# What is the dimension/unit of a spinor?

I am interested in getting the physical units of a spinor for the usual $(1,3)$ Minkowski spacetime. I am getting 2 different answers, using 2 different approaches!

On one hand, using the Lagrangian (density) of QED, which has units:

$E L^{-3}$

with $E$ and $L$ referring to energy and length respectively. I obtained that the unit of a Dirac spinor is $L^{-3/2}$.

On the other hand, since a Dirac spinor is a direct sum of 2 Weyl spinors, and since $x^\mu$ can be written as a tensor product of 2 Weyl spinors (as is done for instance in Penrose and Rindler's 2-volume book), it does seem that the unit of a Weyl spinor, and therefore of a Dirac spinor is $L^{1/2}$, unless Penrose and Rindler set various constants such as $c$, $\hbar$,... to $1$.

Can someone please clear my confusion?

• Both coordinates and momenta are vectors but have different dimensions. What does that tell you? – OON Jan 4 '18 at 13:17

The representation the object belongs to doesn't tell you anything about its dimensionality at all. Even if you fix $c=1,\hbar=1$ you will work with objects with all sorts of dimensionalities (and those are "natural" objects without any extra arbitrary dimensionful factor):
• Scalars: canonically normalized scalar field $\phi\sim L^{-1}$, mass $m\sim L^{-1}$, Lagrangian density $\mathcal{L}\sim L^{-4}$, action $S\sim 1$, auxiliary fields in SUSY $F,D\sim L^{-2}$.
• Vectors: coordinates $x^\mu\sim L$, 4-velocity $u^\mu\sim 1$, momentum $p^\mu\sim L^{-1}$, canonically normalized vector fields $A^\mu\sim L^{-1}$, 4-current $J^\mu\sim L^{-3}$
• 2-tensors: metric $g_{\mu\nu}\sim 1$, canonically normalized spin-2 field (e.g. used to describe graviton) $h_{\mu\nu}\sim L^{-1}$, stress-energy tensor $T^{\mu\nu}\sim L^{-4}$.
• Spinors: canonically normalized spinor field $\psi\sim L^{-3/2}$, supercharge and Grassmanian coordinates $Q,\theta\sim L^{-1/2}$.