Understanding Weinberg's Symmetries and Rays

[emma83]

Hello,

I am reading Weinberg's book and in the part on symmetries he speaks about rays, and says basically that 2 vectors $$U,V$$ which are on the same ray can only differ by a phase factor $$\phi$$, so that $$U=e^{i\phi}V$$.

Is "ray" meaning "direction" here ? Can I rephrase it and say that 2 colinear vectors can only differ by a phase factor ?

Thanks for your help!

 

[tiny-tim]

Is "ray" meaning "direction" here ? Can I rephrase it and say that 2 colinear vectors can only differ by a phase factor ?

Hi emma!

From pp. 49-50:

A ray is a set of normalised vectors with Ψ and Ψ' belonging to the same ray if Ψ' = ξΨ, where ξ is an arbitrary complex number with |ξ| = 1

So a ray is an equivalence class of normalised vectors in Hilbert space …

two normalised vectors "are" the same ray if they only differ by a phase factor.

(but i don't think thinking in terms of "directions" is helpful, when these things are more like functions )

 

[emma83]

Thank your very much!

Ok, now I think I also understand why he infers $$e^{i\phi}$$ as proportional factor (and not just e.g. a $$k \in \mathbb{C}$$) between the 2 vectors: because it is the most general complex proportional factor which is normalized to 1...