Interpolating Asymptotic Constants for the Poincaré Group, in Particular on Fock Space

Abstract

Starting from a representation of the restricted POINCABE group 8 : on a HILBERT space X and from arbitrary bounded POINCARE invariant operators X, and Xwe describe a method for the construction of bounded operators X whose time evolutions ertHXe-itH tend "very fast" to the given operatom X, and Xas t goes to +and --, respectively. Additionally these operators X (interpolating asymptotic constants) are LOREXTZ (or ErrcLmean) invariant and satisfy D certain boundedness condition with respect to H. t i s mlid. I n particular, if S = I X I a d p H ( X ) exists, then also p U A ( X ) exists and P -~(~Y ) -.I p R ( X ) I . I n this ra.ye and if $ ( X ) is Poincrcre'invariwnt, t h e n a l s o p u H ( 9 )its YOINCARE inwwinnt, henre S is asyrnptotical1;y P O~N C A R~ invnriunt. Proof. Obvious. IKecjideR the POINCARI~ invariance we introduce some other invariance not,ions. Let S~9('dt).Note that the property of X to be translationally invariant is independent of the frame. i.e. this property is POINCAR& invariant (this is implied hy the formula 1 U,,U,. mhereAa denotes the image of a under the Lorentz transformation = U,,,,,,X for all a € R?. S is called rotat,iom@ invariant, if ,Y (7, = MAX for all -1 = (t L,~)ahereA,-,ESO(:l). S is cdletl E w l i d m n inwrcricint. if S is spatially trnnslationally wid rotationwdly invtlriant. A ) . , Y is called s p a h l l y translalionally invariant, if X1 X is called I-invariant, if X I = I X . Furthermore, we need t h e following definitions.1)efinition 2. Lct ( 23 be CI 1inPti.r mmaifold i n 2 being dense in this syur:e. Then .\ is Bai~m~Lrtcl/~oIIenber.g. Interpoltitirig .Asymptotic ('onstants I 9 rwllcd (c 9-smooth asymptotic constar&. if , Y is (112 u.symptotic constanf m c h that ! I t ) q(X--A-+) e -~X E B , U I !~C~, I t I -l l, + t z l , n = l . 2, ... for trll t~ t 3. Definition 3. Let A . B E "X), AE,=Ea,A = A . BE,,=E,,R= R. The 0 .~4~z . ptotic. c-onstctnt A' is. cc~,llcd interpolatiq with .rpspect to A owl. B if p H ( X ) -A mud Definition 4. S F S ( X ) is called m>-energPtic. boundec? (m i!, ncr.turitl ?Lumber). if Remark 1. Note t.hat an operator XE 9(%) is rn-energet.ic hounded if and only if (N-A)mX (H-3.)-m is hounded for all I from t.he resolvent set res H . IfCorresponding to llefinition 1 we call an operator X E $ ( X ) an asFmptotic constant for the strongly rontinuous and unitary representation [ -( =c-irH of R (time translations) if 8-lim e"tHXe-~H'E$= : X* exist (for both signs. i and -). L-5- 18, :

A class of nontrivial weakly local massive wightman fields with interpolating properties

“…The next proposition deals with the construction of A n shifted to the space (3.23). This proposition is the analog of Lemma 7 in [7]. V. B is also an interpolating asymptotic constant with the same limits and it is @ 0 -smooth where the smoothness estimate is uniform with respect to the orbits defined by (t,0), i.e.…”

“…Simultaneously, the smoothness manifolds 3l w n = 2,3,..., with respect to A n : = A \ Jf π are introduced. We should remark that the essential steps for the construction of the operators A n and the smoothness manifolds Q) n are already described and developed in [7] where θ corresponds to A via θ = ι~ίΛι (see [7]). We form rf/c is an invariant measure on 0 Ml (with respect to θ) and where ρ(-) is an absolutely continuous measure on (nm 0 , oo) with suppρ = (nm 0?…”

“…We do not prove this lemma. See for example [7] where the proof is performed for the case n = 2. See also Ruijsenaars [16, p. 427] where such diffeomorphisms are explicitly written down by the formulas {p 1?…”

“…This approach is useful because the well-developed technique of interpolating asymptotic constants [6] can be applied to the construction of V. An important step in this construction is to find Lorentz invariant asymptotic constants interpolating between zero and the identity. This is done in [7], We remark that our construction of the field A( ) is a pure operator-theoretical one starting with the mentioned ansatz and using some results of Fredenhagen and Hertel [8] on Wightman fields with a regularity condition. Our assumptions on the operator V imply that the quantum field A( ) is necessarily nonlocal.…”

“…where it is assumed that (1 + H) k V(l +H)~kis also bounded. In this case we call V k-energetic bounded (see [7]).…”