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The contribution of physical degrees of freedom to
the one-loop amplitudes of Euclidean supergravity is here evaluated
in the case of flat Euclidean backgrounds bounded by a three-sphere,
recently considered in perturbative quantum cosmology.
In Euclidean supergravity, the spin-3/2
potential has the pair of independent spatial components
(psi_{i}^{A}, {widetilde psi}_{i}^{A'}).
Massless gravitinos are here
subject to the following local boundary conditions
on S**3: sqrt{2} ; {_{e}n_{A}^{; ; A'}}
psi_{i}^{A}=pm {widetilde psi}_{i}^{A'},
where {_{e}n_{A}^{; ; A'}} is the Euclidean
normal to the three-sphere boundary.
The physical degrees of freedom (denoted by PDF)
are picked out imposing the supersymmetry
constraints and choosing the gauge condition
e_{AA'}^{; ; ; ; ; i}psi_{i}^{A}=0,
e_{AA'}^{; ; ; ; ; i}{widetilde psi}_{i}^{A'}=0.
These local boundary conditions are then
found to imply the eigenvalue condition
{[J_{n+2}(E)]}^{2}-{[J_{n+3}(E)]}^{2}=0,
; forall n geq 0, with degeneracy
(n+4)(n+1). One can thus apply again a
zeta-function technique previously
used for massless spin-1over2 fields.
The PDF contribution to the full zeta(0) value is found to be
=-289/360. Remarkably, for the
massless gravitino field the PDF method and
local boundary conditions lead to
a result for zeta(0) which is equal to the PDF value one obtains
using spectral boundary conditions on S**3
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