## Sunday, February 27, 2005 ... //

### Wick rotation

In the discussion under the article The entropic principle about the recent paper by Ooguri, Vafa, and Verlinde, there has been a significant opposition of many participants against the concept of the Wick rotation - one invented by a renowned and virtually unknown physicist Gian-Carlo Wick. They were saying that this mathematical method can't be trusted; they were comparing the use of the Wick rotation to the idea that physical theories should not be tested experimentally. Because I believe that most of this criticism is unfair and the Wick rotation is a useful, and in many cases essential mathematical tool to calculate the physical predictions of a quantum theory, let me dedicate a special article to this issue. Peter Woit added his comments about the Wick rotation, too.

First of all, a summary

The Wick rotation is a calculational trick in quantum theory in which we assume that the energy or the time are pure imaginary. We do the calculations given these assumptions, which are often more well-defined, and then analytically continue the results back the usual real values of time and/or energy. It works. But let's now look at the situation a little bit more closely.

Behavior of path integrals

According to Feynman's approach to quantum mechanics, the probability amplitudes may be calculated as the sum (well, a path integral) over all conceivable classical histories of the physical system. Each of them is weighted by

• exp (i.S/hbar)

where "S" is the classical action calculated for this history. As you can see, the absolute value of this weight is always equal to one as long as "S" is real. From a naive viewpoint, that does not seem to be a good starting point for a convergent integral; the integral keeps on oscillating. Convergence is improved if we add a small negative real part to the exponent. Write the action as

• S = int dt L

and imagine that "dt" has a small imaginary part. You obtain the weight

• exp (i.(int dt (1+i.epsilon)).L/hbar).

Because of the term proportional to "-epsilon" in the exponent (i.e. because of the factor "exp(-epsilon.S)", roughly speaking, the contribution of the configurations with a large action will be exponentially damped, and the convergence will improve. This regularization is applied both to ordinary quantum mechanics as well as quantum field theory. In the latter case, it's the origin of the "i.epsilon" prescriptions for the propagators etc. While the naive Feynman's prescription is obviously reproduced for "epsilon" going to zero, a tiny nonzero value of "epsilon" is essential for making the path integral convergent.

The Wick rotation

This was not the Wick rotation yet, but I hope that the inevitability of this "epsilon" treatment is obvious to everyone: the simple prescription of Feynman is a heuristic inspiration, and the oscillating path integral must be regulated in some innocent way. The "i.epsilon" prescription is the way that preserves all symmetries. Not a big deal. Now let's look at the real Wick rotation.

Imagine that the degrees of freedom in your theory - either quantum mechanics or quantum field theory - are defined not only for real values of time "t", but for complex values. The action is the integral "int dt L". Let's now integrate over a contour in the "t" complex plane, while the time-derivatives in the Lagrangian should also be treated as derivatives with respect to this "dt" which is complex. If the contour is taken to be in the purely imaginary direction, "dt" in the integral will get an extra factor of "i", while the terms bilinear in the time-derivative will flip their sign. One of the results is that the weight of the configuration is effectively changed to

• exp(-S_E/hbar)

where "S_E" is a "Euclidean" action, which is typically a non-negative number; its definition differs from the usual action by changed signs of the kinetic terms that are bilinear in time derivatives, and the overall sign. You see that this exponential dies away if "S_E" becomes large. The contributions decrease very quickly as "S_E" grows and the path integral is even "more convergent" than in the "epsilon" example at the beginning.

What is the physical meaning of these operations? We're essentially continuing the physical results analytically to complex values of time "t". For example, the evolution operator

• exp (H.t/i.hbar)

is continued - if we substitute "t = -i.beta.hbar" - into the density matrix

• exp (-beta.H)

describing the thermal ensemble at temperature "T = 1/beta". Note that the exponential is a holomorphic function. Therefore, the evolution operator "exp (H.t/i.hbar)" is a holomorphic function of the complex variable "t". The matrix elements of it and other physically relevant observables will be holomorphic functions, too. Note that we're not doing anything that would contradict experiments or something like that. We're just using the fact that it is possible to calculate various other functions of a well-defined operator "H". Equivalently, in the path-integral language, it is possible to calculate various other, more convergent quantities out of a formula for the action.

Wick rotation in quantum field theory

In relativistic quantum field theory, the Wick rotation is particularly useful. The analytical continuation of "t" into purely imaginary value effectively converts the Minkowski spacetime into the Euclidean spacetime.

(For time-dependent backgrounds, the nature of the Wick rotation is more subtle. However, locally in spacetime, it's the same problem as in the Minkowski space, and globally, it's likely that we may be forced to learn how to do the Wick rotation even in these more subtle backgrounds in order to get final results. The continuation to imaginary time is definitely important even for time-dependent backgrounds. For example, Maloney, Strominger, and Yin have used the Wick rotation to understand physics of a very specific time-dependent background in string theory.)

Why is it so? Note that Einstein's favorite formula to write down the Lorentz-invariant line interval was

• ds^2 = dx_1^2 + dx_2^2 + dx_3^4 + dx_4^2

where "x_4 = i.c.t". Notice that this formula has the form of the ordinary Pythagorian theorem in four dimensions, except for the pure imaginary value of "x_4". Now it's obvious what we're going to do. If we're interested in the Green's functions, we first calculate them in the Euclidean spacetime where "x_4" is real. We express them using the four-dimensional momenta "k" via the Fourier transform, and analytically continue to a pure imaginary value of "k_4" (to be interpreted as "i.k_0" in the Minkowski space).

The Wick rotation is legal

It is legitimate simply because the physical quantities expressed as functions of the momenta are naturally seen to be holomorphic functions of the momenta. Well, up to some singular points. One can see that the only allowed singular points in the physical quantities expressed as functions of the momenta - in the propagators, for example - are simple poles (corresponding to bound states or quasinormal modes, in the simplest examples) that can perhaps join into a branch cut. However, these functions are locally holomorphic. It's because of the very basic properties how all these functions are understood and calculated - they're treated as holomorphic functions. A physically usable function of the real variable (for example, the energy as a function of the momentum) can be extended into a holomorphic function of a complex variable.

Why are the answers analytical functions of energy

While we defined the relevant integrals for the action to based on complex values of "t", it's actually more important that the observables, such as the Green's functions, are analytical functions of the momenta. The same basic idea applies to quantum field theory and quantum mechanics. In quantum field theory, we usually want to talk about the holomorphic dependence of the Green's functions on the Lorentz-invariant functions of the momenta - for example, the dependence of the propagators on "k_m.k^m" (which typically has poles).

