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I have a question about the definition of the action that Landau defines as:

\begin{equation} S= -\frac{e}{c}\int A^{\mu}dx_{\mu} \end{equation}

he says that the $1/c$ factor is introduced by convenience. My question relates only to the dimensional analysis of the spatial part, because i'm basically doing this: \begin{equation} \frac{e}{c}A^idx_i\equiv \frac{[C][T]}{[L]}\cdot\frac{[V][T]}{[L]}\cdot[L] = \end{equation} \begin{equation} =\frac{[C][T]}{[L]}\cdot\ \frac{[U][T]}{[C]}=\frac{[U][T]^2}{[L]} \end{equation} where $[U]\equiv j~(joules)$, $[C]\equiv (coulomb)$. I can't see how the action could have units of $\textbf{energy x time}$. I missing some basic concept of dimensional analysis? I even thought about canceling [T] and [L] since the context is relativistic, but that should cause a "$c$" to appear.

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    $\begingroup$ Doesn’t Landau use Gaussian units rather than SI? If so, $A^0$ is the electric potential $\varphi$, $e\varphi$ is energy, and $dx_0/c$ is $dt$. Surely the spatial components give the same result! $\endgroup$
    – Ghoster
    Commented Aug 27 at 5:07
  • $\begingroup$ if i use SI the factor 1/c isn't even necessary, becouse in the SI system $A^{\mu}=(\phi/c,\vec{A})$ then both components are correct. But he used the definition in the gaussian system, then $A^{\mu}=(\phi,\vec{A})$, so when introducing the factor 1/c the temporal components are correct but the spatial seems like are not. I'm confused because after this then he sum this term to the free action $S=mc\int ds$ whose units seem to be correct. $\endgroup$
    – MrClapton
    Commented Aug 27 at 16:41
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    $\begingroup$ In Gaussian units, $[e]=[M]^{1/2}[L]^{3/2}[T]^{-1}$ and $[\vec A]=[M]^{1/2}[L]^{1/2}[T]^{-1}$ so $[e\vec A]=[M][L]^2[T]^{-2}=[E]$. And $[dx/c]=[T]$, so everything works just fine. $\endgroup$
    – Ghoster
    Commented Aug 27 at 17:07
  • $\begingroup$ Thank you very much, i was really mixing up some units with dimensions. I think i need to revise these things kkkk. $\endgroup$
    – MrClapton
    Commented Aug 27 at 17:46

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