Acta Physica Academiae Scientiarum Hungaricae 31. (1972)

1972 / 1-3. szám - K. Szegő - K. Tóth: Some Remarks on "Energy-Dependent" Representations

Acta Physica Academiae Scientiarum Hungaricae, Tomus 31 (1—3), pp. 147—151 (1972) SOME REMARKS ON “ENERGY-DEPENDENT” REPRESENTATIONS By K. Szegő and K. Tóth CENTRAL RESEARCH INSTITUTE FOR PHYSICS, BUDAPEST After explaining the meaning of energy dependent representation, we sketch how it can be obtained for the case of the SL(2, C) group. Some physical applications are also treated. Some time ago we examined the problem how one can expand a general two-particle—two-particle scattering amplitude in terms of Lorentz group representations at any s and t values [1]. To do this, first one has to define the scattering amplitude as a function on the group in question. As we noticed in [1], a possible and in some sense desirable way is the following: 0 — P2s2^2 |T| Рз^з^з, P4S4A4> — = <P=Pl+PvPl~Pv MlsA \AT\ P'lP3-Pvs3*3sA} = Ih(a) where Л is a Lorentz transformation acting on two-particle states, its detailed form together with Pi~p2, p3—pt can he found in [1], P' = P(1 3, 2 «-*- 4). If we now perform the expansion using the ordinary |j0ajm ) basis* of the Lorentz group, we get /н(Л) = 2 ®i°m°rmiA)TH(sJo (2) For continuous variables, like a, integration is meant in Eq. (2). The expansion coefficients, i.e. the T functions in Eq. (2) are quite complicated due to <P .... I j0ctjm ....]> type coefficients in it. The main problem with the (i) P = ___—Xs (mi-roif, 0, 0, 1 —}s (mj+m2)2 z|/m1m2 2\lm1m2 if s > (m1+m2)2 m, + m ! 2 Утп1т2 У(т1+т2) —s, 0, 0, ][{m1 — m2)2 —s ТП 2 if s < (m1 m2)2. * In the I jfPjm ) basis, j0 and a characterize an irreducible representations. To label the vectors of a representation space one chooses a subgroup of the Lorentz group; generally it is the rotation group. The | jm ) states are representations of the rotation group. For further details, see, e.g. [3]. 10* Acta Physica Academiae Scientiarum Hungaricae 31, 1972

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