Operator d Alembert

In Eq. (32) we exploited the short-hand notation for the d Alembert operator that contains the second derivatives with respect... 

Some cases refer to a spherical wave, since the spatial coordinate is the radius vector, and in this case, the d Alembert operator is... 

Quantities without any indices such as the mass m or the space-time interval ds, which are not only covariant but invariant under Lorentz transformations, are called Lorentz scalars or zero-rank tensors. They have exactly the same value in all inertial frames of reference. A very important scalar operator for both relativistic mechanics and electrodynamics is the d Alembert operator... 

Since the d Alembert operator is a Lorentz scalar, cf. Eq. (3.51), and the charge-current density fi has been shown to be a Lorentz 4-vector, it is immediately obvious that the gauge field also represents a Lorentz 4-vector and transforms according to Eq. (3.36) xmder Lorentz transformations. 

The four-dimensional generalization of the Laplacian has been identified to be the d Alembert operator... 

The derivation of Eq. (218) from Eq. (206) follows from local gauge invariance, and it is always possible to apply a local gauge transform to the vector A, the Maxwell vector potential. The ordinary derivative of the d Alembert wave equation is replaced by an 0(3) covariant derivative. The U(l) equivalent of Eq. (218) in quantum-mechanical (operator) form is Eq. (13), and Eq. (212) is the rigorously correct form of the phenomenological Eq. (25). It can be seen that Eq. (212) is richly structured in the vacuum and must be solved numerically. The vacuum currents present in Eq. (218) can be computed from the right-hand side of the wave equation (212), and these vacuum currents follow from local gauge invariance. 

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