Solving Exercise I.2 on Supersymmetry Algebras

• AlphaNumeric
In summary, the author is trying to figure out how to introduce the gauge fields in a supersymmetrical theory, but is getting frustrated because he can not seem to find the information in any of the books he has read.
AlphaNumeric
Not sure if this is something you'd put in the homework help area or not, but I've started learning supersymmetry and seem to have hit a bump within the first few pages.

$$\delta_{\lambda}\varphi = \frac{1}{2}\lambda^{\mu\nu}M_{\mu\nu}.\varphi$$

for some field $$\varphi$$, which is one of $$S$$, $$P$$ or $$\psi$$ and $$\lambda^{\mu\nu}=-\lambda^{\nu\mu}$$

$$M_{\mu\nu}.\psi = -(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})\psi - \Sigma_{\mu\nu}\psi$$

Where $$\Sigma_{\mu\nu} = \frac{1}{2}\gamma_{\mu\nu}$$, the Dirac matrix thing. I need to show that $$\delta_{\lambda}(\bar{\psi}\gamma^{\rho} \partial_{\rho}\psi) = \partial_{\mu}(\lambda^{\mu\nu}x_{\nu}\bar{\psi}\gamma^{\rho} \partial_{\rho}\psi)$$

The notes issue a warning that the algebra of operators, of which $$M_{\mu\nu}$$ is a part, only asks on fields, so $$M_{\mu\nu}.(x^{\rho}\varphi) = x^{\rho}M_{\mu\nu}.\varphi$$ and $$M_{\mu\nu}.(\partial^{\rho}\varphi) = \partial^{\rho}M_{\mu\nu}.\varphi$$

My problem is that I can't get the $$\psi$$ field to transform in that nice way. The $$\Sigma_{\mu\nu}$$ term screws it up and I end up with something I can't write as a total derivative!

If I've made zero sense here, I'm trying to do Exercise I.2 here.

Thanks for any help :)

I get the desired result. Just pay attention when you commute the gamma^{rho} with the spin matrix.

Daniel.

That's what I was trying to do but I must be doign some stupid slip somewhere. Ignoring all the other parts of the question, I get it down to showing that

$$\frac{1}{4}\lambda^{\mu\nu}\Sigma_{\mu\nu}\bar{\psi}\gamma^{\rho}\partial_{\rho}\psi + \frac{1}{4}\lambda^{\mu\nu}\bar{\psi} \gamma^{\rho}\partial_{\rho} ( \Sigma_{\mu\nu}\psi) = 0$$

Commuting the Sigma and gamma spin matrices using the Dirac algebra turns this into

$$\frac{1}{2}\lambda^{\mu\nu}\Sigma_{\mu\nu}\bar{\psi}\gamma^{\rho}\partial_{\rho}\psi + \frac{1}{2}\lambda^{\mu\nu}\delta_{\mu}^{\rho}\delta_{\nu}^{\sigma}\bar{\psi}\gamma_{\rho}\partial_{\sigma}\psi$$

I don't see how that all turns to zero (since I've collected all the other terms into the required result). Obviously I'm doing which is pretty stupid and knowing me it's probably right infront of my face so feel free to make me look stupid by pointing it out because it begins to get to me, thanks :)

I've now realized the total pig's breakfast I was making of the above question and how much simpler it actually is. Just needed to think about it.

Unfortunately, now I'm stuck on the conformal superalgebra and this time it doesn't even involve matrices.

$$D.\varphi = -x^{\nu}\partial_{\nu}\varphi - \varphi$$

$$P_{\mu}\varphi = -\partial_{\mu}\varphi$$

The algebra has the result $$[P_{\mu},D]\varphi = P_{\mu}\varphi$$ but I get the negative answer and the same happens with other commutators in the algebra, I get the right terms but signs wrong in places.

$$[P_{\mu},D]\varphi = P_{\mu}(D\varphi) - D(P_{\mu}\varphi)$$

$$= -\partial_{\mu}(-x^{\nu}\partial_{\nu}\varphi - \varphi) + D(\partial_{\mu}\varphi)$$

$$= x^{\nu}\partial_{\mu}\partial_{\nu}\varphi + \partial_{\mu}\varphi + \partial_{\mu}\varphi - x^{\nu}\partial_{\nu}\partial_{\mu}\varphi - \partial_{\mu}\varphi$$

$$= \partial_{\mu}\varphi = -P_{\mu}\varphi$$

It's the wrong sign but I can't see where I've made the mistake?!

Last edited:
It could be a mistake in the book.

Daniel.

It's both in those BUSSTEPP lecture notes I linked to before and it's in 'Introduction to Supersymmetry and Supergravity' by West.

West defines $$P_{\mu} = \partial_{\mu}$$ but that should still give the same conformal algebra (-1 factor cancels) just a slightly different representation.

Via similar, simple, computation I also get $$[P_{\mu},K_{\nu}] = -2\eta_{\mu\nu}D - 2M_{\mu\nu}$$ instead of $$[P_{\mu},K_{\nu}] = 2\eta_{\mu\nu}D - 2M_{\mu\nu}$$.

If I just got a totally different answer, I'd know I'm doing something fundamentally wrong, but the fact it's just a sign error here and there and all I'm doing it taking derivatices of fields and $$x^{\mu}$$ it's not complicated algebra. It's more frustrating that just getting it totally wrong!

This would be the week the entire SUSY group in my department go away for a conference!

It could be not polite to ask for help in this thread considering I have not got to help to the original poster, but in some sense it is a continuation. What I am banging my head against is at the observation that the supercharges are in some sense square roots of the momentum,
$$\{Q, \bar Q\} = ... P_\mu$$
Because then, if one thinks that minimal coupling amounts to replace the momentum operator by the minimally coupled
$$(P_\mu) \to (P_\mu - i e A_\mu - ... )$$
then it seems painfully obvious that the way to introduce the gauge fields in a supersymmetrical theory is to find a set of supercharges $$Q^N$$ reproducing the new minimally coupled operator. It makes sense that in this way we are really bypassing Coleman-Mandula.

I would hope this to be stated in the first chapters of any supersymmetry book. I can not find it. I am wrong, or it is there and just they speak different language?

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What is Exercise I.2 on Supersymmetry Algebras?

Exercise I.2 on Supersymmetry Algebras is a problem set focused on solving equations related to supersymmetry algebras, which are mathematical structures that describe the symmetries of certain physical systems.

Why is it important to solve Exercise I.2 on Supersymmetry Algebras?

Solving Exercise I.2 on Supersymmetry Algebras is important for understanding the mathematical foundations of supersymmetry and its applications in physics. It also helps to develop problem-solving skills and deepen understanding of algebraic structures.

What are some strategies for solving Exercise I.2 on Supersymmetry Algebras?

Some strategies for solving Exercise I.2 on Supersymmetry Algebras include breaking down the problem into smaller, more manageable parts, using symmetry properties to simplify equations, and making use of known identities and relationships between different algebraic structures.

What background knowledge is necessary to solve Exercise I.2 on Supersymmetry Algebras?

To solve Exercise I.2 on Supersymmetry Algebras, it is important to have a strong understanding of algebra, group theory, and supersymmetry. Some knowledge of physics and quantum mechanics may also be helpful.

Are there any resources available to help with solving Exercise I.2 on Supersymmetry Algebras?

Yes, there are many resources available to help with solving Exercise I.2 on Supersymmetry Algebras, including textbooks, online tutorials, and study groups. It may also be helpful to consult with a professor or mentor who has experience with supersymmetry and algebraic structures.

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