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 s, 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-s 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 s in d dimensions,
š’Ÿ (d) A(Σ ) S(s,d) = ------s-----d−2--d−2-, (126 ) 6(d − 2)(4π) 2 šœ–
where š’Ÿ (d) s is (with weight 1/2 for fermionic fields) the number of physical (on-shell) degrees of freedom of a particle of spin s in d dimensions. For gauge fields this assumes gauge fixing. In particular one has
[dāˆ•2] š’Ÿ (d) = 2---- (127 ) 1āˆ•2 2
for Dirac fermions,
š’Ÿ1 (d ) = (d − 2) ⋅ dim G , (128 )
for the gauge vector fields (including the contribution of ghosts), where dim G is the dimension of the gauge group,
2-[dāˆ•2] š’Ÿ3āˆ•2(d) = (d − 2) 2 (129 )
for the Rarita–Schwinger particles of spin 3/2 with gauge symmetry (gravitinos), and
d(d-−-3) š’Ÿ2 (d) = 2 (130 )
for massless spin-2 particles (gravitons).

The result for Dirac fermions was first obtained by Larsen and Wilczek [162, 163Jump To The Next Citation Point] and later in a paper by Kabat [150Jump To The Next Citation Point]. 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 Miele7 [110].


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