deal with imaginary parameters, and only switch back to the reals at the very end of the calculation.<BR/><BR/>There are all reasons why imaginary time is more well-behaved. The exponentially decreasing thermal functions are more convergent etc. The opposite map can be useful, too.<BR/><BR/>So this whole concept whether something is "physical" in your sense is simply misguided. In physics, you are asking whether things are right or wrong, and the word "physical" means something different. If something uses mathematics to be calculated, it does NOT mean that it is physically incorrect. Quite on the contrary. On the contrary, abstract maths is often crucial.Luboš Motlhttps://www.blogger.com/profile/17487263983247488359noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-17435137566059269472008-12-16T01:19:00.000+01:002008-12-16T01:19:00.000+01:00Lumo, thanks for the nice article. It really helps...Lumo, thanks for the nice article. It really helps put things in perspective. <BR/><BR/>Since you are quite familiar with this topic, I wanted to ask you the following question. (I do condensed matter QFT for a living - but I am not familiar with relativistic QFT nor string theory, it's just not my bread and butter.)<BR/><BR/>Can you explain to me (no mathematical arguments if possible - I understand most advanced math, but here I am looking for a physical explanation) why is it valid to map temperature into imaginary time, when I am looking for equations of motion.<BR/><BR/>The example I have in mind is that of the Gross-Pitaevskii (GP) equation. If I write out the quantum partition function, convert it into a coherent state bosonic path integral and take the continuum limit (of the time-slicing), call temperature imaginary time, and take the functional derivative of the action, I get the GP equation. (This is also called the stationary phase approximation, and is found in most standard textbooks.) This gives the correct expression. This analytic continuation to imaginary time is what is done in most texts. <BR/><BR/>But why would it be valid physically? How can one call this real time dynamics (equation of motion), when we simply called temperature time. It just doesn't make sense to me. I know the operation is valid mathematically, I don't have a problem with that. What I have a problem with is calling it the real dynamics.<BR/><BR/>The real quantum dynamics can only be obtained by solving the liouville von neumann equation, which gives a completely different answer. I don't see how this Wick rotation adds any new physics. After all, the stationary phase approximation is an approximation. The whole thing seems dubious to me.<BR/><BR/>Anthony Leggett in his book calls this derivation questionable at best. <BR/><BR/>And I know some experimentalists working on bose condensates, who can't verify this equation with experiments. <BR/><BR/>I would appreciate your help in clarifying what appears to be a very murky situation at best. <BR/><BR/>ThanksUnknownhttps://www.blogger.com/profile/05123955073445826494noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-45297426596988864312008-12-16T01:18:00.000+01:002008-12-16T01:18:00.000+01:00Lumo, thanks for the nice article. It really helps...Lumo, thanks for the nice article. It really helps put things in perspective. <BR/><BR/>Since you are quite familiar with this topic, I wanted to ask you the following question. (I do condensed matter QFT for a living - but I am not familiar with relativistic QFT nor string theory, it's just not my bread and butter.)<BR/><BR/>Can you explain to me (no mathematical arguments if possible - I understand most advanced math, but here I am looking for a physical explanation) why is it valid to map temperature into imaginary time, when I am looking for equations of motion.<BR/><BR/>The example I have in mind is that of the Gross-Pitaevskii (GP) equation. If I write out the quantum partition function, convert it into a coherent state bosonic path integral and take the continuum limit (of the time-slicing), call temperature imaginary time, and take the functional derivative of the action, I get the GP equation. (This is also called the stationary phase approximation, and is found in most standard textbooks.) This gives the correct expression. This analytic continuation to imaginary time is what is done in most texts. <BR/><BR/>But why would it be valid physically? How can one call this real time dynamics (equation of motion), when we simply called temperature time. It just doesn't make sense to me. I know the operation is valid mathematically, I don't have a problem with that. What I have a problem with is calling it the real dynamics.<BR/><BR/>The real quantum dynamics can only be obtained by solving the liouville von neumann equation, which gives a completely different answer. I don't see how this Wick rotation adds any new physics. After all, the stationary phase approximation is an approximation. The whole thing seems dubious to me.<BR/><BR/>Anthony Leggett in his book calls this derivation questionable at best. <BR/><BR/>And I know some experimentalists working on bose condensates, who can't verify this equation with experiments. <BR/><BR/>I would appreciate your help in clarifying what appears to be a very murky situation at best. <BR/><BR/>ThanksUnknownhttps://www.blogger.com/profile/05123955073445826494noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109964211153597852005-03-04T20:23:00.000+01:002005-03-04T20:23:00.000+01:00Fyodor,
