What's wrong with using a vielbein to define Wick rotation?

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)?

• Isn't (1) true only through 1st order in neighborhood size? I don't think this answers your question, but the first possible objection you suggest is stronger than you seem to realize. The $\mathbf{e}^i$ only satisfy (1) approximately, over sufficiently small regions. – Daniel Aug 24 '20 at 0:42
• @Daniel In (1), I'm using a non-coordinate basis (that's what I meant by "vielbein" in the title, aka tetrad), so it's not just a first-order approximation. It's exact, for a suitable choice of $1$-forms $\mathbf{e}^a$, which typically cannot be coordinate $1$-forms. For example, the Kerr metric can be written this way (section 6 in arxiv.org/abs/0706.0622). Related Phyiscs SE post: Non-coordinate basis in GR – Chiral Anomaly Aug 24 '20 at 2:48
• Oh, I see. I guess there's no reason to need them to correspond to coordinates. – Daniel Aug 24 '20 at 2:51

1 Answer

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.

• Thank you for the answer! I'm not sure how to interpret "the same field content," and maybe that's part of what I'm missing. In a path-integral formulation, I think of the symmetries of a theory as being determined by the action, while the components of the fields are just integration variables with no implied symmetry attached to them. But the key seems to be what you suggested about convergence properties: maybe (2) defines a Euclidean version that doesn't have the desired convergence properties. I guess I need to think more about how convergence is related to the spinor representations. – Chiral Anomaly Aug 23 '20 at 22:27
• @ChiralAnomaly: I have to admit I'm not quite sure what you imagine happening to the spinors in your version of Wick rotation. In the end we want to be able to compute something like the propagator $\langle \psi_E^\mu(x)\psi_E^\nu(y)\rangle$ in the Euclidean version and obtain the Lorentzian $\langle \psi_L^\mu(x)\psi_L^\nu(y)\rangle$ from it. But your eq. (2) gives us no recipe for that - if there is no straightforwardly corresponding Euclidean spinor, something like $(\mathrm{i}\psi_L^0, \psi_L^1, \psi_L^2, \psi_L^3)$ just isn't a properly covariant object in Euclidean space. – ACuriousMind Aug 23 '20 at 22:35
• I'm thinking of $\langle \psi^a\psi^b\rangle$ as $\propto\int [d\psi]\ F[\psi]\psi^a\psi^b$ where $F[\psi]$ is the exponential of the action, and I'm thinking of Wick rotation as affecting only $F$, not the integration variables $\psi^{a/b}$. In that view, the choice between $\psi^{a/b}$ being grassmann-even or grassmann-odd is independent of $F$, but everything else comes from $F$, including internal/spacetime symmetries and covariance properties. Maybe that view misses what analytic continuation is supposed to accomplish, though... Your answer is making me doubt my worldview (+1)! – Chiral Anomaly Aug 24 '20 at 3:18
• @ChiralAnomaly Wick rotation is supposed to give us a proper Euclidean field theory, not just to affect the $F[\psi]$ term. We really want to have the relationship $\langle \psi^a_L(t,x), \psi^b_L(t,x)\rangle = \langle \psi^a_E(it,x) \psi^b_E(it,x)\rangle$, where the r.h.s. is computed in a consistent Euclidean field theory. I'll edit the answer in a bit to make that point more clear. – ACuriousMind Aug 24 '20 at 10:49
• Now I see that the approach I described doesn't work for Majorana spinors. If we want a purely-real or purely-imaginary representation of the Dirac matrices on the Lorentzian side, then it generally won't have that property on the Euclidean side, so we can't enforce a Majorana condition on the spinor field on both sides of the Wick rot'n -- we can't maintain "the same field content," exactly like you said. I haven't finished working through the paper yet to see how their version of Wick rotation solves this problem, but I see what's wrong with my version now, and that's what I needed. Thanks! – Chiral Anomaly Aug 30 '20 at 21:13