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Wick rotation is supposed to be a relationship between field theories with spacetime metrics of Lorentzian and Euclidean signature. I thought the definition of Wick rotation was settled, until I came across this paper — by one of the pioneers of supergravity — which proposes what appears to be an entirely different defintion. That probably means I'm missing something important, so I'm asking this question to help me understand what I'm missing.

Here's the definition that I thought was standard. Consider a lagrangian field theory, either classical or quantum, possibly involving spinor fields, with a prescribed (not dynamic) spacetime metric that is globally hyperbolic. Any point in the spacetime has a finite neighborhood in which the metric can be written $$ \newcommand{\bfe}{\mathbf{e}} g = \sum_{ab}\eta_{ab} \bfe^a\otimes \bfe^b \tag{1} $$ where the $\bfe^a$ are one-forms and $\eta$ is the Minkowski metric. Writing the metric this way facilitates constructing an action for spinor fields in curved spacetime. Maybe naïvely, Wick rotation can be defined as the replacement $$ \bfe^0\to i\bfe^0, \tag{2} $$ where $0$ is the "time" index. This changes the signature of $g$ from Lorentzian to Euclidean, or conversely. As far as I know, this definition is unambiguous, as long as we make the replacement (2) everywhere $\bfe^0$ appears in the action.

Question: What's wrong with the definition (2)?

  • One possible objection is that one-forms satisfying (1) cannot always be globally defined in a curved spacetime. Okay, but is that really necessary? They can be globally defined in flat spacetime, and they can be defined in finite regions of a curved spacetime, which seems like the most we can reasonably expect from such a fundamental modification of the metric. Maybe this is an obstacle for quantum gravity, but there are lots of obstacles for quantum gravity, and I don't see why that should stop us from using the simple definition (2) if it's adequate for ordinary quantum field theory.

  • Another possible objection is that the properties of spinor representations are sensitive to the signature of spacetime: if we change the signature, then we fundamentally change the properties of the spinors. Okay, but why is that a problem? Isn't this exactly what we should expect? I mean, isn't this potentially an important source of insight rather than a problem (even if it disrupts supersymmetry)?

So... why would one of the pioneers of supergravity propose a different definition than (2)?

Maybe related: Time Reversal, CPT, spin-statistics, mass gap and chirality of Euclidean fermion field theory

Chiral Anomaly
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1 Answers1

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When we "Wick rotate" we're usually not interested in studying the properties of the transformation you write as eq. (2), but in obtaining a Euclidean field theory whose quantities of interest (usually correleation functions) when analytically continued back to Minkowski space yield the corresponding quantities of the Lorentzian field theory. The reason we want to do this is to a large extent because the convergence properties of Euclidean field theory are better and better understood.

So losing/gaining/changing spinor representations during naive Wick rotation (your eq. (2)) is a problem for this goal, because the argument for how/why analytic continuation works usually relies on the Euclidean and the Lorentzian field theory having "the same field content". Since this doesn't work for spinors in general, you need to do something else, and that's what van Nieuwenhuizen is looking for in the paper you link - a Euclidean field theory from which you can get the Lorentzian correlation functions by analytic continuation (and minor changes to the spinor indices). We're not looking for insight about what "Wick rotation" does to spinors, we're looking for a transformation that obtains the "correct" Euclidean field theory for us to do our computations in.

ACuriousMind
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