Three dimensional gravity is only topolog...Fyodor,<br /><br />Three dimensional gravity is only topological if there is no matter. With matter fields, it will no longer be topological. True, there are no gravitons in three dimensions, but that doesn't mean four dimensional gravity is "better behaved" with respect to Wick rotations. It's purely a matter of faith that Wick rotation works in four dimensions when it does not in three.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109928904548397362005-03-04T10:35:00.000+01:002005-03-04T10:35:00.000+01:00Henry wrote:
Why should we assume our four dimensi...Henry wrote:<br />Why should we assume our four dimensional universe is any different qualitatively from a three dimensional universe?<br /><br />Because there is no dynamics in 2+1, gravity is topological. It's utterly different. *maybe* we can learn something useful from it, though frankly I doubt this. I think that we should study 2+1 and 1+1 just to get some inspiration, as in the Ooguri-Vafa-Verlinde paper.<br /><br /> If Wick rotation fails in three dimensions, isn't that disturbing enough?<br /><br />No, it does not disturb me at all, since I would not expect gravity in 3 d to be anything like gravity in the real world!<br /><br />FyodorAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109889286770119442005-03-03T23:34:00.000+01:002005-03-03T23:34:00.000+01:00Fyodor,
Why should we assume our four dimensional...Fyodor,<br /><br />Why should we assume our four dimensional universe is any different <I>qualitatively</I> from a three dimensional universe? If Wick rotation fails in three dimensions, isn't that disturbing enough?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109856573585913502005-03-03T14:29:00.000+01:002005-03-03T14:29:00.000+01:002+1 Gravity is an example where the Lorentzian the...2+1 Gravity is an example where the Lorentzian theory and the Euclidean theory yield manifestly different results. <br /><br />I would conclude from this that 2+1 gravity [in this particular incarnation] has absolutely nothing to teach us about the real world. We study lower-dimensional gravity in the hope that it will teach us something about 4 dimensions. Sometimes it does, more often it doesn't. <br />Fyodor UckoffAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109854526783517802005-03-03T13:55:00.000+01:002005-03-03T13:55:00.000+01:00One of the anonymous comments asked for an example...One of the anonymous comments asked for an example. Well reread the thread.<br /><br />2+1 Gravity is an example where the Lorentzian theory and the Euclidean theory yield manifestly different results. I think we all must agree that the real world is Lorentzian, so the latter must at first glance be incorrect (or we have made a blunder somewhere).<br /><br />Ironically (at least to me), one of the ways to rectify the two different theories in that particular case is to use Ambjorn's regularization prescription. I'm sure Lubos won't be pleased that he's siding with some of his arch enemies =)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109783278056514902005-03-02T18:07:00.000+01:002005-03-02T18:07:00.000+01:00No, I don't. It's still true that locally one can ...No, I don't. It's still true that locally one can set (-det g)=1 by a diffeomorphism. It's still true that all these potential problems are harmless in all physically meaningful quantities, as demonstrated by stringy S-matrix. <br /><br />It's still true that the Euclideanization does not make things *worse*. And it's still true that any spacetime understanding of these processes and path integrals will probably require *more* continuation and complexification, not less.<br /><br />And by the way, I encourage the participants not to strengthen their cases by posting under various nicknames, including "anonymous".Luboš Motlhttps://www.blogger.com/profile/17487263983247488359noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109782727145192182005-03-02T17:58:00.000+01:002005-03-02T17:58:00.000+01:00By retracting your comment, you are essentially ad...By retracting your comment, you are essentially admiting Distler's objections are perfectly valid.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109729756335207192005-03-02T03:15:00.000+01:002005-03-02T03:15:00.000+01:00To all the anti-Euclidean guys out there, show us ...To all the anti-Euclidean guys out there, show us an explicit model where Wick rotation does not work and spell out all the relevant details. The excuse that you don't know how to do it does <I>not</I> count. You have to prove that Wick rotation cannot work in any way, no matter how clever we are. Do you accept my challenge or are you unable to?<br /><br />All this anti-Euclidean sentiment is getting out of hand. For once, I am with Lubos on this issue.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109681837779808732005-03-01T13:57:00.000+01:002005-03-01T13:57:00.000+01:00Thanks for your insightful comments, ladies and ge...Thanks for your insightful comments, ladies and gentleman (I assume that the ratio is about 50:50).<br /><br />Let me retract the comment that the negatively-normed scale factor is pure gauge. ;-)<br /><br />On the other hand, I would continue to say that this apparent problem is resolved if quantum gravity is done properly, and if we want to do so using a spacetime perspective, it's almost essential to use a proper analytical continuation to complex values of coordinates (and perhaps the metric).<br /><br />All the best<br />LubosLuboš Motlhttps://www.blogger.com/profile/17487263983247488359noreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109647269498739362005-03-01T04:21:00.000+01:002005-03-01T04:21:00.000+01:00> the divergent scale factor in
