If one tries to Wick rotate a quantum field theory with spinor fields, it’s well-known that problems arise, something first recognized in Schwinger’s earliest papers on the subject. I’ll try and outline here the 1972 proposal by Osterwalder and Schrader (see here, here and here), which is the best known way to deal with the problem. Over the years there have been a large number of other efforts to address this issue, and I’ve put together a bibliography of those here. Many of these I’ve never been able to completely understand. I’ll concentrate on the Osterwalder-Schrader proposal since I do now understand it (I didn’t in 1984…), and it seems to correspond best to the conventional wisdom of the subject.
There’s a first indication that something funny is going on when you look at any QFT textbook discussing spinor field theory in the standard “Dirac spinor” formalism where spinors take values in $\mathbf C^4$. A good example is Pierre Ramond’s book, where chapter 5 deals with this, in both Minkowski and Euclidean signature. The Lagrangian in both cases can be written the same way, as
$$\overline \psi (i\gamma^\mu\partial_\mu -m)\psi$$
with
$$\psi=\begin{pmatrix}\psi_L\\ \psi_R\end{pmatrix}$$
If you read more closely you find out that the notation is hiding things:
- In Euclidean signature, $\overline \psi =\psi^\dagger$, but in Minkowski signature $\overline \psi =\psi^\dagger \gamma_0$ (the “Dirac adjoint”).
- In Euclidean signature the spin group is $Spin(4)=SU(2)_L\times SU(2)_R$, $S_L$ is the spin representation of $SU(2)_L$, and $S_R$ is the spin representation of $SU(2)_R$. In Minkowski signature the (time-orientation preserving) spin group is $Spin(3,1)=SL(2,\mathbf C)$, $S_L$ is the spin representation of $SL(2,\mathbf C)$, and $S_R$ is the complex conjugate of the spin representation.
Osterwalder and Schrader propose that Wick rotation of spinor fields involves a doubling of the number of degrees of freedom, giving up the Dirac adjoint relation, and taking $\psi$ and $\overline\psi$ to be independent fields (which they call $\psi_1$ and $\psi_2$). They show that one can then do the same kind of OS reconstruction argument as in their paper dealing with scalar fields. The OS reflection operator that in the scalar case both complex conjugated fields and reflected in imaginary time now also interchanges $\psi_1$ and $\psi_2$, as well as having a $\gamma_0$ factor that interchanges $S_L$ and $S_R$.
This proposal does what it is advertised to do, reconstructing the Wightman functions and state space of the usual Minkowksi spacetime theory, but the way it does this is somewhat unsettling. Wick rotation is not just a matter of putting some factors of i in the right place, but involves a significant change in the degrees of freedom of the theory when one passes from Minkowski to Euclidean. For scalars, OS showed that the Wick rotation of complex conjugation surprisingly also now involved a reflection in spacetime. For spinors this becomes an even more intricate piece of structure one must add to the Euclidean theory to do reconstruction.
Osterwalder-Schrader and most later authors ignore something even more problematic about Wick rotating spinors, something pointed out by Ramond in his book: it doesn’t work for a Weyl spinor field. The basic building blocks of matter fields in the Standard Model are two-component spinor fields, with the simplest building block the theory of a chiral (say right-handed) massless Weyl fermion. This theory is simple to write down, and at first glance has a simple Wick rotation, just by taking time to be complex and proceeding as for scalars.
But this runs into a fundamental issue with how the transformation properties under spacetime rotation change as one goes from Minkowski to Euclidean. It appears that if one wants to describe a massless neutrino of one chirality in Euclidean QFT, one must quadruple the number of degrees of freedom (first double the degrees of freedom to get four-component Dirac spinors, then double again according to Osterwalder-Schrader).
I’ll leave for another time discussion of the details of how spacetime rotations change under Wick rotation in the usual formalism. I’ve outlined a proposal for a very different way of understanding this issue in my Spacetime is Righthanded paper.
A possibly dumb question: The doubling issue does not arise in perturbation theory just because we calculate S-matrix elements, so we can choose physical DoFs in the initial and final state?
lun,
The doubling issue arises not only in perturbation theory, but even in free field theory. The problems occur when you try and set up a theory using fields in Euclidean spacetime that will give you the Minkowski spacetime QFT by the OS reconstruction.
You see a version of the problem also with the path integral. There you end up needing twice the number of integration variables you might have expected.
