# How does the "Space from Hilbert space" program incorporate Lorentz invariance?

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?

• Hi Jacob! Welcome to Physics SE and thanks for asking a question I've been hoping to be explained to me for a while :) A general suggestion: always include links to the arXiv homepage of a preprint rather than to its PDF. People often don't want to straightaway download the paper (this is an issue especially for phone where a link to the PDF downloads the paper straightaway) ;)
– user87745
May 3, 2020 at 14:29
• Ah, my bad. Thanks for pointing that out. May 3, 2020 at 14:45
• You may be interested in this answer of mine: physics.stackexchange.com/a/521712/30833 May 3, 2020 at 17:19
• May 11, 2020 at 1:31