Momentum Operator In Curved Spacetime

[Physicist97]

Hello, I'm sorry if this question sounds silly, but in QM the Momentum Operator is ${\hat{p}}=-i{\hbar}{\nabla}$ . In Relativistic QM in Flat Space, this operator can be written ${\hat{P}_{\mu}}=-i{\hbar}{\partial}_{\mu}$ . Would it be correct, then, to say that in curved spacetime the momentum operator would be ${\hat{P}_{\mu}}=-i{\hbar}{\nabla}_{\mu}$ ? Here ${\nabla}$ represents the gradient, ${\partial}_{\mu}$ is the Four-Gradient, while ${\nabla}_{\mu}$ is the covariant derivative. Again, sorry if this is a naïve question and please correct me if I am mistaken in my line of thinking :) .

 

[fzero]

One immediate problem with trying to set $\hat{P}_\mu = -i \hbar \nabla_\mu$ is that $[ \hat{P}_\mu, \hat{P}_\nu] \neq 0$ (it's actually related to the Riemann tensor). For special relativistic systems one can still do quantum mechanics with momentum and position operators and special relativistic wave equations, but for quantum physics in a classical curved spacetime, one generally needs to study full quantum field theory.

 

[Physicist97]

Yes, but isn't the commutators of the covariant derivatives equal to $R^{\rho}_{\sigma\mu\nu}V^{\sigma}-S^{\lambda}_{\mu\nu}{\nabla}_{\lambda}V^{\rho}$ . It is only equal to the Riemann Tensor when the torsion ( $S^{\lambda}_{\mu\nu}$ ) is zero.

 

[fzero]

Sure, if you add torsion, it is even more complicated. But the point was that that operator will not satisfy canonical commutation relations. It's not a proof that it cannot work, but it is a hint that something different is probably required.

A more fruitful place to start would probably be with the action for a test particle on a curved spacetime. Then one could use path-integral quantization to develop the quantum mechanics. It should be possible to derive the curved-space version of the Klein-Gordon equation, for example.

 

[Physicist97]

I've never heard of path-integral quantization. Do you know of any site or book that could explain it, in a mathematical way?

 

[fzero]

I've never heard of path-integral quantization. Do you know of any site or book that could explain it, in a mathematical way?

Kleinert has a book that he seems to have mostly online at http://users.physik.fu-berlin.de/~kleinert/b5/ I've only just looked at Ch 10 which is on curved space, so I don't know how accessible the early chapters are.