It’s been taking me forever to sort out and write down the details of implications of the proposal described here. While waiting for that to be done, I thought it might be a good idea to write up one piece of this, which might be some sort of introductory part of the long document I’ve been working on. This at least starts out very simply, explaining what is going on in terms that should be understandable by anyone who has studied the quantization of a spinor field.
I’m not saying anything here about how to use this to get a better unified theory, but am pointing to the precise place in the standard QFT story (the Wick rotation of a Weyl degree of freedom) where I see an opportunity to do something different. This is a rather technical business, which I’d love to convince people is worth paying attention to. Comments from anyone who has thought about this before extremely welcome.
Matter degrees of freedom in the Standard Model are described by chiral spinor fields. Before coupling to gauge fields and the Higgs, these all satisfy the Weyl equation
$$(\frac{\partial}{\partial t}+\boldsymbol\sigma\cdot\boldsymbol\nabla)\psi (t,\mathbf x)=0$$
The Fourier transform of this equation is
$$ (E-\boldsymbol \sigma\cdot \mathbf p)\widetilde{\psi}(E,\mathbf p)=0$$
Multiplying by $(E+\boldsymbol \sigma\cdot \mathbf p)$, solutions satisfy
$$(E^2-|\mathbf p|^2)=0$$
so are supported on the positive and negative light-cones $E=\pm |\mathbf p|$.
The helicity operator
$$\frac{1}{2}\frac{\boldsymbol\sigma\cdot \mathbf p}{|\mathbf p|}$$
will act by $+\frac{1}{2}$ on positive energy solutions, which are said to have “right-handed” helicity. For negative energy solutions, the eigenvalue will be $-\frac{1}{2}$ and these are said to have “left-handed helicity”.
The quantized field $\widehat{\psi}$ will annihilate right-handed particles and create left-handed anti-particles, while its adjoint $\widehat{\psi}^\dagger$ will create right-handed particles and annihilate left-handed anti-particles. One can describe all the Standard model matter particles using such a field. Particles like the electron which have both right-handed and left-handed components can be described by two such chiral fields (note that one is free to interchange what one calls a “particle” or “anti-particle”, or equivalently, which field is $\widehat{\psi}$ and which is the adjoint). Couplings to gauge fields are introduced by changing derivatives to covariant derivatives.
The Lagrangian will be
\begin{equation}
\label{eq:minkowski-lagrangian}
L=\psi^\dagger(\frac{\partial}{\partial t}+\boldsymbol\sigma\cdot\boldsymbol\nabla)\psi
\end{equation}
which is invariant under an action of the group $SL(2,\mathbf C)$, the spin double-cover of the time-orientation preserving Lorentz transformations. To see how this works, note that one can identify Minkowski space-time vectors with two dimensional self-adjoint complex matrices, as in
$$(E,\mathbf p)\leftrightarrow M(E,\mathbf p)=E-\boldsymbol \sigma\cdot \mathbf p=\begin{pmatrix} E-p_3& -p_1+ip_2\\-p_1-ip_2&E+p_3\end{pmatrix}$$
with the Minkowski norm-squared $-E^2+|\mathbf p|^2=-\det M$.
Elements $S\in SL(2,\mathbf C)$ act by
$$M\rightarrow SMS^\dagger$$
which, since it preserves self-adjointness and the determinant, is a Lorentz transformation.
The propagator of a free chiral spinor field in Minkowski space-time is (like other qfts) ill-defined as a function. It is a distribution, generally defined as a certain limit ($i\epsilon$ prescription). This can be done by taking the time and energy variables to be complex, with the propagator a function holomorphic in these variables in certain regions, giving the real time distribution as a boundary value of the holomorphic function. One can instead “Wick rotate” to imaginary time, where the analytically continued propagator becomes a well-defined function.
