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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?

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  • $\begingroup$ Both coordinates and momenta are vectors but have different dimensions. What does that tell you? $\endgroup$ – OON Jan 4 '18 at 13:17
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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 $Q,\theta\sim L^{-1/2}$.

Note that I repeatedly use canonically normalized field. This is the requirement to normalize kinetic term of the field so that it was multiplied on the dimensionless constant only. This is the only reason why we assign certain dimensionality to a field. If the object of interest is not the field with a canonically normalized kinetic term... there is no reason for it to have that dimensionality!

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