D’Alembert equation

In U(l) electrodynamics in free space, there are only transverse components of the vector potential, so the integral (158) vanishes. It follows that the area integral in Eq. (157) also vanishes, and so the U(l) phase factor cannot be used to describe interferometry. For example, it cannot be used to describe the Sagnac effect. The latter result is consistent with the fact that the Maxwell-Heaviside and d Alembert equations are invariant under T, which generates the clockwise... 

Whittaker s early work [27,28] is the precursor [4] to twistor theory and is well developed. Whittaker showed that a scalar potential satisfying the Laplace and d Alembert equations is structured in the vacuum, and can be expanded in terms of plane waves. This means that in the vacuum, there are both propagating and standing waves, and electromagnetic waves are not necessarily transverse. In this section, a straightforward application of Whittaker s work is reviewed, leading to the feasibility of interferometry between scalar potentials in the vacuum, and to a trouble-free method of canonical quantization. 

To an excellent approximation, the four Klein-Gordon equations (443) are d Alembert equations, which are locally gauge-invariant. 

The helicity is gauge-invariant. In the Coulomb gauge, it is obvious that A and C satisfy the d Alembert equation, whose solutions can be written in terms of Fourier transforms... 

When the propagation of the planar waves defines parallel planes that are normal to the propagation line of the wave, these planes are called rays. In any case, the general solution to the d Alembert equation is... 

Vol. 442 C- H. Wilcox, Scattering Theory for the d Alembert Equation in Exterior Domains. Ill, 184 pages. 1975. 

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