But without loss of generality, we may talk about the continuation in the time direction only. Therefore we want to see how the Green's functions behave if we continue the energy (the complementary variable to the time, via the Fourier transform) to the complex values. And this problem can already be addressed in quantum mechanics. Just take the evolution operator "exp(H.t/i.hbar)", which encodes all dynamical information, and Fourier-transform it with respect to the c-number variable "t". (A technicality: multiply it first by "theta(t)" so that it's only nonzero for positive "t".) You will get another operator-valued function of "E" (the dual variable to "t") that encodes all dynamics. It's not hard to see that this operator will be essentially - up to some "i.epsilon" in the denominator

• 1 / (H-E)

where "H" is the Hamiltonian (an operator) while "E" is a c-number parameter. Note that this operator-valued function of "E" is clearly a holomorphic function of "E" - up to the simple poles that correspond to the eigenvalues of "H" (when "H.psi=E.psi", then "H-E" can't be inverted) or branch cuts that arise from a continuum of eigenvalues of "H". Nevertheless, the function is a locally holomorphic function of "E". This fact is generalized to quantum field theory where "1/(H-E)" is generalized to the more general Green's functions, and it is the real mathematical reason that shows why the Wick rotation is legitimate.

Emotions and prejudices vs. reality

Someone may dislike these mathematical operations and continuations. But it's not important whether someone dislikes them. The important question is whether they can be done and whether they're useful. Whether they can be done is a mathematical question about a very broad class of physical theories, and the answer to this mathematical question is Yes.

Loops in the Euclidean spacetime are more well-defined

The answer to the question whether the continuation to the Euclidean spacetime is useful is also Yes. Let me enumerate several basic advantages:

• The convergence properties of the path integral are better; exp(i.S/hbar) is replaced by exp(-S_E/hbar) as we discussed above
• When we evaluate loop Feynman diagrams, we obtained - in the momentum representations - nice integrals over the 4-dimensional Euclidean momenta; they can be written in polar coordinates and the non-trivial, radial part can be evaluated to give us results that preserve the SO(4) symmetry; consequently, the analytical continuation back preserves the Lorentz symmetry SO(3,1)
• In the Minkowski space, it would be much more subtle to decide how the divergent integrals should be regulated in such a way that the Lorentz invariance is preserved; the best definition how to regularize the Minkowski loop diagrams properly is probably to say that the methods should follow the calculations in the Euclidean spacetime

Non-perturbative physics and the priceless Wick rotation

While the loop diagrams are manifestly better in the Euclidean setup, there are other aspects of our calculations for which the Euclidean setup is almost inevitable:

• Instantons are non-perturbative contributions to various real processes. They can be visualized as topologically non-trivial field configurations in the Euclidean spacetime that are localized in all directions including the Euclidean time. I think that no one (or almost no one) knows how to calculate the effects induced by the instantons directly in the Minkowski space
• Instantons appear not only in field theory, but also in string theory - string theory also adds new types of instantons such as the D-instantons, and the importance of the language of the Euclidean spacetime is not reduced at all
• In perturbative string theory, the perturbative S-matrix is calculated as the path integral over all Euclidean two-dimensional Riemann surfaces that represent histories of interacting strings or Euclidean worldsheets embedded into the Euclidean spacetime; the Euclidean character of the worldsheet is very important for the covariant calculations because almost no one knows how we should even talk about the topology expansion if the Riemann surfaces were Lorentzian (the Minkowski worldsheets are most natural in the light cone gauge)
• The Euclidean path-integral calculations are also extremely helpful for calculating the thermal properties of a quantum field theory, because of the relation between the thermal density matrix and the evolution operator continued to imaginary times that we mentioned above
• The Euclidean path integral may also be necessary for the understanding of the initial conditions of the Universe, as shown by Hartle and Hawking; this state has also been described in various minisuperspace approximations in string theory, for example by Ooguri, Vafa, Verlinde, and by Karczmarek, Maloney, Strominger
• Most of this text is about quantum mechanics and non-gravitational quantum field theory, but it is reasonable to believe that the path integral in the Euclidean spacetime is gonna be even more important in quantum gravity than it is in quantum field theory. The Euclidean version of a black hole offers a nice explanation of its thermodynamics (the Hawking temperature is determined by the vanishing deficit angle at the horizon). The gravitational instantons, such as Witten's bubble (which is a related solution to the Euclidean black hole), are important for our understanding of instability of various backgrounds. The Hartle-Hawking state from the previous point is another example.

I hope that the text above shows that the technique of the Wick rotation is a legitimate - and in many cases inevitable - tool to find the predictions of a physical theory. If we don't want to jump to the difficult waters of calculating the loop Feynman diagrams, instantons, and other effects directly in the Minkowski spacetime, we may even consider the Wick rotation as a subtle technical part of our definition of quantum field theory.

The results involving the Wick rotation have been tested

What I want to emphasize is that the idea that the Wick rotation is something "extraordinarily suspicious" that deserves "more experimental tests" than other concepts and hypotheses is a completely irrational idea. The Wick rotation is a subtle mathematical tool to make many calculations of ours more meaningful and we understand on theoretical grounds why it should work. We know why this procedure preserves the desirable features of a physical theory that are necessary for its consistency.

Also, even if you assumed that the Wick rotation seems to be a new added element to the structure and definition of quantum field theory, it's completely fine that it is so because the predictions of the (correct) theories, even those that rely on the Wick rotation, have been experimentally tested. In fact, the predictions that have involved the Wick rotation have been more successful than those that did not. This includes the multi-loop corrections to the electron's anomalous magnetic moment. These observables have been calculated in the Euclidean spacetime. The path integral in the Euclidean spacetime is always useful, especially for the questions that can be deduced from the S-matrix. This is true in field theory as well as string theory (including vacua at different dimensions than "d=4").

Failing Wick rotation - a sign of inconsistency

Moreover, if you find a theory in which the Euclidean calculations do not give the results that would seem to reproduce the Minkowskian physics, you should be highly skeptical about such a theory because it is unlikely that this theory will be able to agree with basic physical requirements such as the Lorentz invariance of local physics. An example is loop quantum gravity. There is not just one loop quantum gravity; there are thousands of different proposals what loop quantum gravity should be, and the "Euclidean vs. Minkowski" question is one of many questions that separate different loop quantum gravities to classes. This is definitely another sign of physical inconsistency.

