I am trying to understand something very simple, the einbein field formalism. Can someone write a down to earth step by step play of simple example say the canonical $$L = - m \sqrt{\dot{x}^2}\tag{1.1}$$ case (described in link). I have done some digging around and found this. I don't want to look at it again. How does one come up with an einbein (auxiliary?) field. I am asking because I have seen this sort of strategy at a lot of places recently. If it is possible can we just do an analog with something that looks like a quadratic equation and keep physics lingo away?
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Step 1. Start with a constrained Hamiltonian system.
$$ S[x, p, e] = \int d\tau \left( p_{\mu} \dot{x}^{\mu} - \frac{e}{2} \left( p^2 - m^2 \right) \right). $$
This is just a constrained Hamiltonian system, the constraint being the mass-shell condition $p^2 = m^2$.
Step 2. Solve the (algebraic) e.o.m. for the canonical momenta $p_{\mu}$ and plug it in the action.
$$ p_{\mu} = \dot{x}_{\mu} / e,$$ $$ S[x,e] = \frac{1}{2} \int d\tau \left( e^{-1} \dot{x}^2 + e m^2 \right). $$
This is your desired form of the action I believe. $e$ plays the role of the Einbein field. This can be seen as follows: remember how the diffeo-invariant 4d integrals look in General Relativity? You sort of do the same thing in 1d where the metric tensor has only one component called $g_{\tau \tau} = e^2$. The integral is then
$$ \int d\tau \sqrt{g_{\tau \tau}} \left(g^{\tau \tau} \partial_{\tau} x \partial_{\tau} x + m^2 \right) = \int d\tau \left( e^{-1} \dot{x}^2 + e m^2 \right). $$
Note that $g_{\tau \tau} = e^2;\; \sqrt{g_{\tau \tau}} = e;\; g^{\tau \tau} = e^{-2}$ (I am assuming that $e$ is positive here).
Step 3. Check that the usual form of the action is implied by $S[x,e]$.
Solve the algebraic e.o.m. for $e$ with respect to $e$ and plug the result back in $S[x,e]$. I trust you to do this calculation. The end result is
$$ S[x] = -m \int d\tau \sqrt{\dot{x}^2}, $$
which coincides with what you would expect.
Prof. Legolasov
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