The superscripts C, LS, and T designate the static central, spin-orbit, and tensor parts of the potential. The operators 1*S and Su are the usual spin-orbit and tensor operators. We have written U BE (r, p 2 ) =nr 1 Lp 2 Vs*?(r) + Vs*?(r)p*], Uso and UTE being similarly denned. The symbol m denotes the nucleon mass, r is the relative coordinate | r 3 -r 2 1, and p is the relative momentum \ | pi-p 2 1. The superscript P indicates that the radial form factor is part of the momentum or velocity-dependent potential.
An explicit expression is given for the unit element E of the ring generated by the Duffin-Kemmer-Petiau (DKP) operators βμ. The relation of E to the unit operator I (unit matrix in a matrix representation) is also derived. It is pointed out that one must be careful to distinguish E from I. Bhabha's observation that one may use the irreducible representations (irreps) of the Lie algebra s o (5) to obtain the irreps of the Dirac, DKP, and other algebras is given a concise and general setting in terms of a relation between the Lie algebra s o (n + 1) and a family of semisimple operator rings. We emphasize that for the case n + 1 = 5 this means that there is an underlying relationship between the physical DKP and Dirac algebras and wave equations.
A reduction of the Duffin-Kemmer-Petiau algebra to a direct sum of irreducible subalgebras for spin-0 and spin-1 bosons is presented. The subalgebras are defined by multiplication rules for the linearly independent basis elements. In the representations discussed the spin projection operators are independent basis elements of the subalgebras. The formal utility of these representations is demonstrated by obtaining the reduction of arbitrary operator products and trace theorems. The practical utility is demonstrated by application to the analysis of free and interacting boson field currents. Most importantly, one can understand the differences between DKP nonconserved currents and those obtained from second-order wave equations.
The Brueckner reaction matrix for the ^1 = 18 nuclei is obtained in an accurate fashion from a nonlocal separable potential of the Tabakin type, but possessing a much stronger repulsive core. This potential, as well as those of Tabakin and Yamaguchi, is used in comparative studies of the reaction matrix. Both harmonic-oscillator and plane-wave intermediate states are employed. The results are compared extensively to the work by other authors on the local Hamada-Johnston potential. In the plane-wave treatment, the effect of replacing the kinetic-energy operator by a form (suggested by Baranger) more compatible with the exclusion principle is estimated.
Ins)" M relative to the leading M~ 0 cut contribution for two particles of spin > i. 8 This test is reminiscent (although different from) that proposed for factorization by H. D. I. AbarbanelIt is we 11 known that pseudoscalar particles can be described by at least two different covariant field equations, namely, the Klein-Gordon (K-G) and Kemmer equations, 1 For many classes of processes (such as the quantum electrodynamics of spin-0 mesons) calculations based on either equation yield identical results. 2 Therefore, it has often been tacitly assumed that the two equations will yield identical results in all cases.We note, however, that the K-G and Kemmer fields behave differently under scale transformations [see Eqs.(2) and (3) below]. Since symmetry-breaking effects are sensitive to the scale dimensionalities of the respective fields, 3 it follows that in principle the corresponding field theories could lead to qualitatively different results in processes, such as K ls decays, involving two or more pseudoscalar mesons of different mass. and D. Gross (to be published), to wit, the spin independence of single-particle inclusive distributions. 9 C. DeTar et at., Phys. Rev. Lett. £6, 675 (1971). 10 M. N. Misheloff, Phys. Rev. 184, 1732 (1969.To date this possibility has not been studied in a formal field theory owing to the difficulties involved in treating the strong interactions. Nonetheless, the scaling argument suggests that even in a phenomenological treatment of such a process, the same assumptions could yield qualitatively different results from analyses based on the K-G and Kemmer equations, respectively. If different results were obtained, we could then ascertain which equation gives the better phenomenological description of the particular process in question.We present in this Letter the results of just such an analysis of K l3 decays. Our main conclusions are the following:(1) When compared with experiment, our theory yields a symmetry-breaking parameter p (analogous to the K-G parameter £) whose experimental magnitude is p = 0.28±0.20. Since SU(3)-symmetry breaking is expected to be of this Motivated by scaling considerations we formulate a theory of K IZ decay based on the Kemmer equation. We find that our theory (1) has a symmetry-breaking parameter of order 10-30% which agrees with experiment, (2) makes a definite testable prediction of a kinematic zero at t = (m K +m lx ) 2 in the "effective" scalar form factor, and (3) yields a modified Callan-Treiman relation which improves agreement with experiment.1200
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