# A question about boosts

1. Jul 29, 2015

### naima

I am reading this Jacobson's paper
he writes:
According to the Unruh effect the Minkowski vacuum state of quantum fields—or any state at
very short distances— is a thermal state with respect to the boost hamiltonian at temperature
T.
I know that the accelerated observer is in a thermal environment, but i do not see what is this "boost hamiltonian". Can you write it?
Thanks.

2. Jul 29, 2015

### Finny

This paper may offer insights:

http://arxiv.org/pdf/1205.5325v1.pdf
Horizon energy as the boost boundary term in general relativity and loop gravity
Try the last several paragraphs related to eq (9).

A Google search for 'boost Hamiltonian' brings up lots of hits.

3. Jul 29, 2015

### fzero

Refer to the discussion of Rindler space at http://www.scholarpedia.org/article/Unruh_effect, specifically the coordinate transformation a few lines below eq. (1) which relates $(t,z)$ of Minkowski space to the $(\tau,\xi)$ of Rindler space. The boost Killing vector in the $z$ direction of Minkowski space is
$$z \frac{\partial}{\partial t} + t \frac{\partial}{\partial z} = \frac{\partial}{\partial \tau}.~~~(*)$$
Examining the mode expansion eq. (2), we see that the modes are eigenfunctions of the boosts with eigenvalues $i \Omega_k$. In the thermal expansion, we Wick rotate $\tau$ to $i \beta$, with $\beta$ the inverse temperature, so the eigenvalues become real and positive. Then the thermal distribution (5) is recognized as what we would have computed from a theory where we used the boost (*) as the Hamiltonian.

Last edited: Jul 29, 2015
4. Aug 3, 2015

### naima

Thanks
I found a good paper about the subject.
It describes the semiclassical QFT on curved spacetime. Here $\phi$ is a solution of the motion equation.
Its modes are "planes waves" a) in Minkowski b) in the wedges of accelerated observer.
The Bogoliubov machinery gives an elegant solution to the unruh effect.
It uses no explicit Wick rotation. Is there an analogous theory of statistical mecanics in curved space(time)?
What is the status in this theory or $\phi$ and its modes?

5. Aug 3, 2015

### fzero

The analysis there is completely standard and equivalent to that in the link I provided.

The interpretation of (2.37) as a thermal spectrum does not need an explicit calculation. Your original question asked how does one obtain the Planck spectrum from the "boost Hamiltonian", which is what I answered. According to the methodology of thermal field theory, one Wick rotates to imaginary time. One is left with QFT on a spacetime with Euclidean signature and the time variable is made periodic with period equal to the inverse temperature. One can do this in curved spacetime as well.

The idea is that the Rindler observer would be carrying a particle detector that could detect $\phi$ particles.