Finite energy quantization on a topology changing spacetime

The “trousers” spacetime is a pair of flat two-dimensional cylinders (“legs”) merging into a single one (“trunk”). In spite of its simplicity this spacetime has a few features (including, in particular, a naked singularity in the “crotch”) each of which is presumably unphysical, but for none of which a mechanism is known able to prevent its occurrence. Therefore, it is interesting and important to study the behavior of the quantum fields in such a space. Anderson and DeWitt were the first to consider the free scalar field in the trousers spacetime. They argued that the crotch singularity produces an infinitely bright flash, which was interpreted as evidence that the topology of space is dynamically preserved. Similar divergencies were later discovered by Manogue, Copeland, and Dray who used a more exotic quantization scheme. Later yet the same result obtained within a somewhat different approach led Sorkin to the conclusion that the topological transition in question is suppressed in quantum gravity. In this paper I show that the Anderson-DeWitt divergence is an artifact of their choice of the Fock space. By choosing a different one-particle Hilbert space one gets a quantum state in which the components of the stress-energy tensor (SET) are bounded in the frame of a free-falling observer.

Figure 1

Constructing the trousers spacetime from a flat strip. The left leg ${\mathcal{M}}_L\equiv (t\le 0,x\in [-P,\mu P))$, the right leg ${\mathcal{M}}_R\equiv (t\le 0,x\in [\mu P,P))$, and the trunk ${\mathcal{M}}_T\equiv (t\ge 0,x\in (-P,P])$ are obtained by gluing together the rays bounding the corresponding strips. The white circles depict the removed points. They cannot be returned back after the relevant identifications are performed and thus a naked quasiregular singularity appears.

Figure 2

The slanted rays are the null geodesics at which the derivatives of functions of $\stackrel{\circ }{\mathcal{C}}$ are allowed to have jumps. In the dark and in the hatched regions ${\varphi }_{Rk}$ with, respectively, positive and negative $k$ are constant. In the complements to those regions ${\varphi }_{Lk}$ with $k$ of the same sign are constant.