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Measure for momentum in curved space

  1. Dec 22, 2011 #1
    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?)
  2. jcsd
  3. Dec 22, 2011 #2


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    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.
  4. Dec 22, 2011 #3
    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
    and on the arxiv
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