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By B. Renner, D. ter Haar

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To show that there is no contradiction, it is sufficient to give a model which is relativistically invariant and contains current algebras. We choose a simple quark model,(164_65) described by a Lagrangian density J? 20) Here q{x) is a triplet of free fermion fields, described by Dirac spinors. As vector and axial vector currents we take Ux) = q(x)yßi\iq(x). 22) By simple commutation! 26) UoWJt(y)] = WM*)e»(jc-;y). 15). The fact that we have taken different quark masses (m' # m) and so destroyed SU3 symmetry, has not at all influ­ enced the derivation.

We find significant agreement in all cases where drastic approximations and disputable additional hypotheses can be avoided. There are not many cases of this kind (for instance the Adler-Weisberger relation(9,10'385_6)); sometimes we will find only rough agreement, but very often the predictions will be better than the approximations lead one to expect. (c) Accepting that current algebras are correct, how are they connected to other dynamical theories? We will try to com­ ment on this point in Chapter 10; until then, we shall take the commutators as working hypotheses to be tested.

2m »... 44) (ηΐρ+ηιπ) where the neutrino or antineutrino energy EV(EP) is a free parameter. For W2 > 2mPEv + m% the integrand has to be con­ tinued analytically below threshold. davldilt is the differential forward cross-section for all kinds of inelastic reactions v + P —» (n) + lepton, where (n) is any hadron system with strangeness zero excluding the nucléons. 44). 31 Current Algebras and their Applications A similar sum rule can be constructed for strangeness changing reactions (10). With our present knowledge on inelastic reac­ tions we cannot test the direct sum rules.

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