There is a loose set of ideas going under various names ("It from Qubit"; "Space from Hilbert Space"; "Geometry from Entanglement") which propose that spacetime is not fundamental, but instead emerges from the entanglement structure of an underlying Hilbert space (e.g. https://arxiv.org/abs/1606.08444). Instead of formulating a QFT on a background spacetime, we do away with spacetime altogether and hope that Lorentz-covariant QFT emerges in some appropriate limit.
These ideas are not new but seem to be gaining traction recently, with Sean Carroll being a vocal supporter.
However, if this approach were correct, it would seem to render Lorentz covariance a "coincidence": given a Hilbert space with some factorization into spatial sites, we shouldn't expect a randomly chosen Hamiltonian to generate anything like Lorentz invariant dynamics. Even if we constrain the Hamiltonian to be local (with respect to the factorization) we'll almost always end up with a non-Lorentzian evolution.
As a concrete example, consider the quantization of a real scalar field φ. We know that the Hamiltonian density $$ H = \Pi^2+|∇φ|^2+m^2φ^2 $$ gives us Lorentz-covariant dynamics, but unless we had the explicit goal of Lorentz-covariance, how would we know to choose this Hamiltonian over any other? This problem only gets worse when multiple fields are present, which we want to all be Lorentz-covariant with the same speed of light c. On the other hand, the Lagrangian/path integral approach is manifestly Lorentz-covariant from the start and doesn't face the same problem.
I'd be grateful if anyone can explain how "emergent spacetime" approaches deal with this issue - is there some "natural" constraint on the Hamiltonian (i.e. something stronger than just locality) which ensures Lorentz covariance? Or do these approaches just treat SR as a happy coincidence?
TL;DR - "Space from Hilbert Space" doesn't mesh naturally with SR. How should we expect to get Lorentz covariance to emerge?
