Edge state quantization: vector fields in Rindler

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.

Fine structure of Jackiw-Teitelboim quantum gravity

“…The special case of β L = 0 can be interpreted as a Wilson line stretching from the holographic boundary to the black hole horizon, the fact that the resulting amplitude vanishes is a manifestation of the fact that bulk operators do not couple to horizon degrees of freedom [47,40]. Taking the inner boundary to be the Schwarzian instead, we find: 45…”

“…The appearance of these volume factors have been subject to critique [25,49], hindering a genuine Hilbert space interpretation of such symplectic path integrals. 36 Its norm is indeed the Schwarzian partition function Z, when using δ(k − k) = V C and dim k = V V C ρ(k), as explained in more detail in appendix C. 37 See for example [43,47]. 38 Given the degree of freedom f , one has no information whatsoever on the precise microstate underlying this state.…”

“…The boundary surface can be made transparent, or equivalently the manifolds can be glued together by taking the trace in the extended Hilbert space associated with the edge degrees of freedom (see e.g. [41,42,43,44,45,46,47,40,48] and references therein). As a byproduct we establish that the spectrum of JT gravity contains one-sided black hole states; unlike the Schwarzian theory which on its own is insufficient to describe the Hilbert space of one-sided JT black holes.…”

Clocks and rods in Jackiw-Teitelboim quantum gravity

“…The radar definition of the bulk points allows us to only access the exterior of the bulk 27 Explicit formulas are easily obtained. 28 For related comments in gauge theories, see [70,71,29]. 29 This is a manifestation of the well-known pathological IR properties of the 2d scalar propagator.…”

“…. (D.9) 70 One should be careful in applying this formula to the connected geometries to Taylor expand only this part, and not for example the factors cos 2πµ √ E associated to the geodesic boundaries in (D.3). Indeed, the Riemann-Lebesgue inspired argument we are using based on factors such as exp(iEt) says that for large t, the integrals over E are dominates by E ∼ 1/t 1.…”