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.
Highlights
One of the most important open problems in black hole physics is to provide for an understanding of black hole entropy in terms of microscopic degrees of freedom
Assuming that string theory is the fundamental theory of nature, the microscopic degrees of freedom responsible for black hole entropy must be of a stringy nature [1]
One way to quantize the edge sector that results in the correct final partition function (7.13) is as follows: we further extend the Hilbert space of the theory to allow states associated with a radial canonical flux that varies linear in time, which is still a solution of the classical scalar equations of motion of the field φω,k composing the vector field (7.4)
Summary
One of the most important open problems in black hole physics is to provide for an understanding of black hole entropy in terms of microscopic degrees of freedom. This is in unison with the wellknown fact that 1 + 1 dimensional CFTs have well-known formulae for the entanglement entropy, irrespective of the possible presence of a gauge symmetry Before discussing these important aspects of Maxwell theory in Rindler, we first scrutinize an issue that arose in the literature for the 1 + 1 case. We commence this work by presenting a canonical quantization of Maxwell theory in 1+1 dimensional Rindler in Lorenz gauge and demonstrate that the Faddeev-Popov ghosts exactly cancel the two unphysical photon polarizations, as they should by construction. Canonical quantization of Lorenz gauge Maxwell and Proca theory in d + 1 dimensional Rindler space, construct the Hilbert space, including edge states and give physical meaning to this sector. A complementary path integral perspective on the problem is given in [50], where it is generalized beyond the Abelian Rindler set-up
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