Tom Mattson

Staff Emeritus

Science Advisor

Gold Member

- 5,474

- 20

## Main Question or Discussion Point

The title says it all. I'd like to have an ongoing discussion about Witten's SUSY QM for N=2. At the moment I'm going through a book called

The first, and most basic, is: Can anyone out there recommend another good book, but one that has

Here's the next question. Junker cites Witten's definition of the supercharges thusly.

[tex]Q_1:=\frac{1}{\sqrt{2}}\left(\frac{p}{\sqrt{2m}}\otimes\sigma_1+\Phi(x)\otimes\sigma_2\right)[/tex]

[tex]Q_2:=\frac{1}{\sqrt{2}}\left(\frac{p}{\sqrt{2m}}\otimes\sigma_2-\Phi(x)\otimes\sigma_1\right)[/tex]

He then goes on to say that you can get the N=2 Hamiltonian by [itex]H=2Q_1^2=2Q_2^2[/itex]. Now when I go ahead and compute [itex]2Q_1^2[/itex], I get the following.

[tex]H=\frac{p^2}{2m}\otimes\sigma_1^2+\Phi^2(x)\otimes\sigma_2^2+\frac{1}{2m}\{p,\Phi(x)\}\otimes\{\sigma_1,\sigma_2\}[/tex]

This is wrong. Since [itex]\sigma_1^2=\sigma_2^2=1[/itex], first two terms are OK. But the anticommutators in the third term should in fact be commutators. Why doesn't the normal distributive law work here?

*Supersymmetric Methods in Quantum and Statistical Physics*by Georg Junker, and I've got some issues that I'd like to discuss.The first, and most basic, is: Can anyone out there recommend another good book, but one that has

*exercises*in it? This book is meant to be read by grad students, but it has no problems to cut your teeth on.Here's the next question. Junker cites Witten's definition of the supercharges thusly.

[tex]Q_1:=\frac{1}{\sqrt{2}}\left(\frac{p}{\sqrt{2m}}\otimes\sigma_1+\Phi(x)\otimes\sigma_2\right)[/tex]

[tex]Q_2:=\frac{1}{\sqrt{2}}\left(\frac{p}{\sqrt{2m}}\otimes\sigma_2-\Phi(x)\otimes\sigma_1\right)[/tex]

He then goes on to say that you can get the N=2 Hamiltonian by [itex]H=2Q_1^2=2Q_2^2[/itex]. Now when I go ahead and compute [itex]2Q_1^2[/itex], I get the following.

[tex]H=\frac{p^2}{2m}\otimes\sigma_1^2+\Phi^2(x)\otimes\sigma_2^2+\frac{1}{2m}\{p,\Phi(x)\}\otimes\{\sigma_1,\sigma_2\}[/tex]

This is wrong. Since [itex]\sigma_1^2=\sigma_2^2=1[/itex], first two terms are OK. But the anticommutators in the third term should in fact be commutators. Why doesn't the normal distributive law work here?

Last edited: