Generation of axion one-particle states by coherent oscillations of axion field

Question

Let's have axion field action in expanding universe. For a case of spatially homogeneous classical axion field the solution of corresponding EOM is oscillating solution, $\theta \sim \frac{cos(m_{\theta}t)}{a^{\frac{3}{2}}(t)}$.  Then scientist say that because of corresponding energy density behaves as $\frac{1}{a^3}$, where $a$ denotes the scale factor, and the pressure of such field is equal to zero, oscillating classical axion field is equal to coherent state of quantum axions with zero momentum. But this statement is not obvious for me, since I don't understand how we transform classical description of background field to the quantum description of creation and destruction operators.

Comments

In general, the quantum states that are closest to the classical ones are coherent states. In this regard the QFT situation is not much different with the QM case. 

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@JiaYiyang : but why this mechanism is called "axion production" mechanism? I think that axions are already produced before axion field begins to oscillate. In which sense misalignment mechanism produces axions?

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JiaYiyang : I've read that only after generation of axion mass we may quantize axion classical field modes. Maybe, massless field can't form VEV in coherent state? I see here analogy with static constant magnetic field, in which photons are off-shell.

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Sorry but I don't have any knowledge on "axion production" mechanism.

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@JiaYiyang : I now understanding this mechanism as a consequence of a fact that vacuum in theory of massless theory isn't determined uniquely, so that field may have arbitrary nonzero VEV (which corresponds to zero mode coherent state). This VEV isn't observable. However, if we turn on mass (by nonzero VEV of Chern class caused by instantons), then VEV (corresoinding coherent state) becomes to make contribution in observable quantities since energy of zero mode now is not zero. This is what is called vacuum misalignment mechanism of axion production.

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VEVs are observable as particle masses. They are a property of the elementary excitations of the physical vacuum state. All the other vacua correspond to different superselection sectors of the universe, and are therefore unobservable.

(It is a bit misleading to talk in this context about an "axion production mechanism", as production is usually understood as a process creating particles. The VEV is responsible for the existence of the axions, not for their production. For the latter one has to specify specific reactions, which follow from the form of the effective action.)

Answer 1

In quantum field theory, classical background fields are created in the following way:

Given a quantum field $\phi(x)$, the transformation that changes the field to $\phi(x)+\phi_{cl}(x)$, where $\phi_{cl}(x)$ is a classical field, is called a Bogoliubov transformation. By introducing a cutoff and a Fock space description of the field with the cutoff, one can see that the Bogoliubov transformation is a unitary transformation that transforms the vacuum state of the Fock representation into a coherent state whose expectation is the classical field. If the cutoff is removed, the transformation remains unitary if the classical field decays fast enough at spatial infinity; otherwise the transformation maps the vacuum sector to a different superselection sector. 

This can be seen by working with free fields only. In this case, the Lagragians before and after the Bogoliubov transformation are both quadratic, and everything is exactly solvable.

This is also the way spontaneous symmetry breaking works - the field in an unstable (false) vacuum state undergoes spontaneous fluctuations that drive the system exponentially fast away from the vacuum state until it settles in a new (physical) vacuum state, characterized by a nonzero vacuum expectation value. This nonzero vacuum expectation value is the classical background field.

Comments

Thank you! But why this mechanism (oscillations of axion EV after turning on mass) is called "axion production" mechanism? I think that axions are already produced before axion field begins to oscillate. In which sense misalignment mechanism produces axions?

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I've read that only after generation of axion mass we may quantize axion classical field modes. Maybe, massless field can't form VEV in coherent state? I see here analogy with static constant magnetic field, in which photons are off-shell.