Future and speculations

This article has mostly discussed the aspects of the Wick rotation that have been established. It remains to be seen how the future understanding of physics will view the Wick rotation and continuation of various quantities to complex values of the real observables. The Wick rotation may remain a calculational trick, but the complexified time or energy may also offer us some new important insights about quantum gravity - for example about the black hole information paradox. There are new things to be learned in quantum gravity. In quantum gravity, for example, one can argue that one should work with complex values of the metric in the Planckian regime in order to cure the unboundedness of the Einstein-Hilbert action from below (even the Euclidean action, one that is usually bounded in other theories). Note that these comments are largely moral in character; in the consistent theory of gravity, namely string theory, we don't compute these things by direct path integrals over metrics; instead, string theory cures most of the potential problems without telling us how it was done. ;-)

And finally, the Wick rotation is able to be more controversial than the Iraq war and innate differences. A discussion in which the old alliances will be rearranged and in which the smallest Euclidean and Minkowskian biases will be magnified is getting started, so enjoy. ;-)

#### snail feedback (69) :

The usual reason to be suspicious of Euclidean quantum gravity (in d≥4) is that the Euclidean action is unbounded from below. This is an infrared problem, and is not cured by replacing the Einstein-Hilbert action by its ultraviolet completion (string theory). Beyond the semiclassical approximation, it is unlikely that there's any sense whatsoever to Euclidean quantum gravity.

Whether that tarnishes your "love" for the Hartle-Hawking wave function is up to you.

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.Hermann Minkowski,When looking at the dynamical nature of a loop using Greg Egan's boxed coordinates it seems very real this movement recognizing the loops color changes, with context of the frame of reference.

In this same way, I find it relevant if you were to define the gravitational waves in regards to the graviton, as mentioning the hologrpahical detail of a spot on that bubby structure.

You of course had to leave the confines of spacetime and higher to realize, that it could be done, from spacetime to a euclidean reference?

Gluon perception(feynman's path integrals) is still relevant here, and speaks to the realization of the energy in that early universe.

How dynamcial this environment is, is still very troubling issue, but from another point of view, graviton scattering is not?:)

A laymen perspective and hopefully I did not detract from the issues.

First of all, my article was focused on QFT, not quantum gravity, but despite this fact, I disagree with your comment, Jacques.

The reason why I don't think that your reasoning makes any sense is that it is the type of reasoning that says "A", but does not say "B", and then it "deduces" the statement "not B".

I agree that the Euclidean action is unbounded from below and it's a naively infrared effect; I refine this point at the end of this answer. (That's the statement "A".) But the Minkowskian action is *also* unbounded from below. (That's the statement "B" that you did not mention.)

The negative sign of the kinetic term of the scale factor is a physical, objective fact that has nothing to do with the choice of the signature where one wants to do calculations. The Minkowski signature certainly does not make these things *better*. And if you decided to use the opposite i.epsilon prescriptions for the scale factor and the rest, or something along these lines, be sure that you will break the diffeomorphism invariance.

And if you wanted to claim that because of this problem with the negative factor, the S-matrix for gravity in d>=4 can't exist, then I hope that everyone would agree that such a statement has already been falsified. String theory does reduce to GR in the infrared, and its calculation of the S-matrix is obviously not invalidated by the kind of problems that you describe.

This means that the problem you described is either fixed by string theory, or irrelevant for physical calculations.

Incidentally, when you talk about UV and IR, you should be much more careful about distinguishing the proper distances and coordinates. The fact that your described problems look like the IR problems in the coordinate space does not mean that they're insensitive to the UV completion of the theory - where the adjective "UV" refers to the proper distances. It's exactly because the scale factor, which is the source of the potential problems you describe, can change the scales a lot, especially if it goes to infinity (which is what you need to allow if you want to use its unboundedness from below).

Here is Greg Egan's site and the link I was speaking too.

Play with the coordinates:)

I forgot to say one more simple reason why Jacques' criticism is most likely vacuous:

the divergent scale factor in the path integral is pure gauge. One can always choose a gauge of diffeomorphisms in which (det g)=-1. There's enough freedom to do it. The divergence goes away.

Any good treatment of the GR path integral does the calculations in such a way that this potential problem is harmless. String theory does it automatically.

The Lorentzian action is never bounded from below. Nor is that what you want, as the integrand of the Lorentzian functional integral is oscillatory. The Euclidean functional integral, in any case where the Euclidean QFT makes sense, is bounded from below.

This has nothing to do with whether the S-matrix for quantum gravity exists. No one was suggesting that it doesn't. We were talking about the Hartle-Hawking wave function, and whether it makes sense.

(Do you always have to throw up these straw-man arguments? It makes discussions with you considerably more tedious than they have to be.)

And, if you think you have an argument that the "IR" nature of this sickness of Euclidean quantum gravity is, in any sense, an artifact of considering coordinate versus proper distances, please spell it out. It will be entertaining to hear it.

There are lots of reasons to believe (most lucidly explained in Witten's treatment of the 2+1 d case) that the Euclidean and Minkowskian quantum gravity theories are utterly independent. In some cases, both might make sense; in others, only the Minkowski theory does. When they both make sense, they're not, generally, related by Wick rotation.

You say that the physical quantities (like S-matrix elements) are analytical in momenta and add the caveat "except maybe for singular points". Right you are, but you understate the importance of this caveat. Look at QFT in flat space with a constant electric background. You find that there is an infinite series of poles on the imaginary time axis. They correspond to multi-instanton contributions. Thus when you try to rotate the time contour by 90 degrees you will hit these poles and you are in trouble. Should I be suspicious of this theory? Not at all, it has been given a satisfactory physical interpretation as it stands. Nevertheless, our understanding of it is incomplete. For example, we can never couple such a theory to gravity even at the most basic semi-classical level. To do this, one must understand the theory directly in Minkowski space, and we don't know how to do that. Wick rotation has no hope or promise of rectifying this. It is what it is: a method of making things doable with a quite limited realm of applicability.

When you say that in time-dependent backgrounds we need to understand how to define the Wick rotation -- isn't it a polite way of saying that we don't understand it? If you look at the mathematical difference between, say, Euclidean and Minkowskian 4-manifolds, there is every indication to believe that, in going from the latter to the former, a substantial amount of physical information is lost. Unless you invent a *new* method of preserving this information, you don't understand the theory well. This has not been done yet.

Also, unlike the Euclidean manifolds, the Lorentzian ones are very hard to classify. This makes the definition of path integrals difficult -- how do you sum over a set of things when you don't really know what they are? I know, gravity wasn't the main subject of your article. But nevertheless, as a string theorist you should worry about gravity. Perturbative string theory is largely defined through Wick rotations and it is time to realize that this can only be a first step. There is more to the story than naive Euclidean physics.

