An Email Conversation with Prof. Edward Witten

When I proposed the linear version of the Schrodinger equation, one of the things that I tried was to find a connection with supersymmetry. Since this equation is the equivalent of the Dirac equation and introduces spin in the non-relativistic limit it can lead to formally introducing supersymmetry in non-relativistic quantum mechanics. This is actually an interesting problem to work on for someone who is interested in theoretical problems (although it wont help you get a postdoc). I spent some time on this problem but due to a busy teaching schedule and job uncertainties I have not been able to give this problem its due time. Another reason is the lack of evidence for supersymmetry. In any case I do feel that there should be a connection, at least mathematically.

This is what several luminaries like Prof. Juan Maldacena and Prof. Warren Siegel suggested I do when I emailed them. I then emailed Prof. Ed Witten for suggestions. Prof. Ed Witten is a well known string theorist and it was a pleasure to have this conversation with him. Here you go, following is the unmodified email exchange between us:

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Me:
Dear Prof. Witten,

In two recently published papers I have introduced an equation in non-relativistic quantum mechanics:

http://link.springer.com/article/10.1007%2Fs10701-015-9944-z
http://www.ijqf.org/archives/3480

The square of this equation gives the Schrodinger equation. I have seen several of your papers on non-relativistic susy. Prof. Warren Siegel and Prof. Juan Maldacena have suggested I connect the approach in the above two papers with non-relativistic susy. I have tried but not succeeded yet. I would really appreciate if you can guide me in this matter and suggest how to seek a possible connection of the two approaches.

Thank you.

Best Regards,


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Prof. Ed. Witten:

Dear Dr. Ajaib,

Have you included a potential in your constructions?
I.e. do you get a Schrodinger equation with a potential?
Supersymmetric quantum mechanics leads to special potentials
so if you were getting a potential, that would probably make it obvious
whether one could compare to supersymmetric quantum mechanics


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Me:

Thank you for your reply Prof. Witten

I did include a potential. For example, in section 3 of the paper I referenced earlier (http://arxiv.org/pdf/1502.04274.pdf) I solved potential step problem. Sorry but I am not sure how this makes the connection obvious.

Best Regards,

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Prof. Ed. Witten:

Dear Dr. Ajaib,

I am not sure if there is a connection. If you can incorporate a completely general potential
then what you have does not correspond to supersymmetric quantum mechanics, in which only
certain rather special potentials are possible.


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Me:

I see. Thank you so much for this insight I really appreciate it.

Best Regards,