Current conservation matrix elements

The vanishing of this matrix element is, in fact, independent of the assumption of current conservation, and can be proved using the transformation properties of the current operator and one-partic e states under space and time inversion, together with the hermiticity of jn(0). By actually generating the states q,<>, from the states in which the particle is at rest, by a Lorentz transformation along the 3 axis, and the use of the transformation properties of the current operator, essentially the entire kinematical structure of the matrix element of on q, can be obtained.15 We shall, however, not do so here. Bather, we note that the right-hand side of Eq. (11-529) implies that... 

The matrix elements of r, and a are of rank-one, r is the exact analogue of the isospin-changing El operator and the conserved-vector-current (CVC) theorem can be invoked to relate the matrix element of a to that of r. Further, [r, spin-dipole operator. The helicity operator 75 is the time-like component of the rank-zero axial current - also known as the axial charge - discussed in Section 1. We thus have five independent matrix elements, two each of rank zero (RO) and rank one (Rl) and one of rank two (R2) which we denote as... 

The states a < 5, may be called the hydrodynamic states since they are associated with the conserved variables of number density, longitudinal and transverse components of the current, and kinetic energy. The other two states, correspond to the stress tensor and heat current, respectively. Therefore, the diagonal matrix elements involving these states must be related to the transport coefficients of shear viscosity and thermal conductivity as is well known in conventional transport theory. We will see below that these elements are important in formulating kinetic models. Besides the matrix elements shown in Table 1, we will include one additional element, namely. 

It is now postulated that (1.2.12) holds for the entire current, not just for the nucleon piece of it, and therefore V" is a conserved current. But precisely because it is conserved its matrix elements can be shown to be uninfluenced by the strong interactions, as is explained in some detail in Appendix 3, so that... 

This unwelcome discovery is potentially catastrophic for our unified weak and electromagnetic gauge theory. There we have lots of gauge invariance, many conserved currents, both vector and axial-vector, and hence many Ward identities. Moreover the Ward identities play a vital role in proving that the theory is renormalizable. It is the subtle interrelation of matrix elements that allows certain infinities to cancel out and render the theory finite. Thus we cannot tolerate a breakdown of the Ward identities, and we have to ensure that in our theory these triangle anomalies do not appear. 

It is these relations together with the exact results for the hadronic matrix elements of conserved currents discussed in Appendix 3 that provide the real basis for the sum rules quoted in eqns (16.4.2) to (16.4.4). 

If we consider non-diagonal matrix elements of vector currents between a spin-half baryon octet the most general form allowed Lorentz invariance and parity conservation is... 

The matrix elements Sij describe the probability of transition from state i to state j. For a complete solution of the S-matrix one needs M linear independent boundary conditions a. The transition probability is simply given by Pin = 5/np. The current conservation requires J2n = 1> which can be used as a test of the numerical accuracy of our calculation. 

The conservation of the matrix current along the connector, I = const, following from Eq. (4) and the boundary condition in Eq. (36), results in conservation of the vector current, I = IN = IB = const. This implies that for all elements of the connector, the unit vectors p coincide, therefore the... 

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