# 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. Aug 24, 2020 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 Aug 24, 2020 at 2:48
• Oh, I see. I guess there's no reason to need them to correspond to coordinates. Aug 24, 2020 at 2:51

• @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. Aug 23, 2020 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)! Aug 24, 2020 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. Aug 24, 2020 at 10:49