The hamiltonian in quantum relativity

Question

The energy in general relativity is:

$$E=\gamma m c^2$$

where $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

so that it can be quantized by the Hamiltonian:

$$H=mc^2\frac{1}{\sqrt{1-\frac{\Delta}{c^2}}}$$

where $\Delta$ is the Laplacian, if $\Delta<c^2$. In the case $\Delta>c^2$, we have to take:

$$H=mc^3\frac{1}{\sqrt{1-\frac{c^2}{\Delta}}} \sqrt{\Delta^{-1}}$$

Can we make a theory of quantum relativity with this Hamiltonian?

Comments

Can we make a theory of quantum relativity with this Hamiltonian?

No, we cannot. Hamiltonians are not constructed with such a replacement. Your construction is not local, has a wrong dimension in the ratio with velocity, etc.

Some (relativistic or not) Hamiltonians do not represent the system energy, see, for example, an interaction picture or similar constructions.

Answer 1

Easily only for a single particle. Hamiltonian multiparticle relativistic theories are quite complicated. See the comprehensive survey article

B.D. Keister and W.N. Polyzou,
Relativistic Hamiltonian dynamics in nuclear and particle physics,
Adv. Nuclear Physics 20 (1991), 226--479.