  Classical equations of motion that are first-order in time and conserve
energy can only be quantized after their variables have been transformed to
canonical ones, i.e., variables in which the energy is the system's
Hamiltonian. The source-free version of Maxwell's equations is purely
dynamical, first-order in time and has a well-defined nonnegative conserved
field energy, but is decidedly noncanonical, an issue that has attracted little
attention. The opposite parities of the electric and magnetic fields and
consequent curl operations that typify Maxwell's equations are especially at
odds with their being canonical fields. Transformation of the magnetic field
into the transverse part of the vector potential helps but is not sufficient;
further simple nonnegative symmetric integral transforms, which commute with
all differential operators, are needed for both fields; such transforms also
supplant the curls in the equations of motion. The canonical replacements of
the source-free electromagnetic fields remain transverse-vector fields, but are
more diffuse than their predecessors. Combined as the real and imaginary parts
of a complex field, the canonical fields prove to be the transverse-vector wave
function of a time-dependent Schroedinger equation whose Hamiltonian operator
is the quantization of the free photon's square-root relativistic energy. Thus
proper quantization of the source-free Maxwell equations is identical to second
quantization of free photons that have normal square-root energy. There is no
physical reason why first and second quantization of any relativistic free
particle ought not to proceed in precise parallel, utilizing the square-root
Hamiltonian operator. This natural procedure leaves no role for the completely
artificial Klein-Gordon and Dirac equations, as accords with their grossly
unphysical properties.
