Special Relativity and Origins of Quantum Wavelength?

L. DeBroglie stressed a strong connection with special relativity when predicting that a quantum particle (electron) should exhibit wave behaviour. Wave behaviour is already found in the photon which is a quantum particle arising from Maxwell’s equations which are Lorentz invariant. In fact, there are statements in the literature that dE/dt= grad x B and dB/dt = - grad x E (sourceless Maxwell’s equations) are in fact quantum equations for a photon wavefunction of E+iB. Furthermore, in exp[ipx-iEt), px-Et is a Lorentz invariant, although for a bound state exp(-iEt) remains, but exp(ipx) is replaced with a momentum distribution W(x)=wavefunction = Sum over p a(p)exp(ipx). In a previous note (1), we argued that one may take the relativistic Lagrangian L= -mo sqrt(1-vv) and write v=X/T. With exp(i Action) = exp(i Integral dt1 L) and taking id/dt yields E time exp(i Action) and -id/dx exp(iAction), pe xp(i Action). The relationship EE=pp + momo may then be converted into a differential equation, namely the Klein-Gordon equation. For a free particle, a solution is exp(ipx-iEt). exp(ipx) immediately suggests a physical wavelength proportional to 1/p. The question is how does this wavelength originate? For a photon which has spin 1, one may create circularly polarized light with E and B moving in a plane perpendicular to photon motion with frequency px if t is held fixed. This motion of E and B is said to account for spin and may show a physical reason for a photon wavelength i.e. one related to E and B fields. The Klein-Gordon equation, however, holds for spin 0 particles. We argue in this note that the physical reason for a quantum wavelength may be related to the factor sqrt(1-vv/cc) from special relativity