In:
Journal of Mathematical Physics, AIP Publishing, Vol. 30, No. 3 ( 1989-03-01), p. 594-600
Abstract:
Boson operator realizations of su(2) and su(1,1) are obtained. Scalar products are introduced on ‘‘Fock spaces’’ (Verma modules) spanned by generators of the Heisenberg algebra H and by generators of su(2). These scalar products unitarize certain of the representations of H, or of su(1,1). It is shown that the Gel’fand–Dyson realization of su(1,1) implies a scalar product that unitarizes H, while the Primakoff–Holstein realizations imply a scalar product that unitarizes su(1,1). The relationship between the Gel’fand–Dyson boson operators a° and the Primakoff–Holstein boson operators b° is obtained making use of the two distinct scalar products. Generalized ‘‘vacuum states’’ are defined that are formed by polynomials in the creation–annihilation operator pairs a°a. A representation ρ of H and su(1,1) on the states a°m and an is discussed. For this representation a1=a‖0〉≠0, but rather (a°a)‖0〉 =0. The states of this representation space consist of boson states and boson-hole states. All the familiar results of H and su(1,1) representation theory are preserved within the representation ρ.
Type of Medium:
Online Resource
ISSN:
0022-2488
,
1089-7658
Language:
English
Publisher:
AIP Publishing
Publication Date:
1989
detail.hit.zdb_id:
1472481-9
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