  We explore the canonical description of a scalar field as a parameterized
field theory on an extended phase space that includes additional embedding
fields that characterize spacetime hypersurfaces $\mathsf{X}$ relative to which
the scalar field is described. This theory is quantized via the Dirac
prescription and physical states of the theory are used to define conditional
wave functionals $|\psi_\phi[\mathsf{X}]\rangle$ interpreted as the state of
the field relative to the hypersurface $\mathsf{X}$, thereby extending the
Page-Wootters formalism to quantum field theory. It is shown that this
conditional wave functional satisfies the Tomonaga-Schwinger equation, thus
demonstrating the formal equivalence between this extended Page-Wootters
formalism and standard quantum field theory. We also construct relational Dirac
observables and define a quantum deparameterization of the physical Hilbert
space leading to a relational Heisenberg picture, which are both shown to be
unitarily equivalent to the Page-Wootters formalism. Moreover, by treating
hypersurfaces as quantum reference frames, we extend recently developed quantum
frame transformations to changes between classical and nonclassical
hypersurfaces. This allows us to exhibit the transformation properties of a
quantum field under a larger class of transformations, which leads to a
frame-dependent particle creation effect.
