Abstract
A definition for the entanglement entropy in both Abelian and non-Abelian gauge theories has been given in the literature, based on an extended Hilbert space construction. The result can be expressed as a sum of two terms, a classical term and a quantum term. It has been argued that only the quantum term is extractable through the processes of quantum distillation and dilution. Here we consider gauge theories in the continuum limit and argue that quite generically, the classical piece is dominated by modes with very high momentum, of order the cut-off, in the direction normal to the entangling surface. As a result, we find that the classical term does not contribute to the relative entropy or the mutual information, in the continuum limit, for states which only carry a finite amount of energy above the ground state. We extend these considerations for p-form theories, and also discuss some aspects pertaining to electric-magnetic duality.
Highlights
For an Abelian theory, the middle term in eq (1.1), depending on dia, is absent
For the electric centre case,2 by studying the correlation functions of the electric field on the boundary, we will find, both for the Abelian and non-Abelian cases, that this distribution is typically determined by very high momentum modes localised close to the boundary
In the continuum, the different sectors in the electric centre or the EHS definition are specified by the value of Tr En2 and other Casimirs on the boundary, where En is the normal component of the electric field
Summary
Let us begin by considering a 3+1-dimensional free U(1) theory. The entanglement entropy in this theory for a spherical region of radius R was studied in [5, 11,12,13,14,15]. One interesting possibility is that the theory flows to a non-trivial fixed point in the UV In this case, the behaviour of the two-point function of the electric field would be determined by its anomalous dimension at the UV fixed point with the short distance contribution taking the form. Suppose that the theory is strongly coupled in the UV and flows to a weakly coupled one, which is close to the free non-interacting theory, in the IR at an energy scale of order E ∼ Λ In such cases, one would still expect that the contribution to the classical term from modes with energies E ≤ Λ is approximately governed by the two-point function discussed above and the contribution of these modes should drop out in the mutual information or the rela-. When the z direction is non-compact though, Ez must vanish to keep the energy finite and such a contribution to the mutual information or relative entropy will not arise
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