Best,
Dan

Whoops! That's "The Euclidean action, ... is bounded from below"

And, just to drive the point home, if you think that String Theory somehow makes sense of Euclidean quantum gravity, please tell me this:

What are the observables of Euclidean Quantum gravity for d≥4?

If Euclidean String Theory made sense, it would tell us what the observables are.

Good discussion of the Wick rotation. Yes it works in so many cases. However, it is only the causality properties of the Lorentz signature QFT on flat space (essentially Feynman's iepsilon prescription) that underlies and gives any justification to the
use of a Wick rotation. The
corresponding Euclidean QFT is then assured the property of Osterwalder-Schrader(OS)positivity from the casual structure of the underlying QFT which is Lorentz invariant. For flat space QFTs like Yang Mills this is well established of course. When you Wick rotate gravitational theories this underlying structure is simply absent so you are walking on thin ice. You then also have the thorny technical issues of defining the path measure over manifolds and topologies etc. Hawking's semiclassical Euclidean quantum gravity has produced neat results--black hole entropy, the no boundary HH models--but the limitations are obvious. The information paradox of black holes is certainly most likely an artifact of the semi-classical approximation and you can't push it any farther. The term "wave function of the universe" has never made any sense to me though--and many other people--at most you have some sort of functional that seems to solve the WDW, which is just a constraint condition. The whole realm of "quantum cosmology" that arises from EQG is pretty murky to say the least. I think Jacques recently described the "wave function of the universe" as the physics equivalent of Godwin's law and he may be right:) Anyway, Wick works well for theories on flat Lorentzian space. Don't need convincing of that. For gravitational theories--dodgy.
Best
Steve

You misunderstood my objections to Wick rotation for Euclidean quantum gravity (as usual, you always misunderstand anyone you disagree with).

I never claimed Wick rotation doesn't work for quantum field theories defined over Minkowski spacetime. In fact, there is a theorem, the Osterwalder-Schrader theorem which states that provided the Euclidean theory satisfies certain properties like the mysterious reflection positivity, it's possible to analytically continue it to an ordinary quantum field theory. But that relies on certain special properties of Minkowski spacetime like the commutativity/anticommutativity of spacelike separated fields and the restriction of the energy-momentum spectrum to the positive light cone so that the Wightman DISTRIBUTIONS can be extended to analytic FUNCTIONS over a domain which can be extended by complexifying the Lorentz group and further extended due to the commutativity/anticommutativity of the fields to the symmetric Schwinger functions. (but you don't like axiomatic quantum field theory, I know)

Similarly, for conformal field theories, there are theorems telling us Wick rotations are OK.

But you wish to justify Wick rotations for quantum gravity, which is an entirely different matter. There are no corresponding theorems telling us it is OK to Wick rotate. In fact, if we try the same trick of looking at the permutated extended tube, we not only find that Euclidean metrics aren't in the domain but also it doesn't make sense to insist spacelike separated fields commute/anticommute when the whole concept of spacelike separation depends upon a dynamical quantized metric field.

According to Feynman's approach to quantum mechanics, the probability amplitudes may be calculated as the sum (well, a path integral) over all conceivable classical histories of the physical system. Each of them is weighted by

exp (i.S/hbar)
where "S" is the classical action calculated for this history. As you can see, the absolute value of this weight is always equal to one as long as "S" is real.
That's not true. There are some theories, like nonlinear sigma models where the absolute weight isn't 1.

Hey, can you tell me more about quantum field theory with a constant electric field background? I've never seen it related to instantons before. Isn't F wedge F zero for a constant electric field (with no corresponding magnetic field)?

Dear Jacques, thanks for your message. I think that you're confused. Your arguments remotely remind me of Peter Woit's logic "you can't build a Planckian collider, and therefore string theory must diverge at three loops". ;-)

I don't think that your statements follow from each other by anything that could be called "logic".

You argue that the Euclidean path integral for gravity is ill-defined. The gravitational S-matrix from string theory, continued to Euclidean variables, exists, as you admit. This either means that your argument is incorrect as it is, or it means that you must explain why the S-matrix is OK while the HH state is not.

You probably can't prove such a statement - especially because the statement is probably not true and the explicit construction of Ooguri, Vafa, and Verlinde is a counterexample of your statement in a specific setup.

If there is a disagreement between the Euclidean and Minkowski description of a physical systems in d>=4, the quantities calculated in the Euclidean setup are more likely to be the correct ones.

In fact, it's not just d>=4. We have a similar situation in d=2. The worldsheet description of string theory is a 2D theory of gravity. Using very naive arguments like yours, one could also argue that the worldsheet should be Minkowskian because it is a real worldsheet embedded in Minkowski spacetime etc. Nevertheless, it is the Euclidean path integral that correctly describes the theory, clearly falsifying your prejudice.

Finally, in quantum cosmology the path integral in Euclidean spacetime is likely to become even more important than in quantum field theory, not less, and the HH state has been the first glimpse why it is so.

If you think that what you wrote can be used as a general criticism against the paper like Vafa+Ooguri+Verlinde, then it's not easy to get too impressed by this criticism.

Dear Jacques, your second criticism - about the action - is even less relevant. Of course that you're right that we usually don't formulate string theory using a spacetime action principle (string field theory is a counterexample). However, string theory has its generalized tools how to define physically meaningful quantities.

The S-matrix, for example, is defined by "thickened" Feynman diagrams that already existed in GR. The HH state is also defined by a generalization of the GR path integral, and the OVV paper is an example how to do it.

If you want to see a similar construction in two-dimensional string theory, see http://arxiv.org/abs/hep-th/0405092.

What you say is simply unconstructive.

Choose a gauge in which the U(1) E&M gauge potential for a constant electric field in, say, the 3-direction looks like

A_3 = E x^0

Then couple this field minimally to a complex scalar \phi and write out the equations of motion. The potential term looks like

V ~ -E(x^0)^2

which is an inverted harmonic oscillator potential.

Schwinger pair production corresponds to tunneling of \phi through this potential. This in turn is naturally described using instantons (see for example Coleman's lectures "The uses of instantons").

The exact classical solutions for \phi are known to be parabolic cylinder functions (see e.g. hep-th0005078). At very late and early times these functions can be described by plane waves which are of the form

\phi(late/early times) ~ \exp (i \pm E t^2)

The point is that, if you start with a purley positive frequency wave at very early times, you will find a linear a combination of positive and negative frequency waves at late times. This mixing is due to pair creation. The coefficients in the linear combination are the Bogoliubov coefficients and you can get them exactly in this simple set up. In this way you can reproduce Schwinger's famous result using instantons.

