### 3.15 Renormalization of entropy due to fields of different spin

The effective action of a field of spin can be written as
The second-order covariant operators acting on the spin- field can be represented in the following general
form
where the matrices depend on the chosen representation of the quantum field and are linear in the
Riemann tensor. Here are some examples [43, 44]
where are gamma-matrices. The coefficient in the small expansion (67) – (69) of the heat
kernel of operator (132) has the general form
where is the dimension of the representation of spin ,
can be interpreted as the number of off-shell degrees of freedom.
Let us consider some particular cases.

##### Dirac fermions ().

The partition function for Dirac fermions is . In this case
and hence
where was introduced in Eq. (127). We note that the negative sign in Eq. (136) in
combination with the negative sign for fermions in the effective action (131) gives the total positive
contribution to Newton’s constant. The renormalization of Newton’s constant due to Dirac fermions is
Comparison of this equation with the UV divergence of entropy (126) for spin-1/2 shows that the leading
UV divergence in the entropy of spin-1/2 field is handled by the renormalization of Newton’s constant in
the same manner as it was for a scalar field.

##### The Rarita–Schwinger field ().

The partition function, including gauge fixing and the
Faddeev–Popov ghost contribution, in this case, is
so that the appropriate heat kernel coefficient is
where is introduced in Eq. (129). The renormalization of Newton’s constant
then, similarly to the case of Dirac fermions, automatically renormalizes the entanglement entropy (126).
However, this property, does not hold for all fields. The main role in the mismatch between the UV
divergences in the entanglement entropy and in Newton’s constant is played by the non-minimal coupling
terms , which appear in the field operators (132).