Notes on Some Entanglement Properties of Quantum Field Theory

TL;DR

Edward Witten's notes investigate entanglement in quantum field theory through the lens of algebraic observables, treating entanglement as a property of local algebras rather than states. By employing Tomita-Takesaki theory, relative entropy, and modular flows, the work provides a rigorous framework for understanding region-based entanglement, monotonicity, and holographic-like phenomena, including the Bisognano-Wichmann wedge and Unruh effect. The text emphasizes that local algebras in QFT are of Type III (often Type III_1), which naturally explains universal UV entanglement divergences and the absence of a simple Hilbert-space factorization, with implications for area laws and black hole thermodynamics. It also connects finite-dimensional analogies via GNS constructions and explores how factorized representations relate through canonical maps and the GNS framework, illustrating the depth and subtlety of entanglement in field-theoretic settings.

Abstract

These are notes on some entanglement properties of quantum field theory, aiming to make accessible a variety of ideas that are known in the literature. The main goal is to explain how to deal with entanglement when -- as in quantum field theory -- it is a property of the algebra of observables and not just of the states.

Figures (8)

• Figure 1: (a) A function $g(u)$ holomorphic in the upper half $u$ plane can be computed by a Cauchy integral formula: any contour $\gamma$ in the upper half-plane can be used to compute $g(u)$ for $u$ in the interior of $\gamma$. (b) If $g(u)$ is continuous on the boundary of the upper half-plane, one can take $\gamma$ to run partly along the boundary. If in addition $g(u)=0$ along part of the boundary -- indicated here by dashed lines -- then that part of the contour can be dropped. In this case, the Cauchy integral formula remains holomorphic as $u$ is moved through the gap and into the lower half-plane, implying that $g(u)$ is holomorphic on that part of the real axis and is identically zero.
• Figure 2: (a) An open set ${\mathcal{U}}$ in Minkowski spacetime, and its domain of dependence $\widehat{{\mathcal{U}}}$ (the union of ${\mathcal{U}}$ with the regions labeled as $\widehat{{\mathcal{U}}}$ in the figure), which in this case is a causal diamond and coincides with the causal completion ${\mathcal{U}}"$ of ${\mathcal{U}}$. (b) The two open sets ${\mathcal{U}}$ and ${\mathcal{U}}'$ are causal complements; each is the largest open set that is spacelike separated from the other. (c) A quite different open set ${\mathcal{U}}$ whose causal completion ${\mathcal{U}}"$ (the union of ${\mathcal{U}}$ and the regions labeled ${\mathcal{U}}"$) is the same causal diamond as in (a).
• Figure 3: The right wedge ${\mathcal{U}}_r$ and the left wedge ${\mathcal{U}}_\ell$ in Minkowski spacetime. They are the domains of dependence of the right half and left half of the initial value surface $t=0$, which are labeled as ${\mathcal{V}}_r$ and ${\mathcal{V}}_\ell$.
• Figure 4: (a) The path integral on the half-space $\tau<0$ as a function of boundary values of the fields gives a way to compute the vacuum wavefunction $\Omega$. (b) To compute the reduced density matrix of the vacuum for the right half of the surface $\tau=0$ by a Euclidean path integral, we use the path integral on the lower half-space $\tau<0$ to compute a vacuum bra $\langle \Omega|$, and the path integral on the upper half-space $\tau>0$ to compute a vacuum ket $|\Omega\rangle$. Then we glue together the left halves of the boundaries of the $\tau<0$ and $\tau>0$ half-spaces, identifying the field variables on those boundaries in the bra and the ket. The net effect -- a path integral on the upper half-space and the lower half-space together with an integral over field variables on half of the $\tau=0$ hypersurface -- produces a path integral on the space depicted here. It can be obtained from Euclidean space ${\mathbb{R}}^D$ by making a "cut" along the half-hyperplane $\tau=0$, $x\geq 0$. (c) Sketched here is a Euclidean wedge of opening angle $\theta$.
• Figure 5: (a) The state ${\sf a}|\Omega\rangle$ can be obtained by a path integral in the lower half plane, with ${\sf a}$ inserted on the right half of the boundary. (b) Acting with $\exp(2\pi \alpha K_\ell) \exp(-2\pi \alpha K_r) {\sf a}|\Omega\rangle$ adds a wedge of opening angle $2\pi\alpha$ to the right boundary and removes one from the left boundary. If we rotate the picture so that the boundary is again horizontal, it looks like this; the operator ${\sf a}$ is now inserted on a ray that is at an angle $2\pi \alpha$ from the horizontal. (c) By the time we get to $\alpha=1/2$, ${\sf a}$ is inserted on the left boundary of the lower half plane. We cannot extend this process farther.