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.

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

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 \(\beta\) tools

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

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,846 answers , 20,597 comments
1,470 users with positive rep
501 active unimported users
More ...

Trouble with Construction of Sphaleron S

+ 1 like - 0 dislike
47 views

Consider SU(2) YMH theory without fermions. Three-space is compactified by adding the sphere at infinity and configuration space is the space of all static finite-energy 3D gauge and Higgs fields in a particular gauge.

As we are looking for finite-energy solutions to the field equations the SU(2) gauge field must tend to a pure gauge and the Higgs field to its vacuum value. This means we can map the sphere at infinity $S^2_\infty$ into the Higgs vacuum manifold SU(2)~$S^3$.

In order to achieve nontrivial topology we consider a loop in configuration space, which in turn induces a loop in the space of the mappings defined above. By setting appropriate constraints we can go from the Cartesian to smash product and our domain space is now:

$$S^1\wedge S^2_\infty\sim S^3_\infty$$ and the map $$S^3_\infty\rightarrow S^3$$ where the target space is the Higgs vacuum manifold three-sphere, now leads to nontrivial topology.

My questions are:

1) If a "point" in configuration space corresponds to some gauge and Higgs field configuration (in a particular gauge), then a path in this space must be a series of continuously varying configurations with respect to some parameter. What is this parameter? Do two neighboring points along this path somehow correspond to a slightly varied field in physical space?

2) Similarly, a "path" in the space of mappings of $$S^2_\infty\rightarrow S^3\sim SU(2)$$ must mean a set of continuously varying maps from spatial infinity to the Higgs vacuum manifold. What exactly is parametrizing this path?

3) How does a loop in configuration space induce a loop in the space of the above maps?

4) Why does considering a loop in configuration space mean considering $$S^1\times S^2_\infty$$?


This post imported from StackExchange Physics at 2017-08-29 09:33 (UTC), posted by SE-user Optimus Prime

asked Aug 2 in Theoretical Physics by Optimus Prime (80 points) [ revision history ]
edited Aug 29 by Dilaton
Pleas see Chapter 11 in Manton and Sutcliffe Topological solitons for a detailed description books.google.co.il/books/about/… I'll try to write an answer in a few days

This post imported from StackExchange Physics at 2017-08-29 09:33 (UTC), posted by SE-user David Bar Moshe
@DavidBarMoshe: Thanks for the input. The issue I have with Manton's book is that given my current level of knowledge, it usually creates more questions than answers! The toy model he considers there, i.e sphalerons on a circle, is basically what I've been trying to visualize for the past few weeks. I may be wrong, but I think the heart of my problem lies in my inability to form a decent visualization of the configuration space of a field. I'm still eagerly looking forward to your response though.

This post imported from StackExchange Physics at 2017-08-29 09:33 (UTC), posted by SE-user Optimus Prime

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):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysic$\varnothing$Overflow
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).
To avoid this verification in future, please log in or register.




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

Your rights
...