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A well-known fact about QFTs in curved spacetimes is that there is a phenomenon of particle production in expanding universes, these being described by the line element $$ds^2=-dt^2+b^2(t)d\vec x^2.$$

This is related to the fact that the action $S$ of a field in this background includes the factor $b(t)$ in some of its terms, and it therefore affects the equations of motion of the mode functions and the notion of particle.

However, for some special cases, e.g. massless scalar fields with trivial coupling to the curvature, the function $b(t)$ is no longer present in the action. As a consequence, no particle production occurs.

Intuitively, I would say that particle production is associated to the breaking of a symmetry under transformations of the scale factor, $b(t_1)\rightarrow b(t_2)$. When this symmetry is respected, no particle production occurs.

Is this intuition correct? If so, which is the associated symmetry (mathematically speaking)?

Edit: from the discussion in the comments, I see two possibilities: that particle production is related to the breaking of time-translation symmetry, or that particle production is related to the breaking of Weyl symmetry. Also, it could be the case that particle production has nothing to do with symmetry breaking. Is there any formal derivation relating particle production with the breaking of any symmetry?

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    $\begingroup$ I would say this is time-translation symmetry $\endgroup$
    – Níckolas Alves
    Commented May 13 at 13:50
  • $\begingroup$ I see... So I guess that from this perspective particle production would just be a manifestation of the lost of energy conservation. However, is there some formal derivation of this, i.e. of the relation between time-translation symmetry breaking and particle production? Could particle production be thought of as an order parameter for this symmetry? $\endgroup$
    – TopoLynch
    Commented May 13 at 14:16
  • $\begingroup$ I also see that in some places this is known as conformal invariance. This is a bit confusing for me, because I understand that conformally-invariant theories are critical ones (i.e. gapless), but there are situations were there is particle production in critical points. $\endgroup$
    – TopoLynch
    Commented May 13 at 14:19
  • $\begingroup$ The behavior of energy is extremely correlated with particle production in curved spacetimes. See this answer, for example. I don't know of a formal derivation of specifically the breaking of time-translation symmetry always leading to particle production (although I do expect this to hold) $\endgroup$
    – Níckolas Alves
    Commented May 13 at 14:32
  • $\begingroup$ Conformal invariance also makes sense (spatially flat FLRW is conformal to Minkowski spacetime precisely with conformal factor $b$ in your notation), but I didn't really understand your comment about them $\endgroup$
    – Níckolas Alves
    Commented May 13 at 14:33

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Actually there is no particle production in QFTCS. Particle states in QFTCS are not well defined, since the definition of a particle relies on a choice of time coordinate to decompose the field into Fock representations* modes.

The phenomenon of particle production is related to diffeomorphisms invariance. The Fock space, and hence the vacuum state, is not diffeomorphism invariant, hence a state that in a reference frame is seen as a vacuum, in another reference frame (not obtained through Lorentz transf.) is seen as a state full of particles.

The concept of symmetry breaking is not very relevant in this sense, since what breaks the definiton of vacuum is not the lack of time translation invariance but the lack of invariance under time-diffeomorphisms. The breaking of time translations for an observer in an FLRW universe has the effect of changing the energy of particles, redshift of photons being a primary example, but does not affect the definition of a vacuum state, which remains a vacuum state also if you don't have time translation invariance.

*Fock spaces rely on a choice of time to define the plane waves, and all inertial frames possess the same Fock representations.

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  • $\begingroup$ Thank you, I see your point. However, as I understand it, you can consider asymptotic flat regions where the notion of particle (Fock space) is well defined. From this perspective, the production of particles arises from the fact that for certain period there has been an expansion, and terms in the action that are not invariant under Weyl transformations evolve non-trivially. So you have a unique notion of in- and out- particles, and a non-trivial evolution that occurs because some terms break a certain symmetry. Considering this, I would argue that symmetry plays a role here... $\endgroup$
    – TopoLynch
    Commented May 20 at 13:14
  • $\begingroup$ The fact is that you have 2 different well defined Fock spaces, one time at -inf and one at time +inf. Being the concept of Fock spaces not generally covariant, the two spaces are not defined w.r.t. Lorentz-equivalent reference frames, so the fact that the 'Out' vacuum is full of 'In' quanta is just a matter of coordinate transformation under which Fock spaces are not well behaved. If you check, in fact, you will find that the 'In' vacuum is full of 'out' states: how would you interpret this duality? The fact is that quanta of Fock spaces are meaningless in curved spacetimes. $\endgroup$
    – LolloBoldo
    Commented May 20 at 16:12
  • $\begingroup$ The above example implies that in the past you see a flux of particles that come at you from the future, so a kind of gravitational antiflux of particles: those are not real, like the incoming flux of particles due to gravitational particle creation. You are rather observing the action of a non inertial system in a space defined as the sets of inertial frame states. The quanta you see at infinity are the quantum/GR equivalent of non-inertial forces of Newton mechanics $\endgroup$
    – LolloBoldo
    Commented May 20 at 16:18

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