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

Please welcome our new moderators!

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

122 submissions , 103 unreviewed
3,497 questions , 1,172 unanswered
4,543 answers , 19,337 comments
1,470 users with positive rep
407 active unimported users
More ...

Mathematically: What is SUSY?

+ 2 like - 0 dislike
79 views

Wikipedia says:

In particle physics, supersymmetry (often abbreviated SUSY) is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners. In a theory with unbroken supersymmetry, for every type of boson there exists a corresponding type of fermion with the same mass and internal quantum numbers (other than spin), and vice-versa. There is only indirect evidence for the existence of supersymmetry [...]

I want a mathematical explanation of SUSY.

This post imported from StackExchange Physics at 2014-03-07 16:41 (UCT), posted by SE-user Neo
asked Feb 9, 2013 in Theoretical Physics by Neo (30 points) [ no revision ]

3 Answers

+ 6 like - 0 dislike

Mathematically, SUSY begins with the supersymmetry algebra, a Lie superalgebra, which is itself a special case of a more general class of algebras called graded Lie algebras. Of central importance is the supersymmetry algebra referred to as the super-Poincare algebra that extends the Poincare algebra to include supersymmetry "charges" and their anticommutators.

In the context of physics, one studies field theories, both classical and quantum, that exhibit invariance under some action of supersymmetry algebras on fields and Hilbert spaces of these theories. As a result, representations of supersymmetry algebras are especially important in physics.

I would highly recommend that you look at THIS set of notes written by Sohnius, one of the original supersymmetry masters and co-discoverers of THIS famous and important theorem which really motivates why supersymemtry is all the rage in physics. The notes talk about representations of supersymmetry algebras in a lot of detail, and the clarity of the prose is top-notch if you ask me.

Addendum. I almost forgot, you also hear the word "superspace" which is a construction that physicists use to, among other things, make constructing manifestly supersymmetric Lagrangians easier. The mathematics behind this is supermanifolds.

Lastly, there is some discussion of these things on math.SE, see for example

http://math.stackexchange.com/questions/1204/why-are-superalgebras-so-important http://math.stackexchange.com/questions/51274/motivation-for-supermanifolds

Hope that helps!

Cheers!

This post imported from StackExchange Physics at 2014-03-07 16:41 (UCT), posted by SE-user joshphysics
answered Feb 9, 2013 by joshphysics (810 points) [ no revision ]
By the way, superspace in the form of super-Minkowski space is right there in the super-Poincare algebra, being the quotient of that by the Lorentz sub-algebra (ncatlab.org/nlab/show/super+Poincare+Lie+algebra).

This post imported from StackExchange Physics at 2014-03-07 16:41 (UCT), posted by SE-user Urs Schreiber
@UrsSchreiber Interesting stuff. Thanks for the link!

This post imported from StackExchange Physics at 2014-03-07 16:41 (UCT), posted by SE-user joshphysics
+ 2 like - 0 dislike

In addition to what Joshua said in his nice answer, may favorite (simplified) point of view is looking at a SUSY transformation as a coordinate transformation (translation) in superspace

$$ x' = x + a + \frac{i}{2}\zeta\sigma^{\mu}\bar{\theta} - \frac{i}{2}\theta\sigma^{\mu}\bar{\zeta}$$

$$ \theta'= \theta + \zeta $$

$$ \bar{\theta}'= \bar{\theta} + \bar{\zeta} $$

with $\theta$ and $\bar{\theta}$ denoting the additional Grassmanian coordinates.

The supersymmetry generators or supercharges, when written down as differential operators, contain momentum operators in both, the "usual" even spacetime coordinates and the odd Grassmann coordinates

$$ Q_a = i\partial_a -\frac{1}{2}(\sigma^{\mu})_{a\dot{b}}\bar{\theta}^{\dot{b}}\partial_{\mu}$$

$$ \bar{Q}^{\dot{a}} = i\bar{\partial}^{\dot{a}} -\frac{1}{2}(\bar{\sigma}^{\mu})^{\dot{a}b}\theta_b\partial_{\mu}$$

Where $\partial_a = \frac{\partial}{\partial\theta^a}$ and $\bar{\partial}^{\dot{a}} = \frac{\partial}{\partial\bar{\theta_{\dot{a}}}}$ are the derivatives along the Grassmanian coordinates.

A nice and very readable introduction to the superspace formalism can for example be found in Ch 11 of this book.

This post imported from StackExchange Physics at 2014-03-07 16:41 (UCT), posted by SE-user Dilaton
answered Feb 13, 2013 by Dilaton (4,175 points) [ no revision ]
+ 0 like - 0 dislike

c.f.

There are two types of ; worldsheet supersymmetry, and spacetime supersymmetry

Worldsheet supersymmetry

The Ramond-Neveu-Schwarz formalism has explicit worldsheet supersymmetry. Since the RNS Action is given by adding the Polyakov Action to the Dirac action, it is given by:

$${{\mathsf{\mathcal{L}}}_ {RNS}}=\frac{T}{2} h^{\alpha \beta} \left( \partial_\alpha X^\mu \partial_\beta X^\nu +i\hbar c_0 \bar{\psi_\mu} \not{\partial} \psi^\mu \right) g_{\mu\nu}$$

The supersymmetric transformations on the worldsheet can therefore be (almost trivially, by taking variations of this above action) shown to be:

$$\begin{gathered} \delta {X^\mu } \to \bar \epsilon {\psi ^\mu } ; \\ \delta {\psi ^\mu } \to - i \not \partial {X^\mu }\epsilon \\ \end{gathered} $$

Spacetime Supersymmetry

The Green-Schwarz formalism, or the , are with explicit spacetime supersymmetry. The supersymmetric transformations on spacetime are (which is rather intuitive if you compare this to the RNS Worldsheet supersymmetry transformations) given by:

$$\begin{gathered} \delta {\Theta ^{Aa}} \leftrightarrow {\varepsilon ^{Aa}} ; \\ \delta {X^\mu } \leftrightarrow {{\bar \varepsilon }^A}{\gamma ^\mu }{\Theta ^A} ; \\ \end{gathered} $$

answered Sep 18, 2013 by dimension10 (1,950 points) [ revision history ]
edited Jan 31, 2015 by dimension10

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$\hbar$ys$\varnothing$csOverflow
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
...