Measure for momentum in curved space

[Judithku]

When I write down a quantum field (for instance to compute T^00 or some expectation value)

I write it as an integral over momentum space.

If I am working in curved space
should this be divided by sqrt [g]?

(and why or why not?)

 

[Demystifier]

The answer is - no. The "momentum" space can be thought of just as a Fourier transform, which has nothing to do with a metric structure of spacetime.

Moreover, expansion of the field in terms of plane waves is not a natural thing to do in curved spacetime, because plane waves are not solutions of the Klein-Gordon equation in curved spacetime.

 

[Judithku]

Thanks -
I realize in general plane waves are inappropriate but I thought in dealing with a very weak potential that they could be used.

In papers written about the density of states in curved space, they say
\int d^3 x \int d^3 p
is invariant.
I thought this meant that the \sqrt g in the space integral cancels out
with its inverse in the momentum integral.
Do you think that is incorrect?
See for instance
http://www.sciencedirect.com/science/article/pii/0375960189905628
and on the arxiv
http://arxiv.org/abs/1012.5421