Best,
Dan

Dear Dan,

thanks for your message. I frankly don't think that I understate the importance of the poles. The more important of the points that are discussed here, I think, is that the Green's functions etc. are holomorphic almost everywhere.

In my opinion, on the other hand, you overstate :-) the importance of the background with an electric field throughout the Universe. You can't couple these things to gravity, for example, and with a little bit more strict definition of the word "physical", one may label these states as unphysical. Note that when you regulate this background in some way, your problem goes away. So the infinite sequence of the poles was a good indication that the background was not quite OK.

I don't think that the summation over the "Minkowski manifolds" - if it can't be reduced to a summation over the Euclidean manifolds - has any quantitative physical meaning. Is it fair for me to say that everything so far suggests that no theory based on summing up Minkowskian manifolds will ever work?

It seems that we outline very similar facts, and you even seem to agree that the Minkowskian manifolds are less well-defined, yet we reach so very different outcomes. I am convinced that the Euclidean path integral - as opposed to the Minkowskian one - will be even more essential for quantum cosmology than it is in non-gravitational quantum field theory, and I don't think that anyone has sketched a piece of evidence to the contrary, except for various unjustified prejudices.

Time-dependent backgrounds are tough in all formalisms we have. But you're wrong if you think that the Wick rotation to the Euclidean spacetime is never doable and never useful. Open for example

http://arxiv.org/abs/hep-th/0302146...

You don't need to like the word "S-brane" - I don't like it - but they study a very specific time-dependent background, and the continuation to imaginary time is essential for their analysis. And there are other papers like that. You create a false impression that the existence of time-dependent backgrounds reduces the importance of the analytical continuation to complex values of time.

All the best
Lubos

Lubos,

I think that the full set of example you give of Euclidean calculations that make sense satisfy the two properties:

1. The relation to Loretzian concepts is transparent, and it is just a convenience to do the calculation is Euclidean space. The prescrition of regularity in Euclidean variables typically encodes some causality properties in Loretzian space.

2. The calculation in Euclidean space is mathematically well-defined. This is because typically
the Euclidean action is bounded from below.

Even in QFT in flat
space there are examples where no Euclidean calculations are NOT possible, for examples calcualting response functions that need real-time finite temp.field theory, which does not have Euclidean formulation.

In many cases of using Euclidean QG, neither one of the conditions 1,2 applies, forexample for calculating the HH state, and one of the comments described it as "thin ice", to which I agree.

Finally, surely at least the observables should be Lorentzian? as far as I know we only have access to that measure Lorentzian quantities...

best,

Moshe

Lubos,

I think that the full set of example you give of Euclidean calculations that make sense satisfy the two properties:

1. The relation to Loretzian concepts is transparent, and it is just a convenience to do the calculation is Euclidean space. The prescrition of regularity in Euclidean variables typically encodes some causality properties in Loretzian space.

2. The calculation in Euclidean space is mathematically well-defined. This is because typically
the Euclidean action is bounded from below.

Even in QFT in flat
space there are examples where no Euclidean calculations are NOT possible, for examples calcualting response functions that need real-time finite temp.field theory, which does not have Euclidean formulation.

In many cases of using Euclidean QG, neither one of the conditions 1,2 applies, forexample for calculating the HH state, and one of the comments described it as "thin ice", to which I agree.

Finally, surely at least the observables should be Lorentzian? as far as I know we only have access to devices that measure Lorentzian quantities...

best,

Moshe

apologies, strange inteface, go ahead and
erase one of those....

M.

I am not saying that we understand how to Wick-rotate the measurable quantities and calculate them from the Euclidean path integral in an arbitrary background in quantum gravity, or anything like that.

What I am saying is that the cases where we know how to do it - which includes the Minkowski background and probably all time-independent backgrounds that can be coupled to gravity - show that the continuation to the Euclidean setup is a useful and important method, and it is most likely to be important even for the backgrounds and questions whose quantum physics we have not understood yet. Sure, there will be many new subtleties and new physics when we do similar things in other cases. Sure, some of the calculations will only be valid approximately.

I am absolutely convinced that these statements should not be controversial, and the only reasons why someone would want to prevent others from using this important and useful technique is an irrational prejudice.

The method has worked beautifully, and in all such cases, one should try to push the method as far as possible and gain as many insights as we can.

It's not just "I can't prove Wick rotation works for quantum gravity so I assume it doesn't work because I'm irrationally prejudiced". Many times, a statement can't be proved because it's false, that is, it has counterexamples. And in fact, there are quantum gravity theories where the Euclidean and Lorentzian versions are NOT related by any Wick rotation. That by itself is enough to show any theorem on the existence of a Wick rotation for quantum gravity needs more "fine print" conditions which might not necessarily hold for all realistic models.

Hi Lubos --

"The more important of the points that are discussed here, I think, is that the Green's functions etc. are holomorphic almost everywhere."

Sure, they can be holomorphic all they want, that's fine. But they are not unique. In Euclidean space the Green's functions are typically unique, fixed by the boundary condition that they have to vanish at infinity. In Minkowski space you still have an infinte set of *different* Green's functions, one differing from another by a solution of the homogeneous wave equation.

"the continuation to the Euclidean setup is a useful and important method"

I am not doubting this. My claim is that there are things not amenable to this otherwise wonderful method.

"You can't couple these things to gravity, for example, and with a little bit more strict definition of the word "physical", one may label these states as unphysical."

There is nothing unphysical here. Surely, the electric field will decay and the initial state is unstable. But nevertheless we should be able *in principle* to describe the time-dependent evolution of this unstable initial state. And the truth is we are not able to do so, even in principle.

"Is it fair for me to say that everything so far suggests that no theory based on summing up Minkowskian manifolds will ever work?"

Why would you say that? Just because it's difficult? I think we have every indication that it should work. In particular, your beloved Wick rotation is nothing but a way to make *some* statement about such sums.

"they study a very specific time-dependent background"

Exactly! They choose a specific one, because they can't do it for a generic one either...

All the best,
Dan

Hey Anonymous! What I'm primarily questioning is not that you can find examples - and let's not discuss how much realistic they can be (not much) - where the usual rules of continuation are altered or disappear. What I'm mostly questioning is your dogma that it is the straightforward "Minkowski" calculation that gives the "more" correct results.

