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

185 submissions , 145 unreviewed
4,718 questions , 1,924 unanswered
5,264 answers , 22,441 comments
1,470 users with positive rep
743 active unimported users
More ...

  Supersymmetry non-breaking $\iff$ no "Goldstone fermion"?

+ 2 like - 0 dislike
55 views

In Supersymmetry and Morse Theory (1982) by E Witten,

Concern whether the supersymmetry is broken by checking whether $$ Q | 0 \rangle=0 $$ exists or not --- Witten said:

SUSY breaking: A solution may be shown not to exist by calculating a reliable, positive lower bound to the energy eigenvalues.

SUSY non-breaking: It may be shown that a solution does exist by showing that the theory has a mass gap so that there is no potential "Goldstone fermion".

my question is that why SUSY non-breaking has something to do with

  • no potential "Goldstone fermion"?
  • theory has a mass gap?

Are these if and only if conditions?

$$ \text{Supersymmetry non-breaking} \iff \text{mass gap and no "Goldstone fermion"?} $$

This post imported from StackExchange Physics at 2020-12-12 20:07 (UTC), posted by SE-user annie marie heart
asked Nov 3, 2020 in Theoretical Physics by annie marie heart (1,155 points) [ no revision ]
A "Goldstone fermion" is a massless fermion associated to a broken fermionic symmetry (in this case, supersymmetry). This is directly analogous to Goldstone bosons, which appear when we have a broken bosonic symmetry. If a theory has a mass gap, it means that any excitation above the ground state has mass, so there can be no Goldstone boson, so supersymmetry cannot be broken. This doesn't answer your question regarding "if and only if", but should at least illuminate the relationship of these ideas.

This post imported from StackExchange Physics at 2020-12-12 20:07 (UTC), posted by SE-user George Hulsey
Goldstone boson requires a continuous symmetry spontaneously broken. Does Goldstone fermion imply continuous SUSY? But boson and fermion Hilbert space are discrete?

This post imported from StackExchange Physics at 2020-12-12 20:07 (UTC), posted by SE-user annie marie heart
Supersymmetry is a continuous fermionic symmetry: it depends on an infinitesimal Grassmann parameter. For example, the complex scalar of the chiral multiplet transforms under infinitesimal SUSY as $\delta_\epsilon \phi = \sqrt{2}\epsilon \psi$. There is a continuous (but fermionic) parameter associated to the transformations. So the situation is exactly analogous to the bosonic case, except the continouous parameter is Grassmann valued rather than c-number valued.

This post imported from StackExchange Physics at 2020-12-12 20:07 (UTC), posted by SE-user George Hulsey
but SUSY quantum mechanics do not require a continuous SUSY parameters? One can just switch between bosonic ($-1^F=+1$) and fermionic ($-1^F=-1$) sector

This post imported from StackExchange Physics at 2020-12-12 20:07 (UTC), posted by SE-user annie marie heart

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$\hbar$ysicsOverf$\varnothing$ow
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
...