### 3.14 Area law: generalization to higher spin fields

In this section, we will focus only on the leading UV divergent term, proportional to the area of the
horizon. The proportionality of the entanglement entropy to the area is known as the “area
law”. As we have discussed already for the case of a scalar field, this term in the entanglement
entropy of a black hole is the same as in flat spacetime. In flat Minkowski spacetime, for a field
of spin , massive or massless, including the gauge fields, the calculation of entanglement
entropy effectively reduces to the scalar field calculation, provided the number of scalar fields
is equal to the number of physical degrees of freedom of the spin- field in question. The
contribution of fermions comes with the weight 1/2. Thus, we can immediately write down the
general expression for the entanglement entropy of a quantum field of spin in dimensions,
where is (with weight 1/2 for fermionic fields) the number of physical (on-shell) degrees of freedom
of a particle of spin in dimensions. For gauge fields this assumes gauge fixing. In particular one has
for Dirac fermions,
for the gauge vector fields (including the contribution of ghosts), where is the dimension of the
gauge group,
for the Rarita–Schwinger particles of spin 3/2 with gauge symmetry (gravitinos), and
for massless spin-2 particles (gravitons).
The result for Dirac fermions was first obtained by Larsen and Wilczek [162, 163] and later in a paper
by Kabat [150]. The contribution of the gauge fields to the entropy was derived by Kabat [150]. The
entropy of the Rarita–Schwinger spin-3/2 particle and of a massless graviton was analyzed by Fursaev and
Miele
[110].