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
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. ;-)