There is a well-developed formalism for working with Wick-rotated scalar fields in imaginary time, but Wick-rotation of a chiral spinor field is highly problematic. The source of the problem is that in Euclidean signature spacetime, the identification of vectors with complex matrices works differently. Taking the energy to be complex (so of the form $E+is$), Wick rotation gives matrices
$$\begin{pmatrix} is-p_3& -p_1+ip_2\\-p_1-ip_2&is+p_3\end{pmatrix}$$
which are no longer self-adjoint. The determinant of such a matrix is minus the Euclidean norm-squared $(s^2 +|\mathbf p|^2)$. Identifying $\mathbf R^4$ with matrices in this way, the spin double cover of the orthogonal group $SO(4)$ is
$$Spin(4)=SU(2)_L\times SU(2)_R$$
with elements pairs $S_L,S_R$ of $SU(2)$ group elements, acting by
$$\begin{pmatrix} is-p_3& -p_1+ip_2\\-p_1-ip_2&is+p_3\end{pmatrix}\rightarrow S_L\begin{pmatrix} is-p_3& -p_1+ip_2\\-p_1-ip_2&is+p_3\end{pmatrix}S_R^{-1}$$
The Wick rotation of the Minkowski spacetime Lagrangian above will only be invariant under the subgroup $SU(2)\subset SL(2,C)$ of matrices such that $S^\dagger=S^{-1}$ (these are the Lorentz transformations that leave the time direction invariant, so are just spatial rotations). It will also not be invariant under the full $Spin(4)$ group, but only under the diagonal $SU(2)$ subgroup. The conventional interpretation is that a Wick-rotated spinor field theory must contain two different chiral spinor fields, one transforming undert $SU(2)_L$, the other under $SU(2)_R$.
The argument of this preprint is that it’s possible there’s nothing wrong with the naive Wick rotation of the chiral spinor Lagrangian. This makes perfectly good sense, but only the diagonal $SU(2)$ subgroup of $Spin(4)$ acts non-trivially on Wick-rotated spacetime. The rest of the $Spin(4)$ group acts trivially on Wick-rotated spacetime and behaves like an internal symmetry, opening up new possibilities for the unification of internal and spacetime symmetries.
From this point of view, the relation between spacetime vectors and spinors is not the usual one, in a way that doesn’t matter in Minkowski spacetime, but does in Euclidean spacetime. More specifically, in complex spacetime the Spin group is
$$Spin(4,\mathbf C)=SL(2,\mathbf C)_L\times SL(2,\mathbf C)_R$$
there are two kinds of spinors ($S_L$ and $S_R$) and the usual story is that vectors are the tensor product $S_L\otimes S_R$. Restricting to Euclidean spacetime all that happens is that the $SL(2,\mathbf C)$ groups restrict to $SU(2)$.
Something much more subtle though is going on when one restricts to Minkowski spacetime. There the usual story is that vectors are the subspace of $S_L\otimes S_R$ invariant under the action of simultaneously swapping factors and conjugating. These are acted on by the restriction of $Spin(4,\mathbf C)$ to the $SL(2,\mathbf C)$ anti-diagonal subgroup of pairs $(\Omega,\overline{\Omega})$.
The proposal here is that one should instead take complex spacetime vectors to be the tensor product $S_R\otimes \overline{S_R}$, only using right-handed spinors, and the restriction to the Lorentz subgroup to be just the restriction to the $SL(2,\mathbf C)_R$ factor. This is indistinguishable from the usual story if you just think about Minkowski spacetime, since then all you have is one $SL(2,\mathbf C)$, its spin representation $S$ and the conjugate $\overline S$ of this representation.
Exactly because of this indistinguishability, one is not changing the symmetries of Minkowski spacetime in any way, in particular not introducing a distinguished time direction.
When one goes to Euclidean spacetime however, things are quite different than the usual story. Now only the $SU(2)_R$ subgroup of $Spin(4)=SU(2)_L\times SU(2)_R$ acts non-trivially on vectors, the $SU(2)_L$ becomes an internal symmtry. Since $S_R$ and $\overline{S_R}$ are equivalent representations, the vector representation is equivalent to $S_R\otimes S_R$ which decomposes into the direct sum of a one-dimensional representation and a three-dimensional representation. Unlike in Minkowski spacetime there is a distinguished direction, the direction of imaginary time.
Having such a distinguished direction is usually considered to be fatal inconsistency. It would be in Minkowski spacetime, but the way quantization in Euclidean quantum field theory works, it’s not an inconsistency. To recover the physical real time, Lorentz invariant theory, one need to pick a distinguished direction and use it (“Osterwalder-Schrader reflection”) to construct the physical state space.
Besides the preprint here, see chapter 10 of these notes for a more detailed explanation of the usual story of the different real forms of complexified four-dimensional space.
