# Is this interpretation of quantum fluctuation in eternal inflation in Wikipedia correct?

Although new inflation is classically rolling down the potential, quantum fluctuations can sometimes lift it to previous levels. These regions in which the inflaton fluctuates upwards expand much faster than regions in which the inflaton has a lower potential energy, and tend to dominate in terms of physical volume.

But from Sean Carroll’s article,

Eternal inflation is a different story. The idea there is that the inflaton field slowly rolls down its potential during inflation, except that quantum fluctuations will occasionally poke the field to go higher rather than lower. When that happens, space expands faster and inflation continues forever. This story relies on the idea that the “fluctuations” are actual events happening in real time, even in the absence of measurement and decoherence. And we’re saying that none of that is true. The field is essentially in a pure state, and simply rolls down its potential

So, I asked a friend of mine who knows QFT and he said

I never liked the concept of quantum fluctuations, especially when it comes to cosmology. In QFT, the fields always roll down to the exact minimum of the potential. It doesn't fluctuate in any meaningful sense. But the potential is the quantum mechanical one, not the classical one. The quote in the wikipedia may be a vague way to say that the classical potential acquires quantum corrections. Whether that picture is useful or not is beyond me.”

Is the interpretation of quantum fluctuation in Wikipedia correct from the point of view of QFT? Or does the inflaton field just simply roll down its potential without any effects from quantum fluctuation?

• The quantum fluctuations during inflation are not corrections to the classical potential as your friend posits. They are in addition to any such corrections. As a quantum field, the inflaton exhibits quantum fluctuations about its classical trajectory, $\phi(x,t) = \phi_0(t) + \delta \phi(x,t)$. These fluctuations cause inflation to end at different times at different places in the universe, giving rise to density perturbations in the post-inflationary universe. – bapowell Mar 14 '18 at 1:33
• @bapowell That looks like it should be an answer – David Z Mar 14 '18 at 5:20
• OK, I'll expand on it slightly and post it as an answer. – bapowell Mar 15 '18 at 14:35
• FWIW, the charge being levied in that blogpost is rather serious indeed. It is essentially accusing a great many cosmologists of basically misunderstanding the quantum theory - by imagining essentially it to be like that naive somewhat-better-than-Bohr picture of the electron in an atom as "teleporting" around inside its orbital with a definite but stochastic position at each instant of time,as opposed to more mature quantum interpretations where there is not a definite position at all,but rather an ill-defined one only up to the distribution that there would be to measure a specific position. – The_Sympathizer Mar 15 '18 at 14:56
• Hi @parker, please stop making trivial edits to bump the question into the front page. Thank you for your collaboration. – AccidentalFourierTransform Apr 27 '18 at 3:05

The classical inflaton potential does receive quantum corrections from the other fields in the theory, but quantum fluctuations of the inflaton field itself have much greater significance during inflation. As a quantum field, the inflaton exhibits fluctuations about its classical trajectory, $\phi(x,t) = \phi_0(t) + \delta \phi(x,t)$. The result is a sort of fuzziness to the trajectory:
The fluctuations cause the inflaton to roll down the potential at effectively different rates at different places in the universe, with the result that inflation ends at different times, $\delta t$, in different places. Via the continuity equation, this gives rise to a density perturbation, $\delta \rho/\bar{\rho} \propto H \delta t$.
Classically, a particle rolls down a potential viz. $\frac{d}{dt}\mathbf{p}=-\boldsymbol{\nabla}V$. The equivalent for a classical field is $\frac{d}{dt}\pi=-\frac{\delta V}{\delta\phi}$. The first of these equations is quantum-corrected to $\frac{d}{dt}\langle\mathbf{p}\rangle=-\langle\boldsymbol{\nabla}V\rangle$, which allows for quantum tunnelling. Similarly, the second equation becomes $\frac{d}{dt}\langle\pi\rangle=-\langle\frac{\delta V}{\delta\phi}\rangle$.