That's an explanation proposed for why we can barely measure any gravity stemming from the huge energy contained in the zero-point field. The idea is that the fermion field (matter) and the boson field (energy) cancel each other out. This theory is part of supersymmetry.

This answer is slightly misleading. Fermionic fields have negative vacuum energy, while bosonic ones have positive. This is an basic fact of quantum field theory, a consequence of Lorentz invariance and Fermi-Dirac statistics (resp. Bose-Einstein statistics).

Supersymmetry makes use of this fact: When the bosons and fermions come in pairs, their vacuum energies can cancel because they have opposite sign. But the association of minus signs to fermions and plus signs to bosons is older and more elementary than supersymmetry.

I wish I had an elementary conceptual explanation for why this association exists. It's a simple calculation, but hard to describe without resorting to equations. Maybe the best I can say is that when one writes down the energy function for a quantum mechanical system of many particles, it has a form which is _strongly_ constrained by special relativity. The variables which describe the creation of particles are perfectly paired with the variables which describe the annihilation of particles. For bosons, these variables commute, just like ordinary functions. For fermions, these variables anti-commute, picking up a sign when you exchange them. If you repeatedly exchange these variables to rewrite energy function as vacuum energy + energy of 1 particle + energy of 2 particles + ...., you discover that the fermions contribute negatively to the vacuum energy while bosons contribute positively.

Thanks. This is what I was looking for when I originally posed the question.

So, if I understood you correctly, the restriction of Lorentz symmetry on the Hamiltonian in turn requires QFT creation and annihilation operators to commute/anticommute (the anticommute part what then ultimately drives a negative energy contribution from fermionic fields when summing over all possible energy states).

What causes the anticommute properties? Is there a non-abelian symmetry group backing fermion fields somewhere?

That's about right. There's a tiny bit of confusion

The anticommutation of fermions is a consequence (the content, really) of the spin-statistics theorem, which is a consequence of Lorentz invariance & unitarity.