For scalar and spinor fields, the one-loop correction to the
Bekenstein-Hawking entropy is equal to the entanglement entropy. For
gauge fields, the one-loop correction to the Bekenstein-Hawking entropy
contains a negative divergent piece (a "contact term") that makes the
entropy negative. This contact term has no known statistical
interpretation. Most surprisingly, the divergence appears in two
dimensions, where gauge fields have no local degrees of freedom. We show
that in two dimensions the contact term results from an incorrect
treatment of ghost zero modes. In a manifestly gauge-invariant reduced
phase space quantization, the gauge field contribution to the
Bekenstein-Hawking entropy is positive, finite, and equal to the
entanglement entropy.
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