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- W2052802025 startingPage "3234" @default.
- W2052802025 abstract "This paper studies local boundary conditions for fermionic fields in quantum cosmology, originally introduced by Breitenlohner, Freedman, and Hawking for gauged supergravity theories in anti-de Sitter space. For a spin-textonehalf{} field, the conditions involve the normal to the boundary and the undifferentiated field. A first-order differential operator for this Euclidean boundary-value problem exists which is symmetric and has self-adjoint extensions. The resulting eigenvalue equation in the case of a flat Euclidean background with a three-sphere boundary of radius $a$ is found to be $F(E)={[{J}_{n+1}(mathrm{Ea})]}^{2}ensuremath{-}{[{J}_{n+2}(mathrm{Ea})]}^{2}=0$, $ensuremath{forall}nensuremath{ge}0$. Using the theory of canonical products, this function $F$ may be expanded in terms of squared eigenvalues, in a way which has been used in other recent one-loop calculations involving eigenvalues of second-order operators. One can then study the generalized Riemann $ensuremath{zeta}$ function formed from these squared eigenvalues. The value of $ensuremath{zeta}(0)$ determines the scaling of the one-loop prefactor in the Hartle-Hawking amplitude in quantum cosmology. Suitable contour formulas, and the uniform asymptotic expansions of the Bessel functions ${J}_{m}$ and their derivatives ${J}_{m}^{ensuremath{'}}$, yield, for a massless Majorana field, $ensuremath{zeta}(0)=frac{11}{360}$. Combining this with $ensuremath{zeta}(0)$ values for other spins, one can then check whether the one-loop divergences in quantum cosmology cancel in a supersymmetric theory." @default.
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- W2052802025 date "1991-05-15" @default.
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- W2052802025 title "Local boundary conditions for the Dirac operator and one-loop quantum cosmology" @default.
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- W2052802025 doi "https://doi.org/10.1103/physrevd.43.3234" @default.
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