# Is there any correlation between the energy density fluctuations of two separate systems in a vacuum state?

I think the title says it all. What I am curious to find out is if there are any observable changes in the fluctuations of zero-point energy in a vacuum state system that are the consequence of operations performed at a separate vacuum state system. I am not simply asking if there are correlations naturally, I am asking if there is any operation/setup/configuration that could be implemented to derive such a result. -Thanks-

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What do you mean by a "separate vacuum state system"? – Peter Morgan Jun 29 '12 at 11:44

I've assumed that by a "separate vacuum state system" you mean two systems of measurement or preparation operators, $\hat\phi(x_1)\cdots\hat\phi(x_m)$ and $\hat\phi(y_1)\cdots\hat\phi(y_n)$, where the positions in the $x$ group and the positions in the $y$ group are separated by a relatively large space-like distance.

In that case, No, at least not if the vacuum state $\omega$ of the quantum field satisfies Cluster Decomposition, which is the requirement that for any space-like 4-vector $V$,
\begin{eqnarray}\lim\limits_{\lambda\rightarrow\infty}\omega(\hat\phi(x_1)\cdots\hat\phi(x_m)\hat\phi(y_1+\lambda V)\cdots\hat\phi(y_n+\lambda V))=\omega(\hat\phi(x_1)\cdots\hat\phi(x_m))\omega(\hat\phi(y_1)\cdots\hat\phi(y_n)). \end{eqnarray} This ensures that measurements at large space-like separation are independent. Weinberg, for example, emphasizes that for him this is a Fundamental Principle by giving it its own chapter in Volume $\scriptstyle{\mathrm{I}}$ of his "The Quantum Theory of Fields".

The Cluster Decomposition Condition or Principle, however, is precisely introduced to ensure that there are no correlations at large space-like separation in models of quantum field theory. Within the Wightman axioms, we can prove that if the other axioms that define a quantum field are satisfied, then Cluster Decomposition is equivalent to the vacuum state being unique.

We can introduce models that do not have a unique vacuum state, and people have done so, but we then have to look to experiment to determine whether, in the vacuum state, there are the long-range correlations that a particular model predicts. In non-vacuum states, however, there are states such as Bell states in which there are non-local correlations at arbitrary space-like separation, so the simple discovery of long-range correlations may simply mean that we are not in the vacuum state; the observed long-range correlations must also be Lorentz and translation invariant for them to be considered a property of the vacuum.

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Thanks Peter. I should probably be a little more specific though. I recently read a paper that shows that, in principle, two subregions within a single vacuum state system at roughly milliKelvin temps there is a correlation such that a measurement of the electric field in one subregion causes a local squeezing of the fluctuations in another subregion. What I am asking is wheter or not this same result would occur if the measurement was made in one vaccum state system and a separate vacuum state system was observed for the squeezing of the fluctuation. I appreciate your time. – user10120 Jul 2 '12 at 3:41
So instead of two subregions within one vacuum state system, there are two separate vacuum state systems. A measurement made in one and the squeezing observed in the other. Thanks – user10120 Jul 2 '12 at 3:43
If it's at mK temperature, then it's not a vacuum state. It looks as if you're talking about quantum optics rather than quantum fields? I don't have quantum optics at my fingertips, so I think I may have to pass. Before I do, it looks as if you're modeling a quantum field state, approximately, using a finite-dimensional Hilbert space, for which you should talk of ground states, not vacuum states. The vacuum state is translation invariant, which you don't have. Anyway, you should include a reference to the paper you mention here in your question (arXiv and published, if possible). – Peter Morgan Jul 2 '12 at 14:05
The vacuum state has nonlocal correlations, two regions with vacuum are correlated. You can understand this intuitively as follows--- if you have a true vacuum, you know that there are no photons of any wavelength, including enormous wavelengths. But the condition that there are zero photons of long wavelength means that the state in a region smaller than one wavelength is entangled with other regions. The correlations only vanish asymptotically with large separations, and then only as a power. – Ron Maimon Jul 3 '12 at 2:29
Hey thanks for the post, Ron. I understand what you mean about the correlations vanishing asymptotically with increasing distance, but I don't understand what you mean by "and then only as a power". Could you please clarify. Thanks – user10120 Jul 3 '12 at 23:29

The paper you mentioned is following, right? http://pra.aps.org/abstract/PRA/v84/i3/e032336 http://xxx.yukawa.kyoto-u.ac.jp/PS_cache/arxiv/pdf/1109/1109.2203v1.pdf

In the paper, some comments are made on topics discussed in this thread. The Hall edge current system has an effective description by quantum chiral boson fields in 1+1 space-time dimensions. So the ground state of the system is called "the vacuum state", and this correspondence is certainly vaild in the context of effective field theory. The authors claim that thermal noise effect at mK temperature is too small to affect the observation of energy teleportation in the ground state.

The answer of your question seems trivial. Because quantum noises of two separate vacuums are independent each other, a noise measurement of one vacuum does not provide any information about noise of another vacuum. So we cannot make control of the separate noise fluctuaion to suppress its amplitude and extract a part of zero-point energy.

As opposed to the original setup in the paper, quantum energy teleportation will not be attained in your scheme.