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It is often said that an inertial observer in flat spacetime vacuum will see an accelerating observer thermalize (Unruh Effect). If an accelerated observer takes a particle in a box coupled with the quantum field, there will be a nonzero probability that the particle will be in an excited state, which looks like an emission of a particle from the accelerating frame.

I have 2 main questions.

  1. If there’s a particle in a box, then by definition there is no longer a vacuum state measured by the inertial observer, so how can this even explain the Unruh effect?

  2. How does the particle detector work? I have read that the coupling is described by the interacting Hamiltonian $H$ = $\lambda\epsilon\mu\phi$ where $\lambda$ is the scalar coupling. How does this lead to the Unruh temperature?

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  • $\begingroup$ About 1 the point is that the quantum field is in its vacuum (ground) state. The detector is another system to which we couple the first, so the particle used there is not a quantum of the field you are studying. $\endgroup$
    – Gold
    Commented Jan 9, 2023 at 19:47

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It is often said that an inertial observer in flat spacetime vacuum will see an accelerating observer thermalize (Unruh Effect).

This seems a bit misleading. The Unruh effect is the statement that while the inertial observer sees no particles at all and measures zero temperature, an accelerated observer sees particles and measures a temperature proportional to their acceleration. The accelerated observer perceives temperature, the inertial observer does not.

If there’s a particle in a box, then by definition there is no longer a vacuum state measured by the inertial observer, so how can this even explain the Unruh effect?

A "particle in a box" shouldn't really be taken literally. This approach to the Unruh effect uses a particle detector, which is a quantum system chosen in a manner that allows one to "detect particles". One way of thinking about this detector is a particle in a box that interacts with the Unruh bath, but this is more of an intuitive picture. Literally, what one usually does is to consider a two-level system coupled to the quantum field. The total system is then described in the tensor product of the two relevant Hilbert spaces (the two-level system and the quantum field). Hence, the quantum field can be in its vacuum state, regardless of the state of the detector.

In the particle in a box illustration, you can think that the particle in a box is of a different species than the one under consideration for the Unruh effect. For example, you might have an electron in a box to serve as a detector of Unruh photons. Hence, the photon field can be in the ground state, even though there is an electron in the box.

How does the particle detector work? I have read that the coupling is described by the interacting Hamiltonian $H = \lambda \epsilon \mu \phi$ where $\lambda$ is the scalar coupling. How does this lead to the Unruh temperature?

Using the particle detector model, one can compute the probability of a transition happening in the detector. In other words, you can compute the probability that a detector prepared in the ground state will be measured in the excited state and vice-versa. One can then assert that these probabilities obey the detailed balance at the Unruh temperature, meaning the detector is under the influence of a thermal bath. This bath is due to the interaction with the field and we can then interpret the result as meaning that the observer is measuring a non-vanishing temperature. In other words, the observer simply uses the properties of the particle detector to build a thermometer and measure the Unruh temperature.

Some references

The book by Wald discusses the notion of particle detectors in Section 3.3. I'm sure there are many references that compute the transition probabilities for the detector, but the one I know by heart is arXiv: 2012.14912 [hep-th], which discusses the general idea of particle detectors and implements the Unruh–DeWitt detector in terms of a path integral. If I recall correctly, Unruh's original paper does the calculation using particle detectors, but his model had infinite levels, and hence might be a bit more complicated to work with (nowadays, it is more common to work with two-level systems).

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