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If we consider just one mode and we compute the contribution from this mode to the $n$-th moment $\langle\mu_n\rangle = \langle E(0)^n\rangle$ (the state is the vacuum) and we also compute the moments of a Gaussian distribution (zero mean and unit variance) we observe that the values are the same.

Do you think that this is a conclusive proof for saying that the vacuum fluctuations are Gaussian?

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    $\begingroup$ What is a "vacuum fluctuation"? (and please, don't just say "fluctuations of the vacuum", but give a mathematical definition for it) $\endgroup$ – AccidentalFourierTransform Jan 8 '17 at 11:12
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    $\begingroup$ If all the moments are Gaussian, the distn is Gaussian. $\endgroup$ – innisfree Jan 8 '17 at 11:37
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    $\begingroup$ The expectation value of the electromagnetic field is 0 (vacuum state). Nonetheless, if we compute the expectation value of the intensity of the electromagnetic field $\langle E^2\rangle$, the result is not 0. Therefore, exist vacuum fluctuations about a zero average value. $\endgroup$ – user140771 Jan 8 '17 at 11:54
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    $\begingroup$ The fluctuations of an interacting field theory are not Gaussian by definition. $\endgroup$ – Mark Mitchison Jan 8 '17 at 12:28
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As innisfree points out in a comment, a distribution is gaussian if and only if all its moments are gaussian. In the case of a free bosonic field, Wick's theorem guarantees that all the $n$-point functions are gaussian, as can be seen by comparing the theorem to Isserlis' theorem. In the end it all boils down to the fact that $n$-point functions of free fields are formally gaussian moments[1]: $$ G_n\propto \int\mathrm d\varphi\ \varphi_1\varphi_2\cdots\varphi_n\ \mathrm e^{iS[\varphi]} $$ where $S$ is quadratic, and $\mathrm d\varphi$ is the measure over the space of field configurations (see Functional integration).

But there is a very important point to be made: quantum field theory is not statistical mechanics, regardless of the obvious formal analogies. The correlation functions of QFT do not measure correlations in the statistical sense of the word; the $G_n$ above are not moments of a distribution.

Moreover, as Mark Mitchison mentions in another comment above, the real word is not free and thus the true correlation functions are not gaussian. The action $S$ is not quadratic in general. And the "true configuration" of the universe is not the vacuum state. Therefore, the discussion above does not really apply to the real world.


[1] to make the analogy more clear one should Wick rotate the path integral.

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  • $\begingroup$ GaussIan QFT fluctuations are analogous to the small oscillations approximation in classical mevhanics. $\endgroup$ – Lewis Miller Jan 8 '17 at 14:58

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