> the path integra...<I>> the divergent scale factor in<br />> the path integral is pure gauge.<br />> One can always choose a gauge of<br />> diffeomorphisms in which (det g)<br />> =-1. There's enough freedom to <br />> do it. The divergence goes away.<br /></I>Hardly. Diffeomorphically equivalent metrics have the same action. The unboundedness is not due to gauge freedom and you can't gauge fix the unboundedness away. While gauge fixing will take care of divergences due to the "noncompactness" (I know, I know, compactness doesn't really apply to infinite dimensional manifolds) of the diffeomorphism orbits, it has absolutely nothing to do with the divergence due to e^{-S} as S goes to -infinity.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109646059484479882005-03-01T04:00:00.000+01:002005-03-01T04:00:00.000+01:00forget about quantum gravity. Wick rotation can be...forget about quantum gravity. Wick rotation can be tricky enough even in flat spacetime, especially with spinors. See, for example<br /><br /><A HREF="http://arxiv.org/abs/hep-th?9608174" REL="nofollow">http://arxiv.org/abs/hep-th?9608174</A>Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109641980218922342005-03-01T02:53:00.000+01:002005-03-01T02:53:00.000+01:00My dearest Luboš,
It is our luck that our world h...My dearest Luboš,<br /><br />It is our luck that our world happens to be four dimensional. No, no, no, not 10, not 11, not 26. Not even 2! :-þ Current experimental evidence rules them out. :-þ :-þ I am pretty sure someone of your genius would know this already, but I will remind you anyway; three dimensional QFT's admit the possibility of anyons and anyons cannot be Wick rotated. So there!<br /><br />HenryAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109641088558704782005-03-01T02:38:00.000+01:002005-03-01T02:38:00.000+01:00Dear Lubos,
Wick rotation is only possible becaus...Dear Lubos,<br /><br />Wick rotation is only possible because of causality. In fact, both of them are very closely related. Without one, one cannot have the other. This is why Wick rotation fails completely for noncommutative geometries and general relativity. The former only has macrocausality, not microcausality while for the latter, the causal structure depends upon the metric which turns out to be a dynamical field. Because of this, it is probable that causality will break down at the Planck scale.<br /><br />Let me conclude with the following slogan;<br /><br />"No causality, no Wick rotation!"<br /><br />HenryAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109612353904365072005-02-28T18:39:00.000+01:002005-02-28T18:39:00.000+01:00Lubos,
I am not sure why you are writing that str...Lubos,<br /><br />I am not sure why you are writing that string theory has nothing to say about the issue. The vertex ops.in string theory are on-shell in Lorentzian signature, there typically would be none in Euclidean space. The only observables that make sense are scattering amplitudes of such states, which are inherently Lorentzian. The worldsheet being Euclidean is a distraction, the target space is always Lorentzian.<br /><br />There is no doubt that in ordinary second quantized QFT Euclidean methods are sometimes important, sometime not (as in real-time finite temp.). You decide based on whether or not you have a good physical interpretation of what you are calculating. By physical I mean almost by definition Lorentzian, these are the things we measure.<br /><br />The issue with Euclidean QG has to do with whether we should take the path integral over metrics and its Euclidean continuation so seriously as to make it the defining principle of our theory, in regimes where Lorentzian intuition does not apply. Both components of this approach seem to me orthogonal to the way perturbative string theory works.<br /><br /> <br />best,<br /><br /><br />MosheAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109610793507493702005-02-28T18:13:00.000+01:002005-02-28T18:13:00.000+01:00Well, if it's not called "t", then it's probably "...Well, if it's not called "t", then it's probably "tau" and/or "rho", depending on the picture, is not it? ;-)<br /><br />What if it's called theta? :-)<br /><br />In the context of quantum gravity, one must be prepared to complexify and rotate even more things than what we're used to from QFT in the Minkowski space.<br /><br />Joking aside, I would like to say that I, too, am on Lubos's side in this one. Maybe I am not so confident about *why* euclideanization works, but I am sure that declaring that something is junk on the basis of philosophical principles is not the way to do physics. The opponents of Wickism should either come up with a concrete demonstration that it won't work the way Lubos wants it to, or they should let him get on with pushing it as far as it will go. I'm making this declaration because I detect a sort of vague anti-euclidean sentiment in the community and I think that could be very harmful. As Lubos says, we need more Wicks, not less! :-) The OVV paper is a good example of the way to go.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8666091.post-1109606606941567372005-02-28T17:03:00.000+01:002005-02-28T17:03:00.000+01:00Well, if it's not called "t", then it's probably "...Well, if it's not called "t", then it's probably "tau" and/or "rho", depending on the picture, is not it? ;-)<br /><br />The point is that there are many ways to do the calculations. All procedures that are done properly are fine.<br /><br />In the context of quantum gravity, one must be prepared to complexify and rotate even more things than what we're used to from QFT in the Minkowski space.<br /><br />The new subtleties of GR indicate that we will have to spend *more* time with the continuations and rotations, not less.Luboš Motlhttps://www.blogger.com/profile/17487263983247488359noreply@blogger.com