# The true dimension of Dirac field

In natural units with $\hbar=1$ and $c=1$, as we know, the energy dimension of the Dirac field $\psi(x)$ in QED is $\frac{3}{2}$. But in cgs units, what is the true dimension of the Dirac field $\psi(x)$ in terms of cm, g and s? It seems that only the energy dimension of the Dirac field $\psi(x)$ is physical, is it true?

The Lagrangian for Dirac's equation is $$\mathcal L=-mc^2\psi^2+\cdots \tag{1}$$
As we know that $H\sim\mathrm d^3\boldsymbol x\ \mathcal L$ has units of energy, we conclude that $$\psi^2\sim x^{-3}\tag{2}$$ and therefore $\psi$ has units of $[\mathrm{length}]^{-3/2}$. If you use a different convention for $\mathcal L$ instead of $(1)$ you'll get a different result.
• So if we use different conventions as follows, \begin{eqnarray*} \mathcal{L} & = & -mc^{2}\psi+\cdots\\ \mathcal{L} & = & -m\psi+\cdots\\ \mathcal{L} & = & -mc^{2}\hbar^{3}\psi+\cdots \end{eqnarray*} then $\psi$ may have different dimensions (up to some orders of $\hbar$ and $c$). These different dimensions of $\psi$ may not influence physical observables. Is this right? – Ren-Hong Fang May 5 '16 at 3:10