What is meant by fermionic and bosonic "modes"? The paper The Dirac quantum automaton: a short review (pdf) starts off by stating:

The starting point for the construction of space–time and the physical laws therein is an unstructured, countably infinite set, $G$, of local Fermionic modes.

I have seen in other papers mention of bosonic modes as well. Searching google doesn't give any definition. I understand fermions are quark/electrons/etc., and bosons are photons and such. However, I only have basic undergraduate physics experience. So my only sense after a bit of digging is that the word "mode" is sort of casually used when talking about the different types of wave functions that can occur, but I am not sure.
What is meant by the word "mode" in "fermionic modes" and "bosonic modes"? Is there a standard definition?
 A: The term mode is used to define a particular state of a system and may refer for instance to its spin, wavevector, polarisation, charge etc. If we wanted to create a boson at position $x$ with an up-spin and with wavevector $k$, we may use the field operator $\hat{a}^\dagger(x, k, \uparrow)$ on the vacuum state $\vert0\rangle$.
The most clear distinction between fermionic and bosonic modes are that the field operators describing the former obey anticommutator relations, whilst the later obeys commutator relations. These ensure the Pauli-Exclusion principle and the symmetrisation of the wavefunction respectively.
A: The terminology of a mode of a free quantum field $\phi(x)$ comes from writing it as a Fourier transform, often also called mode expansion:
$$ \phi(\vec x) = \int \frac{\mathrm{d}^3 p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_p}}\left(a(\vec p)\mathrm{e}^{\mathrm{i}\vec x\cdot\vec p} + b(\vec p)^\dagger\mathrm{e}^{-\mathrm{i}\vec x\cdot\vec p}\right)$$
where for a spinor or vector field there are additional factors in there that don't play a role here. The crucial question is whether the objects $a$ and $a^\dagger$ (and likewise $b$ and $b^\dagger$) commute or anticommute, since they are the creation and annihilation operators for the particle (resp. antiparticle in the case of $b$) associated to the field.
If they commute, the corresponding particle is a boson - you can pile up arbitrarily many particles into the same state just by applying the creation operator many times. Then, the operators are bosonic modes. If they anticommute, the corresponding particle is a fermion - applying the creation operator twice just gives zero, so you can't ever have more than one particle in the same state. Here, the operators are fermionic modes. They are "modes" because the Fourier transform splits up a classical oscillation into its pure-frequency modes, so by analogy, we also call the objects it produces when applied to a quanutm field "modes".
In a slight generalization, one calls any collection of creation/annihilation operators bosonic or fermionic modes according to their commutation relations, regardless of whether they arose from a quantum field or were just given in some other way.
