# Why is the mass dimension of anticommuting coordinates $[Mass]^{-1/2}$

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I am reading a review ( http://arxiv.org/abs/hep-ph/9709356 ) about supersymmetry. On page 29 I have read that the mass dimension of the Grassmann anticommuting coordinates is $-1/2$. Why this? Why don't they have the same mass dimensions as the bosonic coordinates?

asked May 13, 2015
edited May 13, 2015

You can work out the dimension of any field from the fact that the free action must be dimensionless (in units where $\hbar=1$).

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By definition supersymmetry transformations square to spacetime translations. In a superspace formalism the supersymmetry operator is constructed from the vector field $\partial_\theta$ with respect to the odd coordinates $\theta$. As this operator has to square to the vector field $\partial_x$ with respect to the even coordinates $x$, which is of dimension $1$, the vector field with respect to the odd coordinate has to be of dimension $1/2$ and so the odd coordinate as to be of dimension $-1/2$.

Equivalently, a typical superfield is of the form

$\phi + \theta \psi +...$

where $\phi$ is a scalar and $\psi$ a spinor. In $d$ spacetime dimensions, a scalar is of dimension $(d-2)/2$, a spinor is of dimension $(d-1)/2$ and so $\theta$ has to be of dimension $-1/2$.

answered May 14, 2015 by (5,140 points)
edited May 15, 2015 by 40227

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