Wick rotation changes the spacetime symmetries of a quantum field theory, changing between
- The Lorentz group, by which I’ll mean either $SO(3,1)$ or $SL(2,\mathbf C)$, the double cover of the time orientation-preserving subgroup of $SO(3,1)$.
- The four-dimensional Euclidean signature rotation group, by which I’ll mean either $SO(4)$ or its double cover $Spin(4)=SU(2)\times SU(2)$.
A very long time ago I got interested in the possibility that one of the two $SU(2)$ factors in the Euclidean symmetry could appear as an internal symmetry in Minkowski spacetime. For many years though I had given up on this idea, convinced that in any version of Wick rotation, this could not happen. If you look at Wightman and Schwinger functions, they are restrictions of a single holomorphic function. $SO(3,1)$ and $SO(4)$ show up as the symmetries that preserve the two different restrictions, related through analytic continuation between two different real forms of $SO(4,\mathbf C)$.
Back in 2020 at some point I realized that there is a significant difference between Minkowski and Euclidean spacetime theories. In Minkowski spacetime, the reconstruction of states and operators from the Wightman functions does not break the $SO(3,1)$ symmetry: there is no distinguished time direction. In Euclidean spacetime on the other hand, OS reconstruction of physical states and operators does break the $SO(4)$ symmetry, by the choice of an imaginary time direction, and thus an OS reflection operator. More physically, in Minkowski spacetime you don’t need a choice of time direction to do physics, while in Euclidean spacetime, you must choose a direction, the direction you plan to do Wick rotation in to recover real-time physics.
The realization that the Euclidean theory had to come with this extra piece of structure, which broke $SO(4)$ symmetry to $SO(3)$ symmetry, led me to the ideas in this quite speculative paper. At the time I was quite confused about the details of how Wick rotation worked in rigorous versions of quantum field theory, but by now I’m much less so. In an earlier post I started writing about this, here will explain what happens to spacetime symmetries under Wick rotation, at least for scalar field theories. Things get much more interesting when you look at spinors, which I’ll do in future posts.
What’s relatively easy to understand is what happens when you start with a Minkowski spacetime theory. The Wightman functions have $SO(3,1)$ symmetry and Wightman reconstruction gives a state space with a unitary representation of this group. The $SO(3,1)$ invariant version of the positive energy condition is to assume that the Wightman functions are supported in the positive light-cone (in energy-momentum space). If one then writes the inverse Fourier transform formula, but for complex spacetime coordinates $z=(z_0,\mathbf z)$
$$W(z)= \frac{1}{(2\pi)^2}\int_{\mathbf R^{3,1}} e^{-i(z_0E-\mathbf z\cdot\mathbf p)}\widetilde W(E,\mathbf p) dE d^3\mathbf p$$
one gets $W(x)$ for $z=x$ real, and an analytic continuation into a “tube” which is a subspace of $\mathbf C^4$ of the form $\mathbf R^{3,1}$ plus $i$ times the inside of a lightcone. This is a generalization of the Paley-Wiener theorem argument discussed in the earlier blog post.
Given this analytic continuation to some of $\mathbf C^4$, one acts holomorphically by the group $SO(4,\mathbf C)$ to get an analytic continuation to a larger region (the “extended tube”). One then uses symmetry under interchange of coordinates to get an analytic continuation to an even larger region (the “permuted extended tube”). The Bargmann-Hall-Wightman theorem says that one can do this with a single-valued holomorphic result. The region of holomorphicity now includes the Wick-rotated Euclidean subspace $\mathbf R^4$. One can easily see from the formula above that $SO(3,1)$ symmetry in $\mathbf R^{3,1}\subset \mathbf C^4$ becomes $SO(4)$ symmetry in the Wick rotated $\mathbf R^4$.
When one tries to go the other direction, starting with Schwinger functions with $SO(4)$ symmetry and doing OS reconstruction to get Wightman functions with $SO(3,1)$ symmetry, one runs into the trouble discussed earlier that there’s no good way to invert the formula for $W(z)$ above and get the Wightman function from the Schwinger function. What OS do is start by assuming some property of Schwinger functions and doing an elaborate analytic continuation argument (with results that are not very satisfactory). Even less satisfactory, but much simpler, is to just take as an axiom a property of Schwinger functions that ensures they come from some $\widetilde W(E,\mathbf p)$ as above. This is done for instance in section 5.6 of the book An Introduction to Non-Perturbative Foundations of Quantum Field Theory, by Franco Strocchi. When you do this, you can show that $SO(4)$ invariance of Schwinger functions implies $SO(3,1)$ invariance of Wightman functions.
The relation between the symmetry of Wightman functions and that of Schwinger functions is thus rather straightforward, but there’s still the question of the representation of the symmetry groups acting on states. On the Minkowski side, $SO(3,1)$ acts on test functions, and on the states constructed from these by Wightman reconstruction as a representation of the group. On the Euclidean side, $SO(4)$ acts on test functions on $\mathbf R^4$, but only the $SO(3)$ preserving a chosen imaginary time direction acts on OS reconstructed physical states.
How the $SO(3,1)$ representation on physical states is recovered is discussed by Seiler in section 8b here and by Klein and Landau here. Things are put in a more general representation-theoretical context by Frohlich, Osterwalder and Seiler here. The analytical continuation argument for how one gets an $SO(3,1)$ representation from the $SO(4)$ representation (+other data) is more involved than the argument above for Schwinger/Wightman functions.
My interest in this topic is in what happens for spinors, while these discussions are almost all about the real scalar theory. The original OS reconstruction paper does have a section about arbitrary spin fields, and the OS papers on fermion fields are designed to fit into this context.
In a post to come soon I’ll write about the issues with spinors and the possibility that the usual way to analytically continue them is not what one wants to do.
