Witten and Boltzmann

From my Notion on Witten and Boltzmann

 It is currently 0916 hrs and I am incredibly tired at this point. I really should have slept for more time last night and right now I really should get sleep because it is impossible to function without it. Hopefully I can just go to the library in the next class and not have to worry about the next classes because I don’t have the patience for it. I have loads of work to do, and I am trying to get back into physics because I really have to get to work on the de Sitter algebras work that I was interested in. But whatever, I want to pen some thoughts on EBMs and something a lot of people are overlooking and on de Sitter I guess, with no intention of fully elaborating on either at the moment.

The first thing is that there is a nice relationship between entropy based sampling and min-$p$ directly, which results in a sort of ``sampling redundancy”. Loosely, start by noticing that $p(x)=\frac{\exp (-\beta E)}{Z}$, and the entropy

$$ S(p)=-\int p(x)\log p(x)\;. $$

Expand into

$$ S(p)=\beta ^{-1}\mathbb{E}_{p}[E(x)]+\mathcal{F}_{\beta }\;, $$

where we just substituted for $p(x)$ and identified the Helmholtz free energy. Then, the sampling algorithms are solved in terms of $\min E$: start by computing $E(\Sigma _{i})$, and then find min-$p$ as usual. However, you could instead just seek to minimize the free energy, since it implies selecting higher probability choices and tells you what the entropy is anyway. Working with $\mathcal{F}{\beta }$ rather than just $E$ is in general better because it incorporates $\beta$ and gives a more subtle relationship between the parameters and the entropy. Idk at this point I am too sleepy to continue but I guess I’ll want to do something with the loss functions to better understand Entropix as an EBM.

The other thing I am interested in is the algebra of observables in de Sitter space, and loosely trying to also find what $T\overline{T}+\Lambda _{2}$ deformations do to the OPEs and in general the operator algebras. Loosely, the problem is as follows: in AdS/CFT, we can take the operator algebra of all single-trace operators $\mathcal{O}{i}$, which is a type von Neumann algebra. In the large $N$ microcanonical ensemble, this algebra can be written as the crossed product with its modular group of automorphisms $G_{\mathcal{A}, 0}$, which would give us a type $\mathrm{II}_{\infty }$ algebra. This comes from the Takesaki theorem for the crossed product of type $\mathrm{III}_{1}$ algebras:

$$ \mathrm{III}_{1}\rtimes G_{\mathcal{A},0}=\mathrm{II}_{\infty }\;. $$

Why is this relevant? By obtaining a type $\mathrm{II}_{\infty }$ algebra and not a type $\mathrm{III}$ algebra, we can actually define entropy, traces and density matrices. One result from this is that the bulk generalized entropy $S_{\mathrm{gen}}(X)$ for the bifurcation surface of the two-sided AdS black hole is defined as the entropy $S_{E}(R)$ of the full right boundary algebra. A result that I had in coincidence with implications on Bousso, Chandrasekaran, etc. is that the entropy of appropriate boundary subspaces correspond to the generalized entropies of minimar surfaces, or more generally the apparent horizon. There’s some subtleties with why this construction works — for instance, the reason we use subspaces and not subalgebras is because including the Hamiltonian $H_{R}$ in the algebra $\mathcal{A}_{R}$ means that operator products like $\mathcal{O}_{i}\cdot H_{R}$ cannot lie in a closed subalgebra, or we would be able to reconstruct arbitrary nonlocal operators given access to only a particular entanglement wedge $\mathcal{E}_{W}(R)$. Now whether such a subspace of finite $G{N}$ operators or observables calculates the generalized entropy in de Sitter space is an open question (first remarked in Chandrasekaran, Penington and Witten’s paper on Large $N$ algebras and generalized entropy). Which is vaguely something I am interested in. But more on this later.

November 1, 2024, 1414 Polish time.