Complementarity in Wormhole Chromodynamics ⋆

The electric charge of a wormhole mouth and the magnetic flux “linked” by the wormhole are non-commuting observables, and so cannot be simultaneously diagonalized. We use this observation to resolve some puzzles in wormhole electrodynamics and chromodynamics. Specifically, we analyze the color electric field that results when a colored object traverses a wormhole, and we discuss the measurement of the wormhole charge and flux using Aharonov-Bohm interference effects. We suggest that wormhole mouths may obey conventional quantum statistics, contrary to a recent proposal by Strominger. ⋆ This work supported in part by the U.S. Department of Energy under Grant No. DE-FG0392-ER40701 † [email protected] ‡ [email protected] § [email protected] Introduction: Many years ago, Wheeler [1] and Misner and Wheeler [2] proposed that electric field lines trapped in the topology of a multiply-connected space might explain the origin of electric charge. Consider a three-dimensional space with a handle (or “wormhole”) attached to it, where the cross section of the wormhole is a two-sphere. On this space, the source-free Maxwell equations have a solution with electric field lines caught inside the wormhole throat. One mouth of the wormhole, viewed in isolation by an observer who is unable to resolve the small size of the mouth, cannot be distinguished from a pointlike electric charge. Only when the observer inspects the electric field more closely, with higher resolution, does she discover that the electric field is actually source free everywhere. It is also interesting to consider what happens when a charged particle traverses a wormhole. ¶ (Of course, this “pointlike” charge might actually be one mouth of a smaller wormhole.) Suppose that, initially, the mouths of the wormhole are uncharged (no electric flux is trapped in the wormhole). By following the electric field lines, we see that after an object with electric charge Q traverses the wormhole, the mouth where it entered the wormhole carries charge Q, and the mouth where it exited carries charge −Q. Thus, an electric charge that passes through a wormhole transfers charge to the wormhole mouths. In this note, we will address two (closely related) puzzles associated with this type of charge transfer process. Our first puzzle concerns the quantum mechanics of charged particles in the vicinity of a wormhole. We can compute the amplitude for the particle to propagate from an initial position to a final position by performing a sum over histories. Naively, one would expect this sum to include histories that traverse the wormhole, and that the contribution to the path integral due to these histories should be combined coherently with the contribution due to histories that ¶ Note that we are assuming that the wormhole is traversable, in violation of the “topological censorship” theorem [3] that can be proved in classical general relativity (with suitable assumptions about the positivity of the energy-momentum tensor). This traversability might be enforced by quantum effects. Alternatively, the reader might prefer to envision our space as a thin two-dimensional film, containing objects with Aharonov-Bohm interactions. Such wormholes might actually be fashioned in the laboratory!