9.2.1.6 Quantum focusing conjecture QNEC

^[1]

The SYK model

The SYK model is a model of N Majorana fermions $\psi_i$ with all-to-all $p$-local interactions defined by \begin{align} &H_{SYK} =i^{p / 2} !!!!!!!!\sum_{1 \leq i_{1} <\cdots < i_{p} \leq N} !!!!!!J_{i_{1} \ldots i_{p}} \psi_{i_{1}} \cdots \psi_{i_{p}},
&\left{\psi_{i}, \psi_{j}\right}=2 \delta_{i j},
&\left\langle J_{I}J_{K}\right\rangle =\frac{\mathcal{J}^2}{ {N \choose p} }\delta_{IK}=\frac{N}{2p^2}\frac{\mathbb{J}^2}{ {N \choose p} }\delta_{IK} \end{align} where $\mathcal{J}$, $\mathbb{J}$ are different normalization conventions and $I=(i_{1}, \ldots ,i_{p})$. Some people also normalize the fermions to remove the factor of $2$. By $p$-local, we mean each term in the Hamiltonian contains $p$ Majorana fermions.

Instead of the Hamiltonian, we could equivalently define the model with the Euclidean ($\tau=i t$) action $S_{\mathrm{S Y K}}=\int d \tau\left( \frac{1} {2} \sum_{i=1}^{N} \psi_{i} \frac{d} {d \tau} \psi_{i}+i^{p / 2} \sum_{I} J_{I} \psi_{I} \right)$.

For the Hamiltonian to be Hermitian, $p$ must be even. Let’s also assume that $N$ is even; then $\psi_i$ are square matrices with size $2^{N/2}$. $\psi_i:\mathbb{R}\to \mathbb{C}^{2^{N/2} \times 2^{N/2}}$.

\todo{

Note2

Lorentzian action is $S_{\mathrm{S Y K}}=\int dt\left( \frac{1} {2} \sum_{i=1}^{N} \psi_{i} i\frac{d} {d t} \psi_{i}-i^{p / 2} \sum_{I} J_{I} \psi_{I} \right)$. Note that since the action is linear in terms of $\frac{d} {d t} \psi_{i}$, there will not be a kinetic term in the Hamiltonian. }

For $N=2r$, we can define the following representation using the Kronecker product of Pauli matrices. For $N=2$, they become $\psi_{1}=\sigma_{1}$ and $\psi_{2}=\sigma_{2}$. \begin{align} \psi_{1}& =\sigma_{1} \otimes1 \otimes\cdots\otimes1
\psi_{2}& =\sigma_{3} \otimes \sigma_{1} \otimes\cdots\otimes1
&\;\;\vdots \notag
\psi_{r}& =\sigma_{3} \otimes\sigma_{3} \otimes\cdots\otimes\sigma_{1}
{\psi_{r+1}} & =\sigma_{2} \otimes1 \otimes\cdots\otimes1\ {\psi_{r+2}} & =\sigma_{3} \otimes\sigma_{2} \otimes\cdots\otimes1
&\;\;\vdots \notag
\psi_{2 r}&=\sigma_{3} \otimes\sigma_{3} \otimes\cdots\otimes\sigma_{2} \end{align}

\etocsettocdepth.toc{subsubsection}

\paragraph{Feynman diagrams and the melonic dominance}

\paragraph{Schwarzian theory} \paragraph{Four point function}

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