Weinberg QFT p.65-66: Orthonormality of States with Momentum k

[Rick89]

Homework Statement

He says "choose the states with standard momentum k to be orthonormal", does he mean for example state with momentum (0,0,0,M1) and with momentum (0,0,0,M2) to be orthogonal or states with the same k^2 to be orthogonal if they have different k?
Also where he calculates in the next page, the "scalar product for [states of] arbitrary momenta" why is k' defined there the "same" as that in formula (2.5.12)? I don't get it...

Homework Equations

in the book...

The Attempt at a Solution

if the states of one given k^2 are orthogonal among each other then I don't get why k' as defined in page 66 could have the same square as k (that comes from p)... otherwise I don't get why we expect k' to be of the form of a "standard momentum" according to the way it is defined

 

[mark726]

Based on the information provided, it seems like the person is asking for clarification on the concept of orthonormal states with standard momentum k. They are also confused about the use of k' in the calculation of the scalar product for states with arbitrary momenta.

my response would be:

Thank you for your question. When someone mentions "orthonormal states with standard momentum k," they are referring to states with the same momentum k being orthogonal to each other. In other words, states with different k values should not be orthogonal to each other. This is because the momentum k is a defining characteristic of a state, and states with different k values represent different physical states.

As for the use of k' in the calculation of the scalar product, it is defined as the same momentum as k because in the context of the calculation, we are looking at the scalar product between two states with the same momentum k. Therefore, it makes sense to use k' to represent the momentum of the second state.

I hope this helps clarify your confusion. If you have any further questions, please don't hesitate to ask.