Are quantum state vectors dependent on coordinate systems or observers?

Question

I'm confused about the two following points of view:

1. In Weinberg's QFT book 1 he claims "if an observer O sees a system represented by a ray R..., then an equivalent observer O' who looks at the same system will observe it in a different state, represented by a ray R'...". So it seems that state vectors depend on the observers.

2. We have the Thermofield Double (TFD) $|\Omega\rangle=\frac{1}{\sqrt{Z}}\sum\limits_{i}e^{-\pi \omega_{i}}|i^*\rangle_{L}|i\rangle_{R}$ (see e.g. eq.(3.22) or (3.30) in arXiv:1409.1231v4). The two sides of this equation, however, correspond to different observers. So it seems that state vectors do not depend on the observers.

[Perhaps a clearer way to see point 2 is by this formula:

$\langle0_{M}|N_{R}|0_{M}\rangle=\cdots$

which, in usual discussion of the Unruh effect, describes the expectation value of the particle numbers with respect to Rindler observers. Here the number operator is defined by Rindler observers while the state is the same vacuum state seen by inertial observers.]

I'm wondering how to reconcile these two points. Any comments will help.

Answer 1

The general concept here is covariance: how a description changes, when the position by which it is observed changes.

Let’s say the both of us are in a room. I am standing by the south wall and you are standing by the east wall.

Now I paint an arrow in the centre of the room pointing from south to north and then go back to stand by the south wall.

Now, from where I am, I see the arrow pointing straight ahead of me. Whilst you will see the same arrow point across you going from left to right.

These are obviously different descriptions of the same arrow. However, there is a relationship between these two descriptions, the way that they are related is called covariance. It’s because of this, that given only the two descriptions and knowing where we both are standing, that we can say whether the two descriptions actually describe the same thing - here, the same arrow.