Any approach to pure quantum gravity must eventually face the question of
coupling quantum matter to the theory. In the past, several ways of coupling
matter to spin foam quantum gravity have been proposed, but the dynamics of the
coupled matter-gravity system is challenging to explore. To take first steps
towards uncovering the influence quantum matter has on spin foam models, we
couple free, massive scalar lattice field theory to a restricted,
semi-classical 4d spin foam model, called quantum cuboids. This model can be
understood as a superposition of hypercuboidal (and thus irregular) lattices.
Both theories are coupled by defining scalar lattice field theory on irregular
lattices via discrete exterior calculus and then superimposing these theories
by summing over spin foam configurations. We compute expectation values of
geometric and matter observables using Markov Chain Monte Carlo techniques.
From the observables, we identify a regime in parameter space, in which the
spin foam possesses a finite total volume and looks on average like a regular
lattice with an emergent lattice spacing dependent on the mass of the scalar
field. We also measure the 2-point correlation function and correlation length
of the scalar field in relation to the geodesic distance encoded in the spin
foam. Our results are consistent with the correlation function of ordinary
scalar lattice field theory defined on a fixed regular lattice with the
emergent lattice spacing and the same mass. We conclude that in this regime of
the model, the scalar field is not sensitive to the fluctuations of the spin
foam and effectively behaves as if it is defined on a fixed regular lattice.
