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

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?

• Where did you get the idea that "quantum state vectors transform according to different observers"? The state vector is what it is, regardless of who is observing it.
– Buzz
Nov 27, 2020 at 3:03
• @Buzz It's stated in many QFT books. Indeed different observers see equivalent states, but not the same states. For instance, in Weinberg's QFT book volume 1 the transformation rule of states under Lorentz transformation is given.
– DEDS
Nov 27, 2020 at 3:18
• Please edit the question title to be more descriptive of what you are asking. Nov 27, 2020 at 3:22

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.