The non-classical calculations done directly in the Minkowski setup are largerly untested, and it is not unreasonable to argue that the Euclidean calculations are the more reliable ones for our calculation and understanding of quantum (and especially non-perturbative) effects. Euclidean setup is the context that has given us various precise results; but also pretty reliable qualitative results (Witten's bubble as the instability of the Scherk-Schwarz compactification, for example).

The assumption is that we continue along these lines - otherwise we're kind of abandoning a significant part of physics that has been established.

Dear Dan,

if you can study more general backgrounds than Strominger et al., be my guest! ;-)

Some things are simply difficult, and we probably agree about that. But difficulty is a different thing than the truth. The right calculation of loop as well as non-perturbative phenomena was always found with the use of the Euclidean, not Minkowski, path integral.

I disagree with the viewpoint that the Wick rotation is just an unwanted method that randomly works in some contexts. I see no indication that a direct path integral summation over the *Minkowskian* manifolds should ever define a meaningful theory. As far as I know, it's not possible even in 2 dimensions.

Imagine that the toroidal worldsheet in string theory has signature 1+1. The Teichmuller space will be changed to SL(2,R)/SO(1,1) - instead of SO(2) in the denominator. Have you ever tried to calculate such things? Does modular invariance work? I just feel that "compact Minkowskian manifolds" are always physically inconsistent, for example, not only because they have closed time-like curves, but also because of a very special type of singularities they develop (where the arrow of time flips). The understanding of physics around these singularities is only naturally obtained from the Euclidean space, once again.

More generally, I am convinced that the quantum gravity path integral done directly in the Minkowski signature is always sick in general as long as there are any local degrees of freedom (which allows d=3 as an exception). It's because the quantum fluctuations of the metric, which are always very drastic at short distances, are simply strong enough to change even the signature of the spacetime. The assumption that the signature remains 1+1 locally is incompatible with the uncertainty principle.

The Minkowskian causal structure that appears in the usual spacetime does not give a right impression for a quantum theory because in a quantum theory, even the causal structure must be allowed to oscillate. In the regime of the Planckian quantum foam, you get a causally nonsensical and singular configurations, and I just don't believe that a summation over them can ever be directly well-defined.

String theory does not yet give us exactly a "local description" of the quantum foam, and it cures all these problems in ways that we don't quite yet understand microscopically. Nevertheless, if we ever understand them locally, the understanding will have to be easier in a Euclidean setup where the problems with the time-like curves, singular points where the arrow of time reverses, and so forth - where these things are absent.

I simply view all attempts to define directly the Minkowskian path integral for quantum gravity as a misguided, anti-historic religion whose results are more or less guaranteed to contradict the local Lorentz invariance and other physical principles.

All the best
Lubos

"You probably can't prove such a statement - especially because the statement is probably not true and the explicit construction of Ooguri, Vafa, and Verlinde is a counterexample of your statement in a specific setup."

Lots of people have computed the Hartle-Hawking wave function in various mini-superspace approximations. If you choose your mini-superspace judiciously (so as to avoid the modes for which the action is unbounded from below), then there is absolutely no hint of a problem.

The result, of course, has absolutely zero relevance to well-definedness of Euclidean Quantum gravity.

In this regard, the OVV calculation is no different from any of the others.

And why are you bringing up 2D Euclidean quantum gravity, when you know perfectly well that it does not share the IR pathology of its higher-dimensional cousins?

Is this yet another attempt to kick sand in the air?

Steve reminded me of my maxim that discussions of the Wave Function of the Universe is the Physics-equivalent of Godwin's Law.

In that light, I should probably bow out, and leave you in the capable hands of Dan and the others, who evidently have a greater tolerance for this stuff.

Lubos,

Even if Wick rotation holds for quantum gravity, and that's a big if considering all the known problems, surely you agree a Lorentzian version of quantum gravity must also exist (our universe sure doesn't look Euclidean)? Despite the possible existence of closed timelike loops? Path integrals aren't the only way to define a theory. Lorentzian quantum gravity models have been rigorously constructed in 1+1 and 2+1 dimensions, and in 3+1 dimensions, people are working with Lorentzian models, whether by canonical quantization or with lattice models. Even your beloved Wheeler-deWitt equation is Lorentzian.

Amusingly, this battle has been fought before, and by the same actors too (Motl and Distler).

The discrete charm of quantum gravity

For Jacques: OVV is not just another minisuperspace approximation. Its minisuperspace is chosen in such a way that there are good reasons to believe that all quantities protected by supersymmetry in the dual picture are reproduced correctly by their HH state.

For the anonymous:

The world is Minkowskian classically, and the corresponding symmetries are inherited exactly by the exact quantum mechanical processes. Classically, one can imagine that physics takes place on a background whose signature is Minkowskian.

But it is misleading to imagine the spacetime in quantum gravity having a well-defined Minkowski signature even at the quantum level, i.e. at the Planck scale - i.e. it is misleading to imagine the virtual processes that contribute to observable physics to look like space with the Minkowskian signature at every point.

The more one thinks about the very short distance microphysics and the virtual corrections to observable physics, the more important it is to regulate various expressions correctly - which means in the way dictated by the Euclidean path integral. The physics smoothed out by the Planckian phenomena dictates one to imagine that the quantum foam is essentially Euclidean. It is equally easy to change the signature (or complexify the fields) as to create a Planckian black hole.

The quantum foam is not about geometries that have a Minkowskian signature.

The quantum foam at the Planck scale is about topological defects and microscopic Planckian black holes appearing and disappearing all the time, and these processes can't be imagined or calculated as non-singular processes that take place on a Minkowskian manifold. One can see lots of inconsistencies about the idea of a "Minkowskian quantum foam". First of all, it would have to include an infinite number of branching points where the time is terminated. Second of all, it would contain closed time-like curves all over the place. Third of all, the restriction that the signature remains Minkowskian would be a strong restriction on the allowed fluctuations of geometry - a strong enough restriction that would essentially say that the fluctuations are smaller than allowed by the uncertainty principle.

Because this point does not seem to be appreciated at all, I will dedicate a special article to it sometime in the future.

...

Moreover, WDW is not my beloved equation, and it does not have to be Lorentzian.

I know quite well that some people are working with "Lorentzian (lattice etc.) models" in 3+1 dimensions - this is one of the programs based on misconceptions that I mentioned. I hope that you allow me to think that this direction of research is not going in the right way.

