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For a while now I have heard that the explanation behind the non-zero energy of vacuum space is due to quantum fluctuations. Sometimes this phenomenon has been described with virtual particles, where a particle & antiparticle pair appear and quickly annihilate each other (though this explanation on its own seems to not apply when the particle in question is its own antiparticle, e.g. photon).

So the question is whether it really is all fields that fluctuate or only specific ones. Not sure if there is any reason for why it could not be all. Nevertheless, I am curious to know the specifics of this.

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It is a trivial consequence of being an energy eigenstate that the ground state energy of a quantum system cannot fluctuate (the variance must be zero).

The so-called quantum fluctuations are a by-product of perturbation theory. Since we do not know the exact ground state of a system of interacting quantum fields, we have to use perturbation theory to obtain such ground state in terms of excited states of the free fields.

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As Giorgio says, the quantum fields do not fluctuate. There is a fluctuation, but it's a fluctuation in our measurements not in the fields. For more on this see my answer to Are vacuum fluctuations really happening all the time?

The discussion in that answer applies to all the quantum fields, so measurements made on all the quantum fields will fluctuate in the same way.

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