One-Loop Results for the Spin-1/2 Field with Local Boundary Conditions

Abstract

The PDF one-loop calculation for pure gravity subject to magnetic boundary conditions on S³ enables one to derive in a similar way the one-loop properties of fermionic fields subject to local boundary conditions on the three-sphere. For this purpose, we first study the function F which occurs in the nonlinear eigenvalue condition for the massless Majorana spin-1/2 field subject to these local boundary conditions on S³. Using the theory of canonical products, we prove that, in terms of squared eigenvalues, F still obeys a relation of the kind used for ζ(0) calculations in section 7.3. Using the parameter x → ∞, after the analytic continuation x → ix, and defining as usual $\alpha _m \equiv \sqrt {m^2 + x^2 }$, one can again expand asymptotically a formula of the kind log(Σ) as $\sum {_{n = 1}^\infty \frac{{A_n \left( {m,\alpha _m } \right)}} {{\left( {\alpha _m } \right)^n }}}$. However, the form of the coefficients A _n is more involved. In fact they are polynomials with both even and odd powers of t ≡ m/α. Five infinite sums can contribute to ζ(0). Using the contour formulae of section 7.3 and the uniform asymptotic expansions of the regular Bessel functions J _m and their first derivatives J′ _m , we find for a massless Majorana field: ζ(0) = 11/360. We also prove that no higher-order terms, i.e. A _n (m, α) when n > 3, can contribute to our value of ζ(0).

Using two-component spinor techniques, we finally prove that, if the Dirac operator preserves local boundary conditions involving the normal to the boundary and the spin-1/2 field, this implies another boundary condition involving the normal derivatives of the Majorana field, and the trace of the extrinsic-curvature tensor of the boundary. This is done so as to understand the relation between our results and previous work in the literature, and its foundations lie in the general theory of the Dirac operator.