If you formulate GR as a metric theory, sure, it looks reasonable to have signature changing quantum fluctuations. Why not? But if for nothing else than the existence of spinorial fermions which can only couple to gravity via a spin connection, we're forced to work with vierbeins and spin connections instead of a fundamental metric. Now, GR is a Spin(3,1) gauge theory and signature fluctuations would mean Spin(3,1) and Spin(4) are isomorphic, which they're clearly not. Even string theorists work with vielbeins and spin connections. They've got no choice.

About black holes, you know full well the apparent signature change of the Schwarzschild metric in the Schwarzschild coordinates at the Schwarzschild radius is only an artifact of the coordinate system. There is really no region inside the black hole with a Euclidean signature.

Since you mentioned complexifying the metric, what physical meaning do you attach to that?

Interestingly, if I can trust John Baez in the thread "The discrete charm of quantum gravity", Wick rotation is possible for diffeomorphic covariant theories. Except that it's not the ordinary Euclidean quantum gravity model you'd think of naturally with a Spin(4) spin connection but a perverse "Euclidean gravity" with a Spin(3,1) spin connection. This is so weird that I wonder if I can trust him. A Spin(3,1) Euclidean theory can hardly describe a Riemannian geometry.

Well, I'm off to read the paper Constructing Hamiltonian quantum theories from path integrals
in a diffeomorphism invariant context
he's basing his outrageous claims on.

In this context, I was never talking about the signature change of the black holes - and I agree with your comments that the usual Minkowskian black hole's signature flip is a coordinate artifact. What I mentioned in the main text was the *purely* Euclidean black hole solution, analytically continued in time.

Concerning complexifying metric or anything like that, I only attach the meaning that these things have, namely a mathematical meaning. The virtual processes that contribute to observable phenomena are not directly measurable, and it's not scientifically justified to look for a "physical meaning" of these things.

Whatever mathematics is helpful to extract the predictions for observable phenomena is physically relevant, even if someone thinks that it has no physical meaning. It's not a good idea to imagine that the formalism used to calculate quantum predictions must "look" the same way as the classical, low-energy physics.

The statement that the guaranteed metric may be preserved if the vielbein is taken as the fundamental field is an interesting statement. Even with these variables, the fluctuations must be very large.

Concerning John Baez, it's not surprising that when he continues the kind of models he's thinking about - loop quantum gravity is a gravity interpreted as gauge theory - then the analytical continuation does not change the defining group because the association of this group with the spacetime signature is indirect.

In my opinion, these subtleties are mostly irrelevant because these theories ultimately don't describe gravity anyway. Nevertheless, it's fun to have Baez on our side, even with the bizarre context he's thinking about. ;-)

This is a very fascinating heated debate that you have here.

If I may point out my own observations, which no one else appears to have noticed, your analytic continuation of exp(H t/i hbar) to exp(-beta H) is flawed because H is not bounded from below for gravitational theories. The partition function diverges badly. Someone else has already pointed out the related objection for the Euclidean action. That makes the path integral very divergent. But just to let you know, your Hamiltonian "proof" is also severely flawed.

Signed,
Davru

Lubos,

One can ask the question "which metrics are we to sum over?" within the framework of perturbative string theory, which actually makes sense. The short answer is neither, there is no sense in which we are summing over any set of metrics.

More detailed answer is that that the longwavelength part of any loop integration corresponds (among other things) to deforming the Lorentzian metric by Lorentzian gravitons . But of course no geometric interpretation at all exists at short distances.

I thought that why we are exempt from all the infinitely subtle question to do with a precise definition of metric spaces and integration over them.

best,

Moshe

The fact that H is not bounded is not such a big problem as everybody here seems to think.
(H is unbounded already in QM, e.g. for the hydrogen atom and we know how to quantize in this case)

There are several "cures" for gravity.
Hartle - Hawking use one where the conformal factor gets seperated and handled differently from the remaining degrees of freedom.
Lattice gravity is another way, but I will not elaborate since LM does not like this approach and he is hosting this blog.

But I am mostly with Lubos on this issue. The Euclidean sector is eventually more important than
conventional wisdom might suggest.

Wolfgang
http://yolanda3.dynalias.org/tsm/tsm.html

I forgot to mention:
Kleiner calculated the H-atom in the
Euclidean sector ( H is unbounded
due to V(r) = 1/r^2 of course)

And Padmanabhan et al. explored the
parallels between gravity models and
the H-atom in some detail.

About the Hamiltonian argument, we have to make a distinction between the classical Hamiltonian and the quantum Hamiltonian. For the hydrogen atom, the classical Hamiltonian isn't bounded from below. That's the old problem of why electrons don't radiate away all their energy electromagnetically and fall into the nucleus. However, the quantum Hamiltonian is certainly bounded from below.

If we ask the similar question about gravity, is the quantum Hamiltonian also bounded from below? We have to be careful. What we have is a Hamiltonian constraint, not a Hamiltonian. It's still alright to write down expressions like e^{-iHt} with the understanding that we're gauge fixing the shift vector to zero and the lapse parameter to one. This construction unfolds time from the timeless theory. Since we have no choice but to use vierbeins and spin connections if we are to include fermions, which obviously exist, be careful here, Lubos! You're secretly working with spin foams. :) But at the end of the day, we still have to integrate over all t so that \int dt e^{-iHt}=2 pi delta(H) and we get the Hamiltonian constraint back again, the projection operator of the Hamiltonian constraint to be more precise.

sorry V(r)=1/r of course

Of course, Lubos is dealing with some lattice theory without knowing it 8-)

Bosonic strings, the corresponding matrix model and dynamical triangulation of simplicial lattices are the same thing.
It does not require too much imagination to assume that M-theory will be equivalent to some lattice theory in the end.
Once this is understood, we will finally be able to compute true results from string theory ...

But dont tell Lubos, he gts mad at you for even mentioning the word "lattice" 8-)

Wolfgang

Poles are not the worst things that correlation functions can have in the complex plane. Luis Alvarz Gaume argued that the apparent problems in field theories with time-like non-commutativity (non-unitarity etc) are due to a Wick-rotation through a branch cut.

Lubos, T'Hooft has a good little review paper on path integrals that you might want to (re)read.

http://arxiv.org/abs/hep-th/0208054

I think Witten's 2+1 treatment is cited and he more or less spells out the obvious pathologies in d>4. For the case d = 4, he spells out when and where it is possible to make sense of things.

In any event, it seems clear to me that mathematically the euclidean path integral and the lorentzian path integral do not always coincide. The circumstances where they DO coincide are special and incidentally make QFT manageable.

