Choice of Boundary Conditions in One-Loop Quantum Cosmology
The problem of boundary conditions in a supersymmetric theory of quantum cosmology is studied. For fermionic fields one has a choice of local or nonlocal boundary conditions. N = 1 supergravity is the simplest supersymmetric model, which we study in a background cosmological model which is flat Euclidean space bounded by a three-sphere. We pick out the physical degrees of freedom by imposing the supersymmetry constraints and choosing a gauge condition. A set of naturally occurring nonlocal boundary conditions for fermions is such that the coefficients of the physical degrees of freedom are regular at the origin, and half of them are set equal to zero on S 3. However, supersymmetry does not relate them to Dirichlet or other local boundary conditions for the gravitational field. Studying N =1 supergravity at 1 loop about flat space, this approach is shown to yield a nonvanishing result for the PDF ζ(0). Namely, the PDF contribution to the prefactor due to the spin-3/2 field is proportional to a 289/360 (a being the three-sphere radius), which does not cancel a −278/45 due to the gravitational field subject to Dirichlet boundary conditions for the perturbed three-metric.
We therefore study possible local boundary conditions for both bosons and fermions. One set, related by supersymmetry, was originally introduced by Breitenlohner, Freedman and Hawking for gauged supergravity theories in anti-De Sitter space. It involves the normal to the boundary and field strengths for spins s = 1, 3/2, 2, while for s = 0 it involves a complex scalar field and its normal derivative, and for s = 1/2 the undifferentiated field. Using twistor theory in flat space, the existence is proved of a form of the spin-lowering operator which preserves these local boundary conditions required on S 3 for solutions to the massless free-field equations for adjacent spins s and s + 1/2.
The eigenvalue condition implied by these boundary conditions for the massless spin-1/2 field is found to be : J n +1(Ea) = ± J n +2(Ea), ∀n ≥ 0, with degeneracy (n + 2)(n + 1). Using the formalism of SU(2) spinors in Euclidean four-space and a theorem due to von Neumann, we also prove that a first-order differential operator for this boundary-value problem exists which is symmetric and has self-adjoint extensions.
Moreover, in the case of the spin-1 field strength these local boundary conditions are shown to lead to the following PDF values of ζ(0): −77/180 (magnetic) and 13/180 (electric), whereas for a complex scalar field one finds: ζ(0) = 7/45. Thus the values of ζ(0) are not equal for all spins. As a partial justification of this result, we prove why, in the comparison spin 0 vs spin 1/2, the spin-lowering operator does not lead to the same eigenvalues. Finally, we discuss in detail the issue of the preservation in time of the gauge constraint used in evaluating ζ(0).
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