  A first quantized free photon is a complex massless vector field $A=(A^\mu)$
whose field strength satisfies Maxwell's equations in vacuum. We construct the
Hilbert space $\mathscr{H}$ of the photon by endowing the vector space of the
fields $A$ in the temporal-Coulomb gauge with a positive-definite and
relativistically invariant inner product. We give an explicit expression for
this inner product, identify the Hamiltonian for the photon with the generator
of time translations in $\mathscr{H}$, determine the operators representing the
momentum and the helicity of the photon, and introduce a chirality operator
whose eigenfunctions correspond to fields having a definite sign of energy. We
also construct a position operator for the photon whose components commute with
each other and with the chirality and helicity operators. This allows for the
construction of the localized states of the photon with a definite sign of
energy and helicity. We derive an explicit formula for the latter and compute
the corresponding electric and magnetic fields. These turn out to diverge not
just at the point where the photon is localized but on a plane containing this
point. We identify the axis normal to this plane with an associated symmetry
axis, and show that each choice of this axis specifies a particular position
operator, a corresponding position basis, and a position representation of the
quantum mechanics of photon. In particular, we examine the position wave
functions determined by such a position basis, elucidate their relationship
with the Riemann-Silberstein and Landau-Peierls wave functions, and give an
explicit formula for the probability density of the spatial localization of the
photon.
