SUSY

It’s been nearly three weeks since I promised to write about my activities here. My apologies, in case I actually have any readers. I got swamped in last minute deadline-beating and managed to sneak in my eport just at the last second as the clock was striking 4. I’m writing this post thanks to some encouragement from Tom. So, here goes:

Day 1 : Symmetries have played a very important role as guiding principles in Theoretical Physics. A given symmetry enables us to make some statement about the behaviour of a system (namely the associated conserved charge), thus one would naturally be interested in any possible new symmetry that physical models could have. However, there is a theorem due to Coleman and Mandula, that forbids the possibility of any symmetries that transform as tensors under Lorentz transformations other than Pμ and Mμν. There is a very simple plausibility argument by Witten as to why this should be so. It basically says that, the conservation of Pμ and Mμν, leaves only the scattering angle unknown. Any new conservation law would leave us with only a discrete set of possible angles, but since the scattering amplitude is an analytic function of angle it must be zero for all angles. Thus it would seem that the Coleman-Mandula theorem, rules out all possible symmetries other than the Poincare group(translations and Lorentz transformations). This seems to spoil all our fun, by saying that no new symmetries lie in wait to be found by daring & adventurous experimentalists.

 

Thanfully, there’s a catch! We have only ruled out symmetries, whose generators transform as tensors under the Lorentz group. What about spinors?? That is another representation of the Lorentz group. We can have spinorial conserved charges. This is because, spinorial conserved quantities do not give us any observables, which impose constraints on scattering matrices. (For the more technically minded, the Coleman-Mandula theorem applies to Lie algebras, whereas supersymmetry charges obey a different structure, they have anticommutators instead of the Lie-bracket). What they do is to relate boson-boson and fermion-fermion scattering. Now, the anticommutator of two supersymmetry generators is going to have two spinor indices. This quantity is like a vector, if it consists of a left & a right handed spinor, but this thing being a conserved vector, has to be nothing other than Pμ. All other anticommutators have to be zero, because such a quantity does not correspond to anything that we observe. More importanty P0, the Hamiltonian, can be written as the anticommutator of a Q & its hermitian conjugate Q*. This is the simple case with just one supersymmetry, what is called, N=1 supersymmetry. Therefore, we have:

H = {Q, Q*}

Now, this simple way of writing H, as an anticommutator has several interesting properties. Next day, I think I’ll write a bit about differential forms, and slowly build up to a proof of a special case of the Atiyah-Singer index theorem(for the de Rham cohomology).

 

In case anybody is reading this, I’d really like to know that somebody actually reads the stuff I’m writing. And considering that my audience, would probably be a singleton, I could try to make the stuff I put here more/less technical as you might like it. It would really feel nice to know that the stuff I’m writing here, is (hopefully)useful/enjoyable to someone.

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3 responses to “SUSY

  1. Looking forward to the ‘next day’ that you’ll blog! Include some (readable) online references if there are any.. thanks!

  2. I admit that, in as much as I would love to understand exciting things like SUSY, I am not always motivated enough to carefully go through details of the theory like the Coleman-Mandula theorem in this case. So my comments will reflect a certain levity and an unwillingness to do sufficient research before posting which may not impress someone who is taking this seriously. WIth this warning, I’ll proceed to make the comments :

    I think you need to be more precise and if necessary mathematical in stating the Coleman-Mandula theorem. Especially the reasons why spinors are excluded from these constraints – about spinorial charges not giving observables and that SUSY generators satisfying Lie superalgebras and so on. I checked WIkipedia for the exact statement of the theorem and like I said didnt care to follow all the links given there to fully comprehend it. So it would help if you get into the particulars of the theorem and state it throughly with all its limitations and exceptions.

    And tthe way you have motivated the identification of the Hamiltonian with the anti commutator of the spinors is also not very clear.

    So, if you are really dont mind enetrtaining a casual reader like me, then go ahead and post the next part.

  3. very interesting, but I don’t agree with you
    Idetrorce

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