Local Finite-Dimensionality
The Hilbert spaces considered by physicists are often infinite-dimensional, from a simple harmonic oscillator to quantum field theories. However, there are good reasons from quantum gravity to think that the true Hilbert space of the universe is “locally finite-dimensional” [4].
That is, we can decompose H into a (possibly infinite) tensor product of finite-dimensional factors,
where for each a we have dim(Ha) < If we have factored the Hilbert space into the smallest possible pieces, we will call these “micro-factors.” The idea is that if we specify some region of space and ask how many states could possibly occupy the region inside, the answer is finite, since eventually the energy associated with would-be states becomes large enough to create a black hole the size of the region [5]. Similarly, our universe seems to be evolving toward a de Sitter phase dominated by vacuum energy; a horizon-sized patch of such a spacetime is a maximum-entropy thermal state [6] with a finite entropy and a corresponding finite number of degrees of freedom [7, 8].
There are subtleties involved with trying to map collections of factors in (4) directly to regions of space, including the fact that “a region of space” R might not be well-defined across different branches of the quantum-gravitational wave function. All that matters for us, however, is the existence of a decomposition of this form, and the idea that everything happening in one particular region of space on a particular branch is described by a finite-dimensional factor of Hilbert space HR that can be constructed as a finite tensor product of micro-factors Ha. Given some overall pure state | ) e H, physics within this region is described by the reduced density operator 
In that case, there is no issue of specifying the correct algebra of observables: the algebra is simply “all Hermitian operators acting on HR.” Any further structure must emerge from the spectrum of the Hamiltonian and the quantum state.
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