Constraint in BRST quantization of point particle

Question

On page 130 of Joe Polchinski's String Theory volume 1 book, the Constraint or the missing equation of motion for point particle after gauge fixing is $H = 0$, and the BRST operator is the ghost $c$ times this. I don't understand how can the missing equation of motion be $H = 0$. Missing equation of motion should be the stress tensor of worldline theory which we get after varying the metric in path integral. Stress tensor should get some contribution from ghost part of the action. But $H=0$ don't have any ghost contribution. So what is the reasoning behind this missing equation of motion?

Answer 1

1. The constraint $H\approx 0$ is a primary constraint when trying to Legendre transform the square root Lagrangian for a relativistic point particle. See also e.g. this related Phys.SE post.

2. Alternatively, with inclusion of the einbein $e$ [which is the square root of the world-line (WL) metric], the Hamiltonian [which is essentially the SEM tensor in 0+1D] vanishes $H\approx 0$ because of the EOM for $e$.

3. Finally, the vanishing Hamiltonian $H\approx 0$ is a consequence of world-line (WL) reparametrization invariance, cf. e.g. this Phys.SE post.