I'm interested in understanding the many-body generalization of the canonical commutation relations. I.e. commutators of the form $$ [a^\dagger_I, a_J] $$ where $I,J$ are multi-indices with the same weight $|I|_0=|J|_0=N$ and the $a,a^\dagger$ are boson annihilation/creation operators.

edit: By multi index, I mean, $$a_I = \prod_{j=i}^m (a_j)^{I_j}=(a_1)^{I_1}...(a_m)^{I_m}\ \ \ \ \sum_{j=i}^m I_m=|I|_0=N $$

This seems like a natural question, but I haven't been able to find anything about them. Perhaps "many-body" and "permanent" are not the right keywords.

Without a relevant reference, I'm trying to compute these by hand. Its not hard to see that this result can be constructed by considering a somewhat simple commutator: $$ [(a^\dagger)^N, a^N]$$ Unfortunately, I haven't been able to find or solve this one either. Its tantalizingly close though. Noting that $[a^\dagger, f(a)]=f'(a)$ for analytic functions $f$, we can take higher derivatives by nesting commutators: $[a^\dagger,[a^\dagger,f(a)]]=f''(a)$, etc. But using basic results about ring commutators, solving $ [(a^\dagger)^N, a^N]$ involves nested anticommutators $$\{a^\dagger,\{a^\dagger,\{...,[a^\dagger,f(a)]\} \} $$

I've played around with it a little bit, and it looks like a nested version of the product rule, but I can't quite find the pattern to write down the general solution.

Any advice? either a (partial) solution, a link to a relevant reference, or even some suggested key words would be great appreciated. This is surely a well studied problem, I just don't know how to plug into it.

  • $\begingroup$ To be clear, are you trying to find $[\sum_k M_{ik} a_k^\dagger, \sum_k M_{jk}^*a_k]$, where the $a_k$ are single-site annihilation operators? Because I'm not sure what the phrase "I,J are multi-indices with the same weight" is supposed to mean here. $\endgroup$ – Jahan Claes Sep 11 '17 at 21:59
  • $\begingroup$ I edited to clarify. I'm considering monomials of creation/annihilation operators. $\endgroup$ – Mr.Weathers Sep 11 '17 at 22:36
  • $\begingroup$ A permanent is a polynomial of degree $n$ symmetric is the entries of an $n\times n$ matrix. Can you clarify how you see your $a_I$ as a permanent? $\endgroup$ – ZeroTheHero Sep 12 '17 at 0:11
  • $\begingroup$ Taken as boson operators, $|I>=a^\dagger_I |0>$. If you have a suggestion for better notation, please suggest it. Part of the reason I can't find much on this is that I'm using nonstandard terminology for sure. $\endgroup$ – Mr.Weathers Sep 12 '17 at 17:27

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