# Are vacuum fluctuations Gaussian?

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?

• What is a "vacuum fluctuation"? (and please, don't just say "fluctuations of the vacuum", but give a mathematical definition for it) – AccidentalFourierTransform Jan 8 '17 at 11:12
• If all the moments are Gaussian, the distn is Gaussian. – innisfree Jan 8 '17 at 11:37
• 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. – user140771 Jan 8 '17 at 11:54
• The fluctuations of an interacting field theory are not Gaussian by definition. – Mark Mitchison Jan 8 '17 at 12:28

## 1 Answer

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: $$G_n\propto \int\mathrm d\varphi\ \varphi_1\varphi_2\cdots\varphi_n\ \mathrm e^{iS[\varphi]}$$ 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.

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

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