A Creation Op, Scaled Flows, and Operator Identities

Motivated by operational identities in “Enumerative geometry, tau-functions and Heisenberg–Virasoro algebra” by Alexandrov, I’ve explored the exponential mapping of the creation / raising op , a generalization of the raising op of one family of Hermite polynomials. Both ops have applications in quantum physics in the characterization of related operator algebras.

Draft of notes:

The Creation Op : Scaled flow and operator identities (pdf)

(numerous typos, inadvertent variable switches, corrected 2/11/2022)

An instance of the creation / raising op eqn

is eqn. 269 on p. 75 of “On Topological 1D Gravity. I” by Jian Zhou, related to the Hermite polynomials of OEIS A099174, which in turn are related to A344678. Both are related to the Heisenberg-Weyl algebra, in which the commutator and its generalization play the definitive roles. (The roles of and can be switched in the last identity. Charles Graves and Pincherle made much use of this commutator as well as Lie, Born, Heisenberg, Jordan, Weyl, and others, of course, researching quantum physics.) One realization of the pair of ladder ops–the raising / creation op and the lowering / destruction / annihilation op –is the pair and ; another, and , for the family of Hermite polynomials above. and and with satisfy the basic commutator relation as well.

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Additional orderings and factorizations of the basic operator relation (added 2/22/22):

From the main text,

but also

so, extracting the operators, we have the operator identity

It follows that

and

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Related Stuff :

“An introduction to operational techniques and special polynomials” by Ricci and Tavkhelidze,  eqns. 2.13-2.15 on p. 164 (basic action formulas)

“Commutation Relations, Normal Ordering, and Stirling Numbers” by Mansour and Schork, pp. 391-2 and 398 (basic action formulas)

”Lie theory and special functions” by Miller, eqns. 1.58 and 1.59 on pp. 18 and 19 (basic action formulas)

“Differential posets” Stanley, Thm. 2.5 on p. 925,  equations 10 and 11 on pp. 926 and 927 (basic action formulas, relations to posets also provided)

“Evolution operator equations: integration with algebraic and finite difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory” by Dattoli, Ottaviani, Torre, and Vazquez, eqns. 1.2.22 and 1.2.23 on p. 6 (basic action formulas, no proof, but refs to proofs in earlier literature on Lie theory and differential geometry are provided).

Added 2/22/22:

“Ladder Operators and Endomorphisms in Combinatorial Physics” by Duchamp, Poinsot, Solomon, Penson, Blasiak, and Horzela has a similar result and conjugacy derivation for the action of the exponentiated , but they are rather sparse in crediting earlier work. They do reference Kurbanov and Maksimov on operator expansions, but not Charles Graves, who is deservedly referenced by A. Krzysztof Kwasniewski in several of his papers along with O. V. Viskov (see, e.g., “Extended finite operator calculus – an example of algebraization of analysis” by Kwasniewski and Borak). Two relevant papers published by Charles Graves in the early 1850s, when Lie was a child, are

I)  “A generalization of the symbolic statement of Taylor’s theorem”, Proceedings of the Royal Irish Academy, Vol. 5, (1850-1853), p. 285-287

II) “On the principles which regulate the interhange of symbols in certain
symbolic equations”, Proc. Royal Irish Academy, Vol. 6(144), (1853-1857)

Other relevant papers by Graves are presented on p. 588 of the “The Theory of Linear Operators” by Davis, an invaluable resource on linear operators and the history of the topic, which I stumbled upon eons ago when studying Heaviside’s operational calculus. Search on Graves in Davis. Page 13 of Davis presents the basic flow equation. See also my answers to the MathOverflow question “In Splendid Isolation“. Loeb, in “The World of Generating Functions and Umbral Calculus” also discusses expansions of linear operators and extension of the Sheffer calculus of operators.