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Boson operator ordering identities based on (generalized) Stirling and Eulerian numbers

Mathematical Physics and Probability Seminar

Boson operator ordering identities based on (generalized) Stirling and Eulerian numbers
Series: Mathematical Physics and Probability Seminar
Location: MATH 402
Presenter: Rob Maier, University of Arizona

Single-mode boson creation and annihilation operators satisfy a canonical commutation relation and generate the Weyl-Heisenberg algebra. The same algebra is generated by the noncommuting operators x and D (the derivative with respect to x), acting on any suitable space of functions. A simple ordering identity in this algebra would be the rewriting of a word (a product of x's and D's) as a linear combination of normally ordered words: ones in which D's appear to the right of x's. In the easy `single-annihilator' case, the coefficients will be Stirling numbers, which have a combinatorial interpretation. But ordering identities can be more complicated. In this talk, we introduce triangles of generalized Stirling numbers, Eulerian numbers, and generalized Eulerian numbers, and associated ordering identities in which they appear as coefficients. Some of these number triangles are related to one another by a `binomial transform' operation, and some have as yet no combinatorial interpretation.

(zoom: https://arizona.zoom.us/j/86800353711)