Edge State Quantization: Vector Fields in Rindler

TL;DR

<3-5 sentence high-level summary> This work develops a comprehensive canonical quantization framework for Maxwell and Proca fields in Rindler space to understand horizon entanglement and the associated contact term. In 1+1 dimensions, Faddeev-Popov ghosts precisely cancel the thermodynamic contributions of unphysical polarizations, leaving a trivial bulk and a non-contributing omega=0 edge sector; in higher dimensions, edge states correspond to horizon electric flux punctures and large gauge transformations, reproducing the contact term and connecting to black hole microstates. The construction extends to Proca and tensor fields, linking edge degrees of freedom to asymptotic symmetries and soft hair, and offering a unified boundary-edge picture of black hole entropy via nonfactorization.

Abstract

We present a detailed discussion of the entanglement structure of vector fields through canonical quantization. We quantize Maxwell theory in Rindler space in Lorenz gauge, discuss the Hilbert space structure and analyze the Unruh effect. As a warm-up, in 1+1 dimensions, we compute the spectrum and prove that the theory is thermodynamically trivial. In d+1 dimensions, we identify the edge sector as eigenstates of horizon electric flux or equivalently as states representing large gauge transformations, localized on the horizon. The edge Hilbert space is generated by inserting a generic combination of Wilson line punctures in the edge vacuum, and the edge states are identified as Maxwell microstates of the black hole. This construction is repeated for Proca theory. Extensions to tensor field theories, and the link with Chern-Simons are discussed.

Figures (13)

• Figure 1: Rindler wedges and coordinate systems. At $T=0$, the Minkowski Hilbert space splits into the left and right Hilbert space. The Rindler observer then evolves these states using $H_R$ along the hyperbolic trajectories. This is one way of making sure the right observer does not see anything originating from the left wedge. Likewise, the left Hilbert space evolves upwards with $H_L$.
• Figure 2: Wilson line along $\mathcal{C}$ piercing a surface $\Omega$. The normal on the surface $n^\Omega$ and the orientation of the curve $\mathcal{C}$ determine whether the $\theta$-function evaluates to $+1$ or $-1$.
• Figure 3: Left: Two Wilson lines $\mathcal{W}_1$ and $\mathcal{W}_2$ at the same boundary point. Right: Closed Wilson loop $\mathcal{W}_{o} = \mathcal{W}_1\mathcal{W}_2^{-1}$.
• Figure 4: Cutting open a path integral along a curve, requires summing over the intermediate value of $\mathcal{E}$. In formulas: $Z = \int d\mathcal{E} e^{-A_1\frac{\mathcal{E}^2}{2}}e^{-A_2\frac{\mathcal{E}^2}{2}}$. Fixing the value of $\mathcal{E}$ for a manifold with boundary gives $Z(\mathcal{E}) \sim e^{-\frac{A_1}{2}\mathcal{E}^2}$.
• Figure 5: Volume-regularized path integral. The blue semi-disk and the red rectangle have equal area, and hence the final state can be constructed using both regions.