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I found this paper: On the origin of the LIGO "mystery" noise and the high energy particle physics desert, currently only published on arXiv as far as I can tell.

I do not understand any of the ideas in the paper (due to my own ignorance), but I would like to at least clear up how the word "fluctuation" is used there: For instance, it is stated in the paper that (first page, second column)

"With these motivations, we recently explored how the stress fluctuations of a massive quantum field would backreact on the gravitational vacuum in the IR."

Now, my understanding of quantum fluctuation is this: It is a (somewhat mysterious) name given to the not so surprising fact that: In any quantum mechanical system with states in the Hilbert space $\mathcal H$, if you have a self-adjoint linear (densely defined, maybe?) operator $$\hat{\mathcal{O}}:\mathcal H\supset D(\hat{\mathcal O})\to\mathcal H$$ corresponding to some observable $\mathcal O$ and a state $\vert\psi\rangle\in\mathcal H$ which is not an eigenvector of $\hat{\mathcal O}$, then performing the same measurement on a physical system in the state $\vert\psi\rangle$ will lead to different results. For further discussion of my understanding see [1], [2], [3], [4], [5]. In particular, the word "fluctuation" is misleading since the above fact need not have anything to do with a change in time (or space).

My question is this: Does the paper use the word "quantum fluctuation" differently than how I explained the word above? (I.e. in the sense that there is effectively a change in time or space?) If so, wouldn't this be at odds with the canonical formulation of quantum mechanics?


Bonus: At the end of a similar paper it is stated that

"[...] one can dismiss our bounds as a possible artifact of a UV regularization scheme [...]".

What is this sentence supposed to say?

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  • $\begingroup$ One can have uncertainty/fluctuations due to quantum effects, due to thermodynamic effects, and purely statistical uncertainty associated with measurement precision. So they need to be distinguished. $\endgroup$
    – Roger V.
    Sep 26, 2021 at 5:22

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Your understanding is correct: "fluctuation" is synonymous with non-eigenstateness, because of the implications of non-eigenstateness for measurement. It's not meant to suggest anything changing in time, at least not in the absence of measurement.

The language in the paper refers to something that is consistent with that, but more specific. "Stress" refers to the stress-energy tensor, which is the source for the gravitational field and which should be an operator in a quantum model. Just like in equation (2) of arXiv:0710.3787, the paper arXiv:1911.09384 is using the word "fluctuation" specifically for the quantity $$ \langle A^2\rangle-\langle A\rangle^2, \tag{1} $$ where $A$ is one of the operators corresponding to a component of the stress-energy tensor. The expectation value $\langle\cdots\rangle$ is with respect to the vacuum state. If the vacuum state were an eigenstate of $A$, then the quantity (1) would be zero, so nonzero value of the quantity (1) is a symptom of "fluctuation" (non-eigenstateness) exactly as you defined it in the question.

By the way, in relativistic QFT, the vacuum state cannot be an eigenstate of any local operator, so (1) must be nonzero. The paper isn't quite doing relativistic QFT, because it's trying to account for the backreaction on gravity, but the same principle is still expected to hold: a local operator can't have the vacuum as an eigenstate. That's called the Reeh-Schlieder property.

I glossed over some issues above, like exactly what "vacuum" means in this context, and exactly how the expectation values should be defined in view of the need for regularization/renormalization. The latter issue is just a fancy way of saying that we need to define the thing we're calculating before we can calculate it. I only skimmed through the paper arXiv:1703.05331 that was mentioned in the bonus question, but that's presumably the answer to the bonus question: the authors are respectably acknowledging that the way they chose to define (regulate) the otherwise-divergent quantities is subject to question and that their result might be an artifact of that choice.

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  • $\begingroup$ Thank you for this illuminating answer! As a small following (if it is too cumbersome for you to answer here I can open a new question): In arXiv:0710.3787 it is stated on page 7 that there are „active fluctuations in the intrinsic degrees of freedom of gravity“ and on page 13 that „Quantum stress tensor fluctuations lead to passive fluctuations of the gravitational field, which are in addition to the active fluctuations coming from the quantization of the gravitational field itself.“ Do you know what these active fluctuations refer to? $\endgroup$ Sep 26, 2021 at 7:55
  • $\begingroup$ Here is the follow-up question, so if you want to answer, you can do it there 😄: physics.stackexchange.com/q/668162/233512 $\endgroup$ Sep 26, 2021 at 8:11
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    $\begingroup$ Ahhh I see! So in that case, the terminology used for "fluctuation" is still perfectly consistent with my terminology. You have been very helpful for me, thank you! $\endgroup$ Sep 26, 2021 at 15:21
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    $\begingroup$ @MaximilianJanisch You're welcome! By the way, I'm convinced now that my guess about their active/passive language is correct, so I deleted my tentative-answer-in-a-comment and posted a real answer to the new question instead. And you're right: your definition of fluctuation still holds in that context. $\endgroup$ Sep 26, 2021 at 15:58
  • $\begingroup$ Beautiful answer! See also this one: physics.stackexchange.com/a/21745/226902 $\endgroup$
    – Quillo
    Apr 26, 2023 at 23:49

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