On Smooth Asymptotic Constants for the Poincaré Group, I

Abstract

Stitrting from : L representation of the restricted PomeAa4 group 8 : on i~ HILBRRT space I-x irnd from arbitrary unitary Poracsui invariant opri\toru X+ and Ya method for the construction of unitary PomcmG coverient operators X whose time evolution emXe-i"H tend very fa& to the given operators X+ and X-t goes to +and -00, respectively, is described.Let H be a selfadjoint operator on a separable HILBS.~ space 95. The set of all bounded operators X such that (1) Xtu : = e' "Xe-i' Hu, t E R tend very frrst to limits Xi% as t + fa for vetom zccB* (spin1 linear manifolds) is called the set of smooth asgmptotic ccmsta&s for H (see 0.g. [3] and [lo]). In particular if there is strongly continuous unitary representation U(g) of the restricted POINCAR& group 313g on ' dc such that e-ItH is the corresponding subrepresentation of the time tmnshtion group of 8; and the limits Xi of tbe smooth asymptotic constant commute with U(g), g<$:, then me call X a mnuoth mynptotic constunt for the P o m c d p a p . Such operators were studied in [3], [4], "33 and were used for the construction of special quantum fie& in [4], [S].