Green's Functions on a Wormhole and Induced Operators

抄録

We have evaluated a Green's function of a minimally coupled scalar field on a wormhole background. The Green's function is written in terms of the eigenfunctions and eigenvalues of the Laplacian operator with a spherical symmetry, thus the problem being reduced to a one-dimensional potential barrier problem in quantum mechanics, and is finally solved by a numerical calculation. We have shown that the asymptotic form of the Green's function connected by a wormhole has the τ-dependence 〜 1/|τ|^2|τ'|^2. This τ-dependence is interpreted as an appearance of a pair of local operators in the wormhole ends, 〜ψ(p(x_0)ψ(x'_0). The procedure we used in this paper may be extended to calculate a Green's function of a field with higher spin, in which case the angular variables are separated with appropriate (spinor, vector, tensor, ・ ・ ・) harmonics and the eigenvalue equation is reduced to one-dimensional potential barrier problem just as the calculation of the scalar Green's function. Averaging over the relative angles between the two asymptotic regions is rather complicated, however.