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Particle production in curved spacetime

  1. May 8, 2007 #1
    Is there a coordinate independent method to compute particle production in curved spacetime for some scalar field?

    In all methods, they fix the quantum state of the field (they usually take the 'vacuum') by specifying a complete set of solutions of the classical wave equation of the field. Those solutions are written in a specific coordinate system. Another coordinate system will evoke different complete set of solutions (like say plane waves in cartesian coordinates and spherical waves in spherical coordinates). The results would be different if one chooses different complete sets of solutions.

    Moreover, no matter in what coordinate system one writes the wave equation, any complete set of solutions can be expressed in that system only some sets would look 'natural' (like a set of plane waves would look natural in cartesian coordinates but nobody can stop you to get the spherical waves set in the 'unnatural' for them cartesian coordinates).

    I can't understand why the scalar field wold 'prefer' a specific complet set of solutions or specific coordinate system in which those solutions look 'natural'. Isn't that a total violation of the GR principle that all coordinate systems are equivalent ???
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  3. May 8, 2007 #2

    Chris Hillman

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    Event horizons are global phenomena. Semiclassical approximation is delicate. So you shouldn't be surprised that it pays to use a coordinate chart which simplifies things as much as possible. If you poke around in the arXiv you can certainly find discussion of alternative charts, however.
  4. May 8, 2007 #3


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    There have been some past discussions of related issues.

    I think the first thing to realize is that the standard definition of particle density can't follow the tensor transformation rules, because the particle density can be zero in one coordinate system and non-zero in a co-moving but accelerating coordinate system - i.e. the Unruh effect.

    This behavior isn't compatible with the standard tensor transformation rules.

    Covariant defintions for particle density have been proposed, however, see for instance


    and the PF thread where they were mentioned https://www.physicsforums.com/showthread.php?t=160533&page=3

    unfortunately I don't really understand them. The author of the paper did not suggest any simple means of actually measuring the proposed definition of particle density.

    If there is some (possibly currently unknown) simple way of tying this covariant definition to measurements, I would think that the problem is just that our current definition of particle needs to be revised.

    This is based to some extent on personal philosophy - my personal definition of "real" is that a quantity has to both transform as a tensor, and be easily measured, to be "real".

    Examples of "real" objects by this definition would include the electromagnetic field as a whole (the Faraday tensor) and the acceleration 4-vector.

    [add]Perhaps more importantly in the context of the original question, "coordinate independence" requries that the objects involve transform in "coordinate independent" ways, which means as tensors. If you have something that is not a tensor, you shouldn't be surprised when you have to define the coordinate system to talk about the object. The problem is with the object itself - in this case, the idea of particle density.
    Last edited: May 8, 2007
  5. May 8, 2007 #4


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    I think that the very definition of particle in curved backgrounds is coordinate dependent, so you will not find a method for this I guess.
  6. May 8, 2007 #5
    Are you sure you are not mixing up gauge with the impossibility of applying Wick rotations on certain curved spacetimes?
  7. May 8, 2007 #6
    Tangent question: has the Unruh effect been measured or confirmed?
  8. May 8, 2007 #7
    Unruh hasn't been measured, nor is radiation from black holes (Hawking effect).

    Particle creation in early universe is very important because it is believed to provide the seeds of inhomogeneity that we later observe as large scale structure (galaxies, clusters, super clusters) as well as fluctuations in the Cosmic Microwave Background (power spectrum of CMB).

    The scalar field that is believed to lead to the initial inhomogeneities through particle creation is the inflaton, the field that had driven inflation (allegedly).

    So my qestion is not just academic question but has observational consequences. I noticed they always do the calculation in the usual FRW comoving coordinates, that's why I asked if the answer is coordinate dependent.
    Last edited: May 8, 2007
  9. May 8, 2007 #8


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    I expanded my answer a bit, but the simple version is that the current, accepted, standard definition of particles is coordinate dependent, so it's not terribly surprising that the calculations are, too.

    I think there's been some rather ambitious proposals to measure the Unruh effect, but they are just proposals as far as I know at this stage.

    Google finds http://www.munich-photonics.de/res-areas-B11.en.html [Broken]
    in this regard.
    Last edited by a moderator: May 2, 2017
  10. May 9, 2007 #9


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    The concept of a particle is coordinate dependent because it starts from the assumption that the fundamental quantity is a field, not a particle. In other words, from the point of view of field theory, the concept of a particle is rather artificial.

    A possible alternative is that the particle is a fundamental object, while the field is only an auxiliary mathematical object that helps in a description of particles. In this case, it is a field, not a particle, that should depend on the choice of coordinates. Such a picture seems particularly appealing if one assumes that particles are not exactly pointlike dots but slightly expanded STRINGS. For some results in that direction see
    especially Sec. II.D.
    You may also take a look at Fig. 1 which I particularly like.
  11. May 9, 2007 #10
    Casimir effect, whose calculation depends on the difference between two vacuum states (just like in the computations of particle creation in curved spacetime), has been measured experimentally and matches the theoretical result.

    Apart from that, the only effects that are linked to particle creation and are measurable are large scale structure and CMB fluctuations in cosmology. Unruh effect and Hawking effect have never been measured.

    I agree that in order to put GR and QFT in agreement we either have to make QFT local or have to make GR global somehow.

    Casimir effect is often cited as an example of non-local effect in QFT since the vacuum state of the EM field + metalic plates system is defined by the global configuration of the system (the distance between the plates). That doesn't mean though that the effect can't be derived with local concepts, yet I haven't seen such derivation yet.

    If anyone has seen a local derivation of Casimir effect, let me know. That will be interesting to read.
    Last edited: May 9, 2007
  12. May 10, 2007 #11


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    According to the Jaffe derivation of the Casimir effect
    this effect has nothing do to with vacuum energy, but is a consequence of electromagnetic interactions between particles that constitute the Casimir plates. Is that local enough?
  13. May 10, 2007 #12
    The derivation shown in that paper is obviously global. Jaffe uses Green's functions wich are the vacuum expectation value of two field operators. The vacuum state is defined through a Fock decomposition of the Hilbert space of the system. That decomposition is achieved through specifying a set of complete mode solutions to the field equation for the quantized field. The mode solutions are specified over all space and time and hence are global.

    Any derivation of Casimir effect based on conventional QFT is inevitably global due to the way the vacuum (and the other states) is specified in QFT. The non-locality of state definition in QFT is easily traced back to the non-locality of Quantum Mechanics and the lack of defition of measurement of a local observer there.

    A local derivation of Casimir should use a theory that is equivalet to conventional QFT but defines vacuum and other states for a local observer not over the whole space and time, something like having a separate Hilbert spaces at each spacetime point, similar to the tangent spaces in GR. Such a theory hasn't been written.
    Last edited: May 10, 2007
  14. May 11, 2007 #13


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    You are right, the concept of vacuum in QFT is nonlocal, so in this sense the derivation of the Casimir effect as above is nonlocal as well. However, with such a criterion for (non)locality I would say that ANY calculation in QFT is nonlocal. Would you agree?

    Still, you might like this:
  15. May 11, 2007 #14
    Yes any calculation in QFT or QM that uses the conventional definition of particle states. I think the old derivation of Casimir of the VanderWaals forces between two molecules was done in QM so it still doesn't qualify as local.
    Last edited: May 11, 2007
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