Why is the Hamiltonian of a photon = 0?

Question

I'm studying the motion of light near Schwarzschild black holes, and I was wondering why the Hamiltonian of the Schwarzschild metric $$H = - \left( 1-\frac{2M}{r} \right)^{-1} \frac{p_{t}^2}{2}+\left( 1-\frac{2M}{r} \right) \frac{p_{r}^2}{2}+\left( \frac{p_{ \theta}^2}{2r^2}+\frac{p_{\phi}^2}{2r^2\sin^2\theta} \right)$$ is equal to zero for photons. I know that $H < 0$ is for every other particle and $H >0$ is mathematically possible but physically very unlikely since it would mean the particle is travelling faster than light. Still I don't know why the Hamiltonian (sum of potential and kinetic energies) of photons equals 0. Could anyone please help me?

Answer 1

The Hamiltonian is not always the sum of potential and kinetic energies. Like the Lagrangian, it is a theoretical construct that is not unique for any given physical system, and you can always construct a time-reparametrization invariant formulation of a system where the Hamiltonian vanishes. See this answer of mine for the construction and the first part of this answer of mine for the reason time-reparametrization invariant formulations where the canonical variables are scalars under time-reparametrization have vanishing Hamiltonians.

Answer 2

OP's Hamiltonian of the form $$ H~=~\frac{p^2+m^2}{2}, \qquad p^2~:=~p_{\mu}g^{\mu\nu}(x) p_{\nu}~\leq~0,\tag{1} $$ is called a super-Hamiltonian in e.g. MTW, cf. e.g. this Phys.SE post. (For a massless particle like the photon $m=0$.) The super-Hamiltonian (1) is the $e=1$ gauge of the Hamiltonian $$H~=~ \frac{e}{2}(p^2+m^2)\tag{2}$$ for a relativistic point particle, cf. e.g. this & this Phys.SE posts. Here $e$ is a Lagrange multiplier field that imposes the mass-shell condition $$ p^2+m^2~\approx~0.\tag{3}$$ For the super-Hamiltonian (1) the mass-shell condition (3) must be imposed by hand. The notion of energy is (as usual) tied to the notion of worldline (WL) parameter, cf. Noether's theorem. Because of WL reparametrization invariance, this energy notion is challenged, cf. e.g. this Phys.SE post.

References:

1. MTW; Section 21.1 p. 489.