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Quantum Chaos, Operator Growth, and Holography


The exact role of the internal degrees of freedom (a.k.a. d.o.f.) in holography is not well-understood. Thus, in this thesis, we study a toy model of holography without space: the Sachdev-Ye-Kitaev (SYK) Model. This 0+1 theory of all possible 4-body interactions of N fermion "flavors"/"colors" features a low energy limit reproducing aspects of 1+1 Jackiw-Teitelboim gravity. First, we show that the inherent discreteness of the quantum spectrum results in universal late-time behavior due to eigenvalue repulsion. We then note that the theory's four-point functions probe the phenomenon of operator growth, where an internal d.o.f. goes on to epidemically evolve into larger products of internal d.o.f.s. In this manner where small operators smoothly grow into superpositions of increasing products of operators, we observe a sort of "size" locality, which is intimately tied with the notion of a conformal primary "descending" along its descendants. In fact, we find that the underlying structure of the SYK epidemic limits to that of a probe particle falling into a $AdS_2$ black hole. In other words, similar to how nearest neighbor interactions lead to dynamics on a flat space background, we demonstrate that many internal interactions lead to dynamics on a higher dimensional geometry.

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