Homework Help: Supersymmetry Question

I haven't done any supersymmetry before and am probably going to need it in the future (hopefully anyway).

Anyway, I've been reading through these online notes:

http://www-thphys.physics.ox.ac.uk/people/JosephConlon/LectureNotes/SUSYLectures.pdf

I have a problem with p7:

(i) What is the symbol in the line "However the map is not an isomorphism as N=.... both map to (symbol i don't know) [tex] \in SO(1,3)[/tex]

(ii) I am completely lost trying to do the first exercise. Where should I start?

Thanks for the help.

 

Thanks for both your replies.

Why is X in Spin(3,1) to start with? I think I've confused myself with what Spin(3,1) actually is?

and then the matrix in (16) is 2x2 and has complex entries suggesting it is in GL(2,C)
Why would it be in SL(2,C)? For this to be the case, we would need the det to be 1.
It tells us below that det [tex]=x_0^2-x_1^2-x_2^2-x_3^2[/tex]. Is this necessarily equal to 1? If so, why?

As for the overall proof, are you saying that I should evaluate

[tex]\tilde{X} =N \Lambda^\mu_\nu X_\mu \sigma^\nu N^\dagger[/tex]

or is this completely off base? Even if it's right I'm not really sure what I'm doing - am I checking to see that if we transform X, [tex]\tilde{X}[/tex] transforms in the way described in (18)?

Thanks very much.

 

Thanks for the reply. What is the definition of Spin(3,1)

So this means that [tex]\Lambda \in Spin(3,1) [/tex] right?

And I can't see how proof is going to work. 16 and 18 tell us what [tex]\tilde{X}[/tex] is and how it transforms. How can we use this to establish [tex]\Lambda^\mu{}_\nu[/tex]

Why do [itex] tags not work anymore?

Cheers.

 

I get [tex]\tilde{X}' = N \tilde{X} N^\dagger = N X_\mu \sigma^\mu N^\dagger[/tex]

But [tex]X'_\mu=\Lambda^\nu{}_\mu X_\nu[/tex] and I can't find an X' term in the line above that would allow me to read off what [tex]\Lambda^\nu{}_\mu[/tex] should be.

 

I understand where you got the other expression for [itex] \tilde{X}'[/itex] from so that's fine.

A couple of things:

(i) What was wrong with my indices?

(ii) When we equate our two expressions we get:

[itex]\Lambda^\mu{}_\nu X^\nu \sigma_\mu = N X_\rho \sigma^\rho N^\dagger[/itex]

If we right multiply by [itex]\bar{\sigma}^\kappa[/itex] we get

[itex]\Lambda^\mu{}_\nu X^\nu \delta^\kappa{}_\mu = N X_\rho \sigma^\rho N^\dagger \bar{\sigma}^\kappa[/itex]

which simplifies to

[itex]\Lambda^\mu{}_\nu X^\nu = N X_\rho \sigma^\rho N^\dagger \bar{\sigma}^\mu[/itex]

Because [itex]X \in SL(2, \mathbb{C})[/itex], [itex]\text{det } X = X^\alpha X_\alpha = 1[/itex]

Therefore, right multiplication by [itex]X_\alpha[/itex] gives

[itex]\Lambda^\mu{}_\nu \delta^\nu{}_\alpha = N X_\rho \sigma^\rho N^\dagger \bar{\sigma}^\mu X_\alpha[/itex]

This of course simplifies to

[itex]\Lambda^\mu{}_\nu = N X_\rho \sigma^\rho N^\dagger \bar{\sigma}^\mu X_\nu[/itex]

Is this on the right lines? How do I get the Pauli matrices next to each other so that I can use an identity for them?

 

[itex] \Lambda^\mu{}_\nu X^\nu \sigma_\mu = N X_\rho \sigma^\rho N^\dagger[/itex]
[itex] \Lambda^\mu{}_\nu X^\nu \sigma_\mu \bar{\sigma}^\kappa = N X_\rho \sigma^\rho N^\dagger \bar{\sigma}^\kappa[/itex]
[itex] \Lambda_{\mu \nu} X^\nu \sigma^\mu \bar{\sigma}^\kappa = N X_\rho \sigma^\rho N^\dagger \bar{\sigma}^\kappa[/itex]
[itex] \text{Tr } \left( \Lambda_{\mu \nu} X^\nu \sigma^\mu \bar{\sigma}^\kappa \right) = \text{Tr } \left( N X_\rho \sigma^\rho N^\dagger \bar{\sigma}^\kappa \right)[/itex]
[itex] 2 \Lambda_{\mu \nu} X^\nu \eta^{\mu \kappa} = \text{Tr } \left( N X_\rho \sigma^\rho N^\dagger \bar{\sigma}^\kappa \right)[/itex]
[itex] \Lambda^\mu{}_\nu X^\nu = \frac{1}{2} \text{Tr } \left( N X_\rho \sigma^\rho N^\dagger \bar{\sigma}^\mu \right)[/itex]

How do I move the [itex] \bar{\sigma}^\mu[/itex] to the front on the RHS though?

Cheers.