Could you elaborate on what goes wrong with Weyl spinors?
Alex,
There’s a section of this
https://arxiv.org/abs/2311.00608
that explains this in more detail. I’ll likely write another blog post soon about this.
Brian Dolan,
Yes, I agree. This is one of several things that makes the standard formalism for working with spinors mysterious and confusing (“why are we making a choice of $\gamma^0$ that seems to break Lorentz invariance, in order to get Lorentz invariance???”).
In general, working with the four-component spinor formalism is designed to work nicely in parity-symmetric situations (e.g. QED, QCD). In the case of chiral theories it becomes awkward, and it’s a good idea to think in terms of two-component spinors.
Why do OS double the number of degrees of freedom in Euclidean signature,
I don’t see any need for it?
In Minkowski signature let $\psi_L \in S_L$ and $\psi_R\in S_R$: they transform under complex conjugate representations of $SO(3,1)$ and there are 4 complex numbers.
In Euclidean signature let $\psi_L \in S_L$ and $\psi_R\in S_R$: they transform under independent $SU(2)_L$ and $SU(2)_R$ groups, but there are still 4 complex numbers.
In both Euclidean and Lorentzian there are four invariant bi-linears that can be made out of $\psi_L$ and $\psi_R$.
In Euclidean signature $\psi^\dagger_L \psi_L$ and $\psi^\dagger_R \psi_R$ are two independent $SO(4)$ invariants. But since $SU(2)$ is pseudo-real there are another two:
$\psi_L^T \epsilon \psi_L$ and $\psi_R \epsilon \psi_R$, where $\epsilon= i \sigma_2$ (these are only non-zero for Grassmann variables, of course). They can be made real by a suitable phase transformations on $\psi_L$ and $\psi_R$: once made real we are no longer free to perform phase transformations, they are Euclidean analogues of Majorana masses.
In Lorentzian signature chose a Weyl basis in which $\gamma_5 =\pmatrix{ 1 & 0 \cr 0 & -1}$
and $\bar \psi = \psi^\dagger \beta$ with the spinor metric $\beta=\pmatrix{ 0 & 1 \cr 1 & 0}$ block off-diagonal. Then $\psi_L^\dagger \psi_R$ and $\psi_R^\dagger \psi_L$ are independent Lorentz invariants, or, if you prefer $ \bar \psi \psi =(\psi_L^\dagger \psi_R +\psi_R^\dagger \psi_L )$ and $i \bar \psi \gamma_5 \psi=i(\psi_L^\dagger \psi_R – \psi_R^\dagger \psi_L )$. I think of these as Lorentzian analogues of the Euclidean bi-linears
$\psi_L^\dagger \psi_L$ and $\psi_R^\dagger \psi_R$. Given Weyl spinors $\psi_L$ and $\psi_R$ in Lorentzian signature we can construct $\Psi = \pmatrix{ \psi_L \cr (\psi_L)_c}=\Psi_c$ and $\Psi’ = \pmatrix{ (\psi_R)_c \cr \psi_R }=\Psi’_c$,
where charge conjugate is $\Psi_c = -C \beta \Psi^*$ and $\Psi’_c = -C \beta \Psi’^*$. So the charge conjugate of $\psi_L$ is $(\psi_L)_c=-\epsilon \psi_L^*$ (a right-handed spinor)
and the charge conjugate of $\psi_R$ is $(\psi_R)_c=\epsilon \psi_R^*$ (a left-handed spinor). Then $\bar \Psi \Psi = \psi_L^T \epsilon \psi_L – \psi_L^\dagger \epsilon \psi_L^*$ and $\bar \Psi’ \Psi’ = -\psi_R^T \epsilon \psi_R + \psi_R^\dagger \epsilon \psi_R^*$.
In both Euclidean and Lorentzian signature there are 4 complex degrees of freedom and 4 invariant bi-linears: no need to double the degrees of freedom in Euclidean signature.
Brian Dolan,
The reason for the OS doubling is not spacetime rotation invariance, but self-adjointness. You get a Euclidean invariant theory of fields with positivity of norms of states by the doubling (if you don’t have self-adjointness, you can get it by adding an copy of fields with adjoint properties).
It has always been unclear that you really need to do this, one reason the whole subject is confusing. You definitely need positivity of the physical norm (OS norm, which is NOT SO(4) invariant), unclear whether you really need a SO(4) invariant field theory with positive norms.