A Mystery called Wick Rotation or can we understand the “Action” Formalism?

The Wick rotation pops up as a “mere technical trick” in quantum field theoretical calculations. Making the time coordinate complex $t \to i \tau$ is described as “analytic continuation” and helps to solve integrals. Certainly, there is nothing deep behind this technical trick, right?

Well, I’m no longer so sure.

There is one observation that makes me (and others) wonder:

The difference between the mystical theory of quantum fields and ordinary statistical mechanics is a Wick rotation $t ➝ i /(kT)$.

This is puzzling. On the one hand, we have ordinary statistical mechanics that we understand perfectly well. When we want to make a statement about a system where we don’t know all the details, we invoke the principle of maximum entropy and get as a result the famous Boltzmann distribution $\propto exp(-E/T)$. This distribution tells us the probabilities to find the system, depending on the energy $E$, in the various macroscopic states. There is nothing mysterious about this. The principle of maximum entropy is simply an optimal guessing strategy in situations where we don’t know all the details. (If you don’t know this perspective on entropy you can read about it, for example, here). This interpretation due to Jaynes and the derivation of the Boltzmann distribution are completely satisfactory. It is no exaggeration when we say that we understand statistical mechanics.

On the other hand, we have the mysterious “probability distribution” in quantum field theory that is known as the path integral. I know no one who claims to understand why it works. The path integral is proportional to $exp(iS/\hbar)$, where $S$ denotes the action and even Feynman admitted:

I don’t know what action is“.

In his book QED, when he talks about the path integral, he writes

Will you understand what I’m going to tell you? …No, you’re not going to be able to understand it. … I don’t understand it. Nobody does.

It seems as if all the mysteries of the quantum world are encapsulated in a simple Wick rotation.

I’ve been searching for quite a while but wasn’t able to find any sufficient discussion of this curious fact.

There was some “recent” work by John Baez, which he described in his blog and also a paper. He also tried to make sense of Wick rotations by making use of it in a classical mechanics example. (See the “Homework on A spring in imaginary time” here and additionally, the discussion here). The lesson there was that “replacing time by “imaginary time” in Lagrangian mechanics turns dynamics problems involving a point particle into statics problems involving a spring.” In addition, several years ago Peter Woit tried to emphasize the confusion surrounding Wick rotations in a blog post. He wrote:

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

However, apart from that, there don’t seem to be any good discussions of the “meaning” of a Wick rotation and I still don’t know what to make of it. Yet, it seems clear that something very deep must be going on here. If the Boltzmann distribution can be understood perfectly by invoking the “best guess” strategy known as “maximal entropy”, has the path integral a similar origin? Probably, but so far, no one was able to find it.

In statistical mechanics, our best guess for the macroscopic state our system is in the state that can be realized through the maximal number of microscopic states. This state is known as the state with maximal entropy. Many microscopic details make no difference for the macroscopic properties and therefore, many microscopic configurations lead to the same macroscopic state. We don’t know all the microscopic details but want to make a statement about the macroscopic properties. Hence, we must use the best guess approach, and the best guess is the macroscopic state with maximum entropy.

Aren’t we doing in quantum theory something similar? We admit that we don’t know the fundamental microscopic dynamics. We don’t know which path a given particle takes from point $A$ to point $B$. Nevertheless, when pressured, we make a guess. Our best guess is the path with extremal action.

The observation by John Baez mentioned above that a Wick rotation connects a static description, like in statistical mechanics, with a dynamical description, like in quantum field theory, seems to make sense from this perspective.

Some nice thoughts in this direction are collected by Tommaso Toffoli in his two papers: “What Is the Lagrangian Counting?” and “Action, Or the Fungibility of Computation“. For example, he writes:

just as entropy measures, on a log scale, the number of possible microscopic states consistent with a given macroscopic description, so I argue that action measures, again on a log scale, the number of possible microscopic laws consistent with a given macroscopic behaviour. If entropy measures in how many different states you could be in detail and still be substantially the same, then action measures how many different recipes you could follow in detail and still behave substantially the same.

In addition, I think there could be a connection to recent attempts to understand quantum theory as extended probability theory, where we allow negative probabilities. This line of thought leads to complex probability amplitudes like we know them from quantum theory. For a nice introduction to this perspective on quantum theory, see this lecture by Scott Aaronson. Interestingly he argues that this extension of probability theory is all we need to derive quantum theory. Again, the switch to complex quantities seems to make all the difference.

I think this is a good example of an obvious, but, so far, not sufficiently understood the connection that could yield deep insights into the quantum world. I usually write about things, where I think I have understood something. However, here I mainly wrote this to organize my thoughts and as a reminder to think more about this in the future.

To finish, here is an incomplete list, where Wick rotations are also crucial

1.) We use a Wick rotation to classify all irreducible representations of the Lorentz group. In this context, the Wick rotation is often called “Weyl’s unitary trick”.
2.) A Wick rotation is used to analyze tunneling phenomena, like, for example, the famous instanton solutions in QFT.
3.) People who consider QFT at finite temperatures make heavy use of Wick rotations.

P.S. I wrote a textbook which is in some sense the book I wished had existed when I started my journey in physics. It's called "Physics from Symmetry" and you can buy it, for example, at Amazon. And I'm now on Twitter too if you'd like to get updates about what I'm recently up to.