Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.
W3Counter Web Stats

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public β tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

208 submissions , 166 unreviewed
5,138 questions , 2,258 unanswered
5,413 answers , 23,081 comments
1,470 users with positive rep
822 active unimported users
More ...

  How to calculate the mass spectrum in Lagrangian with SO(3) and SU(2)?

+ 2 like - 0 dislike
2115 views

Consider a Lagrangian:

L=12μϕiμϕiV(ϕ);V(ϕ)=12μ2ϕiϕi+14λϕiϕiϕjϕj.

I understand that the mass spectrum of the particles in the theory can be obtained by:

m2ij=2Vϕiϕj|ϕ=ϕ0.

The minimum of the potential is given by:

Vϕk=μ2ϕk+λϕkϕjϕj=0ϕk=0 or ϕjϕj=μ2λ

Now to obtain the mass spectrum:

2Vϕkϕl=ϕl(μ2ϕk+λϕkϕjϕj)=μ2δkl+λϕjϕjδkl+2λϕkϕl

Substituting the minimum obtained above:

2Vϕkϕl|ϕ=ϕ0=μ2δkl+λδklμ2λ+2λμ2λ=2μ2.

I was expecting to obtain a matrix with the masses of the particles, but I just have one mass. What did I do wrong?

asked Sep 1, 2018 in Theoretical Physics by NewTheorist [ no revision ]

1 Answer

+ 2 like - 0 dislike

The following expression is correct:
 

2Vϕkϕl=ϕl(μ2ϕk+λϕkϕjϕj)=μ2δkl+λϕjϕjδkl+2λϕkϕl

but the following expression is incorrect:

2Vϕkϕl|ϕ=ϕ0=μ2δkl+λδklμ2λ+2λμ2λ=2μ2.

The correct expression is

2Vϕkϕl|ϕ=ϕ0=μ2δkl+λδklμ2λ+2λϕkϕl=2λϕkϕl.

Now you have the correct matrix.  Do you agree?

answered Sep 2, 2018 by juancho (1,130 points) [ no revision ]

well corrected :)

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol in the following word:
pysicsverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...