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Hermitean Oscillator‐like Realizations of Classical Algebras and Superalgebras in Hilbert Space with Positive Definite Metric
Author(s) -
Nowicki Anatol
Publication year - 1986
Publication title -
fortschritte der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/prop.19860341002
Subject(s) - superalgebra , hilbert space , mathematics , creation and annihilation operators , unitary state , pure mathematics , space (punctuation) , positive definite matrix , metric (unit) , mathematical physics , quantum mechanics , algebra over a field , physics , quantum , eigenvalues and eigenvectors , linguistics , philosophy , operations management , political science , law , economics
The hermitean oscillator‐like realizations of classical algebras in terms of bosonic and fermionic creation and annihilation operators are given. The hermitean realizations of classical superalgebras using boson‐fermion oscillators are explicitely described. The assumption of positive definite metric in a Hilbert space of the oscillators states is exploited. Due to this fact, the realizations of superalgebras in the Hilbert space can be constructed only for: the real orthosymplectic superalgebra osp ( N ; 2 M ; R ); the unitary compact superalgebra su ( N ; M ); the unitary noncompact one SU ( N ; K , M ); and the quaternionic unitary superalgebra uu α ( N ; M ; H ).