[AntonioLao]

The electromagnetic force is sometimes called the Lorentz force (L) given by L=eE+ev´B where e is the unit of electric charge, E is the electric field, v is the velocity of the electric charge, and B is the magnetic field.

In vector analysis, the cross outer vector product of v´B is not commutative. That is to say that v´B= -B´v. Therefore the Lorentz force can be written in two valid forms. L(a)=eE(a)+ev(a)´B(a) and L(b)=eE(b)-eB(b)´v(b). Their scalar inner dot product is L(a)·L(b)=e²[ E(a)·E(b)-v(a)´B(a)·B(b)´v(b)]. If L(a)·L(b) is now multiplied by the space-time interval of Einstein’s special relativity then the product is equivalent to the square of energy E² given by E²=ds²L(a)·L(b) where the space-time interval ds² is given by ds²=dx²+dy²+dz²-c²dt². Furthermore by a principle of hidden symmetry |L(a)|=|L(b)| such that [L(g)]²=|L(a)|²=|L(b)|² and E²=ds²[L(g)]². If ds² is light-like then ds² vanishes such that E²=0. However, the Lorentz force remains positive definite. On the other hand if [L(g)]² is multiplied by the absolute square of the probability wave function |y(g)|² then E²=|y(g)|²[L(g)]². These two forms for the square of energy imply the wave-particle duality of E²=ħ²n² and E²=m²c where n is its frequency and m is the rest mass of the particle.