Determine canonical fields of action I'm working on an exercise which asks me to determine the canonical fields, and their equations of motion, of this invariant action:
$$
S = \int d\tau \sqrt{g_{\tau\tau}}\left( \frac{\tilde m}{2} g^{\tau\tau} \frac{d}{d\tau} x^\mu \frac{d}{d\tau} x_\mu - V  \right),
$$
where $\tilde m=1/2$ and $V=m^2$.
I don't know what a canonical field is and I don't know how to determine them from the action. Could anyone help me out here?
 A: I suspect you need to derive the pair of coordinates and their corresponding momenta.
The coordinates are just the variables your action is formulated in so $x^{\mu}$ and $g_{\tau \tau}$.
The definition of canonical momenta is
$$ p_{\mu} = \frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}}, $$
where $\mathcal{L}$ is the Lagrangian density (the integrand in the action), and the dot stands for derivative wrt $\tau$.
It is a simple calculus exercise to derive the explicit formula for $p_{\mu}$ in terms of $x$ and $\dot{x}$ which you of course should do yourself, because we don’t do your homework for you on here ;)
There is no canonical momentum corresponding to $g_{\tau \tau}$, because it doesn’t appear with derivative wrt $\tau$ in the action. We call such variables Lagrange multipliers. They generate constraints on other variables. Your values of $p$ are not all independent. They in fact obey a constraint, which for vanishing $V$ reads $p_{\mu} p^{\mu} = m^2$ which is the famous Einstein’s energy-momentum kinematic relation. It can be derived by requiring that the variation of the action wrt $g_{\tau \tau}$ vanishes:
$$ \frac{\partial \mathcal{L}}{\partial g_{\tau \tau}} = 0. $$
Let me know in the comments if you have further questions!
Update: sorry I didn’t realize $g$ was the internal 1-dimensional metric on the worldline, I thought it was the spacetime metric. I edited my answer accordingly.
