Noncrossing normal ordering for functions of Boson operators

Toufik Mansour, Matthias Schork, Simone Severini

Research output: Contribution to journalArticlepeer-review


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 (a r (a ) s ) n ) and (a r +(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.

Original languageEnglish
Pages (from-to)832-849
Number of pages18
JournalInternational Journal of Theoretical Physics
Issue number3
StatePublished - Mar 2008


  • Lattice paths
  • Noncrossing partitions
  • Normal ordering

ASJC Scopus subject areas

  • General Mathematics
  • Physics and Astronomy (miscellaneous)


Dive into the research topics of 'Noncrossing normal ordering for functions of Boson operators'. Together they form a unique fingerprint.

Cite this