# 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?

## 1 Answer

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
• @RenhongFang yes, exactly :-) – AccidentalFourierTransform May 5 '16 at 8:01
• @RenhongFang anyway, if you think that my post was useful and fully answers to your question, it would be nice for you to click on the accept button (tick mark) so that the question gets the "answered" status. – AccidentalFourierTransform May 5 '16 at 22:25
• I am just a fresh, and thanks you for reminding me of this :-) – Ren-Hong Fang May 6 '16 at 1:16