In this post I’ll discuss Weyl spinor fields and explain why Wick-rotating a single Weyl spinor field appears to be impossible. This motivates a proposal for a different way of thinking about the relation between spinors and vectors, described by the slogan “spacetime is right-handed”.
Written in energy-momentum space, the equation of motion of a right-handed Weyl spinor field is
$$(E-\boldsymbol \sigma\cdot\boldsymbol p)\widetilde \psi_R(E,\mathbf p)=0$$
Here $\psi$ is a two-component complex field. Since
$$(E+\boldsymbol \sigma\cdot\boldsymbol p)(E-\boldsymbol \sigma\cdot\boldsymbol p)=E^2-|\mathbf p|^2$$
solutions will also be solutions of the massless Klein-Gordon equation and satisfy $E=\pm|\mathbf p|$.
Defining helicity as the eigenvalue of the operator
$$\frac{1}{2}\frac{\boldsymbol \sigma\cdot\boldsymbol p}{|\boldsymbol p|}$$
solutions of the equation of motion have either
- Positive energy $E$, so describe particles, and helicity $+\frac{1}{2}$.
- Negative energy $E$, so describe anti-particles, and helicity $-\frac{1}{2}$.
The quantum theory is an infinite collection of complex harmonic oscillators (one for each value of $\mathbf p$), for each of which quantization proceeds as discussed here, with annihilation and creation operators satisfying anti-commutation relations.
The Standard Model description of matter particles is built out of copies of this quantum theory, with interactions determined by gauge symmetry (replacing derivatives by covariant derivatives), and mass terms coming from Yukawa couplings to the Higgs field.
The Wightman function is the two by two matrix
$$\widetilde W_2(E,\mathbf p)=(E-\boldsymbol \sigma\cdot\boldsymbol p)^{-1}=\frac{E+\boldsymbol \sigma\cdot\boldsymbol p}{E^2-|\mathbf p|^2}$$
The obvious way to do Wick rotation would be to analytically continue in the complex $E$ plane, getting a Schwinger function
$$\widetilde S_2(E,\mathbf p)=(iE-\boldsymbol \sigma\cdot\boldsymbol p)^{-1}$$
This runs into a fundamental inconsistency with the usual understanding of the transformation properties of vectors and spinors in Minkowski and Euclidean spacetime. For a detailed discussion of the usual story, see some notes here, but the bottom line is that the operator
$$(E-\boldsymbol \sigma\cdot\boldsymbol p)$$
identifies vectors $(E,\mathbf p)$ with two by two complex matrices, and these matrices are supposed to be maps from the space $S_R^*$ of dual right-handed spinors to $S_L$, the space of left-handed spinors.
In Minkowski spacetime $S_L$ and $S_R$ are complex conjugate representations. $S_L$ is just a name for the conjugate of $S_R$ and one only needs one kind of complex field. The problem is that in Euclidean spacetime $S_L$ has no relation to $S_R$, it’s a different representation. If you want a Euclidean field theory of spinors, with the usual relation of vectors and spinors, you need to add a left-handed spinor field. The OS construction of Euclidean fields does this, then doubles again the number of degrees of freedom in order to get a self-adjoint Schwinger function.
The proposal made here is that one should resolve this problem by only using one kind of Weyl spinor field ($S_R$) to describe spacetime vectors (as maps from $S_R^*$ to conjugates of $S_R$), both in Minkowski spacetime and in Euclidean spacetime. In Minkowski spacetime nothing changes (except not using the inappropriate notation $S_L$), but the Euclidean spacetime one Wick rotates to is different than the usual one. Only the $SU(2)_R$ factor of $Spin(4)$ acts non-trivially on vectors, with the $SU(2)_L$ acting trivially, available for use as a gauged internal symmetry.
Four dimensions is very special in that it’s the only dimension in which the rotation group breaks up into two independent pieces and the geometry can be thought of as decomposing into right-handed and left-handed parts. In the usual formalism, one exploits this in Euclidean spacetime (working with self-dual or anti-self-dual objects), but in Minkowski spacetime the two parts are complex conjugates and can’t be separated. I’m proposing a different point of view, in which both Minkowski and Euclidean spacetime just sees the right-handed part of the geometry, with the left-handed part appearing as internal degrees of freedom.
So far I’ve avoided writing about twistors, which provide a fundamentally chiral context for thinking about the implications of this “spacetime is right-handed” point of view. In twistor theory, a point in spacetime is tautologically the same thing as the space of spinors $S_R$ at that point. $S_L$ is something else. I’ll be away on a long weekend starting tomorrow, but will write about twistors in another posting soon.
