Unruh effect: should Minkowski vacuum transform according to different observers?

Question

It's known that the Minkowski vacuum is observed as a thermal bath for Rindler observers, in paticular:

$\langle0_{M}|N_{M}|0_{M}\rangle=0 \space\space\space\space\space\space\space $ (1)

$\langle0_{M}|N_{R}|0_{M}\rangle \neq 0\space\space\space\space\space\space\space\space$ (2).

However, from eq(1) to eq(2) we change from inertial observers to Rindler observers, so why the $|0_{M}\rangle$ remains the same in these two equations, since we know that quantum state vectors transform according to different observers?

Answer 1

The fact that in your two equations there's $|0_M \rangle$ comes from what you want to measure. The first one is trivial to interpret: It is the number of particles in the Minkowski vacuum for the Minkowski observer. But the second one is a bit tricky: It is the number of particles in the Minkowski vacuum from the point of view of the Rindler observer. You could have for example $\langle 0_R | N_R | 0_R \rangle$ but it is identically null. The Unruh effect is precisely the measure of the number of particles in the Minkowski vacuum from the point of view of the Rindler observer.