Thursday, February 28, 2019

Chemical potential raised states of the quantum vacuum state

We can construct an analog of a thermal state using a number operator instead of the Hamiltonian, which is different from thermal states because the number operator is Lorentz invariantly defined.
This derivation has always rather delighted me. If μ is very large, then this state is arbitrarily close to the vacuum state, then as μ becomes smaller, more and more Lorentz invariant Gaussian noise is added into the system. This "extra quantum noise" state is an infinite energy mixed state that is unitarily inequivalent to the vacuum state and to thermal states.

This post was prompted by my leaving a comment on Azimuth, which included this: "This state could be said to be in a mixed state because it’s interacting with another Lorentz invariant field (which has been traced out) as a zero temperature bath with which it exchanges particles in a Lorentz invariant way. For any algebra of observables that doesn’t include absolutely every field (including dark matter, dark energy, or anything else we haven’t found yet, …), the state over that subalgebra must be a mixed state if there’s any interaction. For the EM field in interaction with a Dirac spinor field, for example, the vacuum state for the EM field with the Dirac spinor field traced out should be a mixed deformation of the free field vacuum state, which is pure. Haag’s theorem, after all, insists that the interacting vacuum sector must not be unitarily equivalent to the free vacuum sector."

I first noticed this kind of structure about 15 years ago: with a much less pretty derivation it can be found discussed in my Phys.Lett. A338 (2005) 8, https://arxiv.org/abs/quant-ph/0411156, with the succinct title, "A succinct presentation of the quantized Klein–Gordon field, and a similar quantum presentation of the classical Klein–Gordon random field". That succinct presentation has been my slowly evolving companion ever since, with the derivation given above, which I can imagine being many, many impenetrable pages in a textbook QFT formalism, part of that evolution.

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