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Register a new userOn Entanglement Entropy of Maxwell fields in 3+1 dimensions
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We consider entanglement entropy between two halves of space separated by a plane, in the theory of free photon in 3+1 dimensions. We show how to separate local gauge invariant quantities that belong to the two spatial regions. We calculate the entanglement entropy by integrating over the degrees of freedom in one half space using an approximation that assumes slow variation of the magnetic fields in longitudinal direction. We find that the entropy is proportional to the transverse area as expected. Interestingly the entanglement properties of the 2D transverse and longitudinal modes of magnetic field are quite different. While the transverse fields are entangled mostly in the neighborhood of the separation surface as expected, the longitudinal fields are entangled through an infrared mode which extends to large distances from the entanglement surface. This long range entanglement arises due to necessity to solve the no-monopole constraint condition for magnetic field.
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We calculate the entanglement entropy of a slab of finite width in the pure Maxwell theory. We find that a large part of entropy is contributed by the entanglement of a mode, nonlocal in terms of the transverse magnetic field degrees of freedom. Even though the entangled mode is nonlocal, its contribution to the entropy is local in the sense that the entropy of a slab of a finite thickness is equal to the entropy of the boundary plus a correction exponential in thickness of the slab.
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