# uncertainty deviations for vacuum astronomy

Since i've done this question, i've been trying to improve and make more precise the statements regarding cosmic squeezed states and how different uncertainties affect the vacuum energy, but as it turns out, it's more difficult than it seems, as i hope to make clear in this question

Intuitively, i can play with some algebra of the uncertainties and write a "minimum" of energy in function of them:

$$H = E^2 + B^2$$

$$H + \Delta H = (E+ \Delta E)^2 + (B + \Delta B)^2$$

$$\langle \Delta H \rangle = 2 ( \langle E \rangle \langle \Delta E \rangle + \langle B \rangle \langle \Delta B \rangle ) + \langle \Delta E^2 \rangle + \langle \Delta B^2 \rangle$$

In the vacuum, $\langle E \rangle = \langle B \rangle = \langle \Delta E \rangle = \langle \Delta B \rangle = 0$ so we can simplify:

$$\langle \Delta H \rangle = \langle \Delta E^2 \rangle + \langle \Delta B^2 \rangle$$

Now, if we accept the uncertainty principle applicability to both the total fields and its modes ( a tall proposition as we'll see) and assume a minimum uncertainty mode:

$$\Delta E = \frac{ \hbar }{ \Delta B}$$

then we are left with

$$\langle \Delta H \rangle = \langle \Delta E^2 \rangle + \frac{ \hbar^2}{\langle \Delta E^2 \rangle}$$

which has a minimum when $\Delta E = \Delta B = \sqrt{ \hbar}$. This is the equation that defines "true" electromagnetic vacuum in terms of the uncertainties of the vacuum.

Now, all of this is cute and nice, but all this applies to a single mode of radiation. As it becomes clear from this other question of mine stuff gets really hairy when we consider a real quantum field.

In any case, i would hope that one can ask in experimental, absolutely concrete terms is: given one region of the cosmos, what deviation from this "true" vacuum, as defined above, can we expect? note that since the electromagnetic spectrum that is relevant for astrophysical phenomena is huge (from gamma ray to microwaves) there might be different deviations at different wavelengths

Now, my question is: are there concrete, existing expressions for the effective field uncertainties in QFT in terms of the uncertainties of the modes that can be used for the purposes of this "vacuum astronomy"? Can we in practice, think in individual EM modes and their uncertainties as "isolated" from the rest of modes?