Understanding Weinberg's Symmetries and Rays

In summary, Weinberg's book discusses symmetries and defines a ray as a set of normalized vectors that only differ by a phase factor. This means that two vectors on the same ray can be expressed as U=e^{i\phi}V. The concept of "direction" is not helpful in understanding this, as these vectors are more like functions. Additionally, the author infers e^{i\phi} as the proportional factor between the two vectors because it is the most general complex proportional factor that is normalized to 1.
  • #1
emma83
33
0
Hello,

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

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!
 
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  • #2
emma83 said:
Is "ray" meaning "direction" here ? Can I rephrase it and say that 2 colinear vectors can only differ by a phase factor ?

Hi emma! :smile:

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. :smile:

(but i don't think thinking in terms of "directions" is helpful, when these things are more like functions :wink:)
 
  • #3
Thank your very much!

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

Related to Understanding Weinberg's Symmetries and Rays

1. What are Weinberg's symmetries and rays?

Weinberg's symmetries and rays refer to the mathematical principles and concepts used to describe the behavior of subatomic particles and their interactions in the Standard Model of particle physics.

2. Why is understanding Weinberg's symmetries and rays important?

Understanding Weinberg's symmetries and rays is important because they provide the framework for understanding the fundamental forces and particles that make up our universe. This knowledge can also lead to new insights and discoveries in the field of particle physics.

3. How do Weinberg's symmetries and rays relate to the Standard Model?

Weinberg's symmetries and rays are a key component of the Standard Model, which is the most widely accepted theory for describing the fundamental particles and their interactions. The symmetries and rays provide the mathematical structure for the Standard Model and help to explain its various predictions.

4. What are the main symmetries and rays described by Weinberg's theory?

The main symmetries and rays described by Weinberg's theory include gauge symmetry, Lorentz symmetry, and chiral symmetry. These symmetries describe the fundamental forces and particles in the Standard Model and how they interact with each other.

5. How has the understanding of Weinberg's symmetries and rays evolved over time?

Weinberg's symmetries and rays have been continuously refined and expanded upon since their initial proposal in the 1960s. The development of new experimental techniques and theoretical advancements have led to a deeper understanding of these concepts and their role in the Standard Model.

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