Exploring the Ambiguity of Particles in Curved Spacetime

[LAHLH]

Hi

I'm trying to learn more about the Unruh effect, and was wondering if someone could comment on how exactly the lack of Poincare symmetry in a general curved space leads ambiguity in the notion of "particles".

Why exactly do we associate particles in QFT with positive frequency modes with respect to some preferred time? this idea is not very clear in my mind.

Thanks for any help or links

 

[marcus]

Hi

I'm trying to learn more about the Unruh effect, and was wondering if someone could comment on how exactly the lack of Poincare symmetry in a general curved space leads ambiguity in the notion of "particles".

Why exactly do we associate particles in QFT with positive frequency modes with respect to some preferred time? this idea is not very clear in my mind.

Thanks for any help or links

I recall finding this helpful:
http://arxiv.org/abs/gr-qc/0409054

 

[LAHLH]

I recall finding this helpful:
http://arxiv.org/abs/gr-qc/0409054

Thanks will have a read

 

[atyy]

Why exactly do we associate particles in QFT with positive frequency modes with respect to some preferred time? this idea is not very clear in my mind.

A "particle" in quantum field theory is not the same thing as a "particle" in classical physics.

In classical physics, a particle is an object that has definite position and momentum.

In quantum field theory, such objects don't exist, and a particle is defined to be one with definite momentum, and no definite position.

In high energy particle experiments, usually the momentum is much more well defined than position. However, there is some uncertainty in momentum, and hence the particle is slightly localized as a track in a cloud chamber or something like that. Such a track, although seeming like a "definite" track to our coarse vision, is a very delocalized thing in terms of the wavelengths of high momentum "particles".

All of that is for quantum field theory in flat Minkowski spacetime, and the Poincare symmetry is essential for the notion of "particles" as definite momentum excitations of a unique ground state (vacuum) that has Poincare symmetry.

In curved spacetime, there is no Poincare symmetry, and maybe no obvious symmetries at all, so it's not clear how to define a unique ground state.

http://arxiv.org/abs/gr-qc/0608018

 

[tom.stoer]

In curved spacetime, there is no Poincare symmetry, and maybe no obvious symmetries at all, so it's not clear how to define a unique ground state.

There is no global symmetry in general, as a specific solution of GR need not respect a certain symmetry. But Loentz and Poincare symmetry are implicitly there in sense of local gauge symmetries of the gravitational force.

 

[sheaf]

Why exactly do we associate particles in QFT with positive frequency modes with respect to some preferred time? this idea is not very clear in my mind.

OK, trying to access my memory from 25 years ago which was the last time I did this stuff...

The Fourier coefficients of the positive frequencies are the objects which become creation operators in QFT and you use these to build the Hilbert (Fock) space of states. I think a key point in thinking of these as representing particles is that this space carries a representation of the Poincare group. Two of the Casimir operators of this group are the mass and angular momentum (spin) operators, and these are essential for interpreting the states in a particle like way.

 

[LAHLH]

My understanding of this at present:

There are several sets of basis solutions to the KG eqn one could pick from. In flat space the KG equation is such that these solutions can be separated into functions of the form X(x)T(t), where T(t)~e^{-iwt}; this allows us to *choose* a preferred set of modes as those that are positive frequency wrt a time coordinate t: $$\partial_t f=-i\omega f$$. The only ambiguity in this is the choice of 't', but because we're in flat space all possible choices of t are related by Poincare transformations, and it's easy to show that this translates into all observers agreeing with the others about whether a mode is positive freq or not (although they will in general disagree on the actual momentum of a mode/particle they nevertheless at least agree that it *is* a particle) and about the vacuum state being empty etc etc. In curved space the story changes because now you can't in general (unless static spacetime) separate solutions into a form T(t)X(x), so you can't pick out of the several possible sets of modes based on which are positive freq anymore; there is no way to distinguish one set of mode solutions as preferred over any others.
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My issue still remains though: why do we use the criterion $$\partial_t f=-i\omega f$$ to define what we mean by 'positive frequency'? secondly why do the positive frequency modes (as defined by this) necessarily end up with the annihilation operator as their coefficient in the field expansion? and finally why do we use (or rather why do we want to use it; since we won't be able to use it in general, only when spacetime is static) this criterion as the distinguishing criterion for which modes out the several sets (all related by Bogolubov transformations) are picked out?