We define the Weyl algebra suitable to represent the quantization postulate for one-dimensional systems whose configuration space is semibounded. It consists of a group {V(beta)=exp(-i beta q),beta is an element of R} of unitary operators and a semigroup {U(alpha),alpha >= 0} of nonunitary isometries. We show that the spectrum sigma(q) is a half-line [x(0),infinity), with an arbitrary x(0)>-infinity, and that the irreducible representations of the Weyl algebra with the same x(0) are equivalent. We also consider the case when the semigroup of translations is substituted with a semigroup of partial isometries of index 1 (particle confined to a segment of unit length). The uniqueness of the irreducible representations of the related Weyl algebra is proved also for this case by exploiting the result for the half-line. (c) 2006 American Institute of Physics.
On the Weyl algebras for systems with semibounded and bounded configuration space
BRACCI, LUCIANO;PICASSO, LUIGI ETTORE
2006-01-01
Abstract
We define the Weyl algebra suitable to represent the quantization postulate for one-dimensional systems whose configuration space is semibounded. It consists of a group {V(beta)=exp(-i beta q),beta is an element of R} of unitary operators and a semigroup {U(alpha),alpha >= 0} of nonunitary isometries. We show that the spectrum sigma(q) is a half-line [x(0),infinity), with an arbitrary x(0)>-infinity, and that the irreducible representations of the Weyl algebra with the same x(0) are equivalent. We also consider the case when the semigroup of translations is substituted with a semigroup of partial isometries of index 1 (particle confined to a segment of unit length). The uniqueness of the irreducible representations of the related Weyl algebra is proved also for this case by exploiting the result for the half-line. (c) 2006 American Institute of Physics.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
