References 1 and 2 define a pure spinor $\psi$ to be a solution of the Cartan-Penrose equation $$ \newcommand{\opsi}{{\overline\psi}} v^\mu\gamma_\mu\psi=0 \hspace{1cm} \text{with} \hspace{1cm} v^\mu\equiv \opsi\gamma^\mu\psi, \tag{1} $$ where $\gamma^0,\gamma^1,...,\gamma^{D-1}$ are Dirac matrices for a $D$-dimensional spacetime with lorentzian signature, and presumably $\opsi\equiv\psi^\dagger\gamma^0$. References 1 and 2 consider real-valued solutions of this equation, so that $v^\mu$ is an ordinary spacetime vector. Equation (1) clearly implies that $v^\mu$ is a null vector (proof: multiply equation (1) by $\opsi$), but reference 1 makes this stronger statement on page 4:

The null direction, and the orientation of ... a space-like $D - 2$ plane orthogonal to the null direction, are both encoded in a pure spinor...

In this context, "plane" means the linear span of a set of $D-2$ linearly independent vectors in spacetime. The null vector $v^\mu$ itself does not uniquely determine the orientation of such a plane. To see why it's not unique, start with any set of $D-2$ linearly independent vectors orthogonal to $v^\mu$. This defines one plane through the origin. Now add some nonzero multiple of $v^\mu$ to one of those vectors. This gives another set of $D-2$ linearly independent vectors that define a different plane through the origin, still orthogonal to $v^\mu$.

In what way does a real solution $\psi$ of equation (1) select a unique orientation for a spacelike $(D-2)$-dimensional plane orthogonal to the null vector $v^\mu\equiv \opsi\gamma^\mu\psi$?

References and notes:

  1. Banks, Fiol, and Morisse (2006), Towards a quantum theory of de Sitter space (https://arxiv.org/abs/hep-th/0609062)

  2. Banks, Fischler, and Mannelli (2004), Microscopic Quantum Mechanics of the $p=\rho$ Universe" (https://arxiv.org/abs/hep-th/0408076)

  3. I posted this question on Physics SE instead of Math SE partly because reference 2 cites Misner, Thorne, and Wheeler's giant book Gravitation (1973). Maybe that book answers my question, but I don't have access to it right now.

  4. Related: Conflicting definitions of a spinor

  • 1
    $\begingroup$ I think taking a look at Misner-Wheeler-Thorne’s explanations on spinors (“Penrose null flags”) and Penrose’s visualizations of null flags and alpha-planes in “Spinors and Spacetime” will be helpful. MTW explicitly constructs a bivector from spinor-helicity variables. $\endgroup$
    – Lightcone
    Nov 12 '21 at 6:31
  • $\begingroup$ Might there be an implicit condition here that the null surface(s) are the spacelike foliation of the associated null hypersurface by surfaces orthogonal to the null direction inside the null hypersurface? (like Valter Moretti constructs it in this answer) $\endgroup$
    – ACuriousMind
    Nov 12 '21 at 12:23
  • $\begingroup$ @Lightcone Thank you for the comment. It helped inspire my self-answer. $\endgroup$ Nov 12 '21 at 15:58
  • $\begingroup$ @ACuriousMind Thank you for the comment. It helped inspire my self-answer. $\endgroup$ Nov 12 '21 at 15:58

I hadn't considered the possibility that the authors of the cited papers are using incorrect or lazy terminology. The issue is resolved by correcting their terminology in either of two equivalent ways:

  • If I change their word "space-like" to "null" and change $D-2$ to $D-1$, then everything makes sense.

  • If I leave their words as-is (space-like and $D-2$) but add the caveat "modulo the null direction $v^\mu$", then everything makes sense. This is equivalent to the previous option.

If this is what they meant, then their claim is consistent with the "flag plane" construction in Penrose's Spinors and Spacetime (which was called to my attention in a comment by @Lightcone). I'm betting that's the same as in Misner, Thorne, and Wheeler, which Banks et al cite (but I can't check this because I don't have access to that book right now). So in hindsight, I'm pretty much convinced that this is just another case of good physicists using incorrect/lazy terminology. The authors know what they're doing, of course, and sometimes that's the problem. When we really know what we're doing, we don't always realize that what we're saying doesn't accurately convey what we're thinking.


Here's an easy example to illlustrate the difference between what the authors said and what I think they actually meant. Work in three-dimensional flat spacetime ($D=3$) with metric diag$(1,-1,-1)$, and consider these three vectors: \begin{gather} v &= (1,1,0)\\ a &= (0,0,1)\\ b &= (1,1,1). \end{gather} The vector $v$ is null (lightlike). The vectors $a$ and $b$ are both spacelike and are both orthogonal to $v$. The pair $\{v,a\}$ spans a plane $P$, and the pair $\{v,b\}$ spans the same plane $P$. There are infinitely many other spacelike codimension-2 "surfaces" (lines) that have those same properties: orthogonal to $v$, and span the plane $P$ when combined with $v$.

Now here's the issue. The authors said that a real solution of the Cartan-Penrose equation encodes both the null vector $v$ and a spacelike direction (codimension-2 surface) orthogonal to $v$. That's problematic because lots of different spacelike directions are all orthogonal to $v$, and their wording suggests that the Cartan-Penrose equation selects one of them. But I think they meant that a real solution of the Cartan-Penrose equation encodes the plane $P$, or equivalently the null vector $v$ and an orthogonal spacelike direction modulo $v$, without intending to single out just one such spacelike direction.


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