Unfortunately gravity is different and much more complicated. Its one of the pitfalls of the field

Lubos, T'Hooft has a good little review paper on path integrals that you might want to (re)read.

http://arxiv.org/abs/hep-th/0208054

I think Witten's 2+1 treatment is cited and he more or less spells out the obvious pathologies in d>4. For the case d = 4, he spells out when and where it is possible to make sense of things.

In any event, it seems clear to me that mathematically the euclidean path integral and the lorentzian path integral do not always coincide. The circumstances where they DO coincide are special and incidentally make QFT manageable.

Unfortunately gravity is different and much more complicated. Its one of the pitfalls of the field

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 "How to put and to the suffering caused by path integrals" at http://matpitka.blogspot.com/.

Quick question: suppose you have a Euclidean space. All coordinates have the same status. How do you decide which of them is going to turn into time when you Wick rotate? Note: the correct answer is *not* "pick the one called t".
:-):-) Actually this comes up in the OVV paper.

Lubos said: "Whatever mathematics is helpful to extract the predictions for observable phenomena is physically relevant, even if someone thinks that it has no physical meaning. It's not a good idea to imagine that the formalism used to calculate quantum predictions must "look" the same way as the classical, low-energy physics."

I think it is a perfectly legitimate mathematical approach to allow imaginary time or such for path intergral, as long as you get the same end result.

But no. The difference of mathematical approaches are physically irrelevant as long as they all arrive at the same end result, i.e., the same observable. The only thing relevant in physics are observables, and predictions of observables by different approaches.

Physically, neither e^(-S/hbar) nor e^(-iS/hbar) are physically meaningful. None of them are more meanful or less meanful than the other, although one implies real time and another implies imaginary time, but that's irrelevant. We can NOT observe the phases in e^(-action/hbar)), we can only observe the end result of path integrals. So only the integration result is physically meanful, the paths are NOT physical meaningful.

It's like in the double split experiment. When we see the interference pattern, it is not physical to ask the question "which split the photon passes through", because that's not an observable!!! Once you try to detect the path (which split) the photon went through, the interference pattern disappears!!!

So it is simply not a physics question to inquire what are the paths it goes through in the path integral, whether it goes through a path of pure real time, or a path in the complex time plane. The path is simply none-observable and none-physical, regardless whether there is a real or imaginary time in the expression of the path.

Another analogy is like for a system to going from one equilibrium thermal state A to another equilibrium state B. We can compare thermal equilibrium states to real time, and none-equilibrium states to imaginary time.

You can ask how does it go from state A to state B. What kind of paths it went through. You can obtain the entropy change by doing a path intergral, and you would though you might obtain a different result if it goes through different paths.

So you worry about whether it should only be allowed to go through a continuous path where thermal equilibrium is always maintained (time always be real), or can it be thermal in-equilibrium in the path (imaginary time). The truth is it does NOT matter at all. Only the end state matters. It's not even a physical question to ask what kind of path the system went through from A to B.

Well, if it's not called "t", then it's probably "tau" and/or "rho", depending on the picture, is not it? ;-)

The point is that there are many ways to do the calculations. All procedures that are done properly are fine.

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.

The new subtleties of GR indicate that we will have to spend *more* time with the continuations and rotations, not less.

Well, if it's not called "t", then it's probably "tau" and/or "rho", depending on the picture, is not it? ;-)

What if it's called theta? :-)

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.

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.

Lubos,

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.

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.

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.

best,

Moshe

Dear Lubos,

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.

Let me conclude with the following slogan;

"No causality, no Wick rotation!"

Henry

My dearest Luboš,

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!

Henry

forget about quantum gravity. Wick rotation can be tricky enough even in flat spacetime, especially with spinors. See, for example

http://arxiv.org/abs/hep-th?9608174

> the divergent scale factor in
> the path integral is pure gauge.
> One can always choose a gauge of
> diffeomorphisms in which (det g)
> =-1. There's enough freedom to
> do it. The divergence goes away.
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.

Let me retract the comment that the negatively-normed scale factor is pure gauge. ;-)

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).

All the best
Lubos

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 not 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?

All this anti-Euclidean sentiment is getting out of hand. For once, I am with Lubos on this issue.

By retracting your comment, you are essentially admiting Distler's objections are perfectly valid.

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.

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.

And by the way, I encourage the participants not to strengthen their cases by posting under various nicknames, including "anonymous".

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).

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 =)

2+1 Gravity is an example where the Lorentzian theory and the Euclidean theory yield manifestly different results.

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.
Fyodor Uckoff

Fyodor,

Why should we assume our four dimensional universe is any different qualitatively from a three dimensional universe? If Wick rotation fails in three dimensions, isn't that disturbing enough?

Henry wrote:
Why should we assume our four dimensional universe is any different qualitatively from a three dimensional universe?

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.

If Wick rotation fails in three dimensions, isn't that disturbing enough?

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!

Fyodor

Fyodor,

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.

Lumo, thanks for the nice article. It really helps put things in perspective.

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.)

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.

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.

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.

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.

Anthony Leggett in his book calls this derivation questionable at best.

And I know some experimentalists working on bose condensates, who can't verify this equation with experiments.

I would appreciate your help in clarifying what appears to be a very murky situation at best.

Thanks

Lumo, thanks for the nice article. It really helps put things in perspective.

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.)

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.

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.

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.

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.

Anthony Leggett in his book calls this derivation questionable at best.

And I know some experimentalists working on bose condensates, who can't verify this equation with experiments.

I would appreciate your help in clarifying what appears to be a very murky situation at best.

Thanks

Dear Pat, thanks for your interest.

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. ;-)

So I will try to reduce it as much as possible.

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.

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

exp(X) = lim (1+X/N)^N,

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.

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.

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. ;-)

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.

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.

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.

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 Feynman lecture. ;-)

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.

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.

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.

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.

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.

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?

Right, crasshopper. But your comment still understates the legitimacy of the procedure.

Even if there were poles etc. and the procedure using the imaginary time led to experimentally verifiable results, it would be well-established physics.

The calculations in the imaginary time and/or Euclidean spacetime are actually more well-behaved, not less behaved, than those in the Minkowski spacettime.

So this is not a silly "detour". It is the canonical, right, straightforward way to calculate things in QFTs.

Lubos.

Thanks for the nice introduction.

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:
http://en.wikipedia.org/wiki/Analytic_continuation

This implies that the expansion holds only for small k_0, i.e. there is an implicit cutoff.
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?

Dear fulig, a good question. It's a topic for a much longer article: singularities in the complex plane.

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.

Such poles have the interpretation of physical states - resonances etc. But one can go on.

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.

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.

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.

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.