  Canonical quantization of gravitational systems is obstructed by the problem
of time. Due to diffeomorphism symmetry the Hamiltonian vanishes: dynamics with
respect to a background time parameter appears "frozen." Two strategies towards
the quantization of such systems are the identification of a clock degree of
freedom before quantization (deparametrization), and quantization on a
kinematical Hilbert space which is subject to constraints (Dirac quantization).
The usual canonical quantization in quantum field theory is analogous to
deparametrization. Here we introduce a frozen formalism and Dirac quantization
for a complex Klein-Gordon scalar field, and show that the resulting theory is
equivalent to usual canonical quantization. We then apply the formalism to the
group field theory formalism for quantum gravity, for which both
deparametrization and a "timeless" quantization have been proposed in past
work. We show how a frozen formalism for group field theory links between these
two existing approaches, and illustrate in particular the construction of
physical observables. We derive effective cosmological dynamics for group field
theory in the new formalism and compare these to previous work. The frozen
formalism could be extended to other approaches to quantum gravity that do not
use a preferred time parameter.
