Noncrossing Normal Ordering for Functions of Boson Operators
نویسندگان
چکیده
Normally ordered forms of functions of boson operators are important in many contexts in particular concerning Quantum Field Theory and Quantum Optics. Beginning with the seminal work of Katriel (Lett. Nuovo Cimento 10(13):565–567, 1974), in the last few years, normally ordered forms have been shown to have a rich combinatorial structure, mainly in virtue of a link with the theory of partitions. In this paper, we attempt to enrich this link. By considering linear representations of noncrossing partitions, we define the notion of noncrossing normal ordering. Given the growing interest in noncrossing partitions, because of their many unexpected connections (like, for example, with free probability), noncrossing normal ordering appears to be an intriguing notion. We explicitly give the noncrossing normally ordered form of the functions (ar(a†)s)n) and (a + (a†)s)n, plus various special cases. We are able to establish for the first time bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths.
منابع مشابه
Noncrossing Normal Ordering for Functions of Bosons
Normally ordered forms of functions of boson operators are important in many contexts mainly concerning quantum field theory and quantum optics. Beginning with the seminal work of Katriel [Lett. Nuovo Cimento 10(13):565–567, 1974], in the last few years, normally ordered forms have been shown to have a rich combinatorial structure [see P. B lasiak, quant-ph/0507206]. In this paper, we apply the...
متن کاملBoson Normal Ordering via Substitutions and Sheffer-type Polynomials
We solve the boson normal ordering problem for (q(a)a + v(a)) with arbitrary functions q and v and integer n, where a and a are boson annihilation and creation operators, satisfying [a, a] = 1. This leads to exponential operators generalizing the shift operator and we show that their action can be expressed in terms of substitutions. Our solution is naturally related through the coherent state ...
متن کاملRepresentations of Monomiality Principle with Sheffer-type Polynomials and Boson Normal Ordering
We construct explicit representations of the Heisenberg-Weyl algebra [P, M ] = 1 in terms of ladder operators acting in the space of Sheffer-type polynomials. Thus we establish a link between the monomiality principle and the umbral calculus. We use certain operator identities which allow one to evaluate explicitly special boson matrix elements between the coherent states. This yields a general...
متن کاملOrdered Expansions in Boson Amplitude Operators
The expansion of operators as ordered power series in the annihilation and creation operators a and a~ is examined. It is found that normally ordered power series exist and converge quite generally, but that for the case of antinormal ordering the required c-number coefficients are infinite for important classes of operators. A parametric ordering convention is introduced according to which nor...
متن کاملMonomiality principle, Sheffer-type polynomials and the normal ordering problem
We solve the boson normal ordering problem for ( q(a†)a+ v(a†) )n with arbitrary functions q(x) and v(x) and integer n, where a and a† are boson annihilation and creation operators, satisfying [a, a†] = 1. This consequently provides the solution for the exponential e †)a+v(a†)) generalizing the shift operator. In the course of these considerations we define and explore the monomiality principle...
متن کامل