1248 Progress of Theoretical Physics, Vol. 81, No.6, June 1989 A Diagrammatic Derivation of Coleman's Vanishing Cosmological Constant Akio HOSOYA Research Institute for Theoretical Physics, Hiroshima University, Takehara 725 (Received February 17, 1989) We present an alternative derivation of Coleman's expression for cosmological constant by summing over topologies of space-time in the wormhole dominance approximation. It is pointed out that the correlation between disconnected spaces, if any, vanishes by Coleman's mechanism. § 1. Introduction Very recently, Coleman l ) has proposed an interesting theory of zero cosmological constant. Suppose wormhole instanton solutions exist in a theory of sub-Planck field theory.2) Then the renormalized effective cosmological constant A relevant at large distances, i.e., the real one vanishes with infinite precision. (Recall that the obser- vationallimit is A/(Planck energy)4<10- l20 .) In his formulation, Coleman used the effective Lagrangian which contains anni- hilation and creation operators*) of baby universes corresponding to disappearance and appearance of wormholes. In the case of single smooth manifold attached by an arbitrary number of wormholes, Giddings and Strominger 3 ) and Coleman 4 ) explicitly proved that the use of annihilation and creation operators of baby universes is equivalent to diagrammatic wormhole instanton calculation. However, when we consider an arbitrary number of smooth closed connected manifolds as well as wormholes as Coleman did, it is not too obvious, at least to the present author, how the effective Lagrangian does the job of summation of such diagrams. It will be instructive if we can do explicit diagrammatic summation over all possible smooth closed connected manifolds bridged by wormhole instantons. In this paper, we are going to give an elementary diagrammatic derivation of Coleman's expression. The author believes that our alternative derivation is beneficial. For example, we can easily see how his conclusion of zero cosmological constant relies on the dilute gas approximation for multiple instantons. (We shall come back to this point at the end of this paper.) As a simple exercise, we show that the same mechanism of zero cosmological constant ensures the vanishing of correla- tion between disconnected spaces. To some extent, our present approach corresponds to Polyakov's5) in string theories, while Coleman's to string field theories. 6 ) They are actually equivalent in string theories and that is the case also in our present wormhole problem. We will not discuss the loss of quantum (in) coherence due to the baby universes, though it may be interesting. 2 )-4),7),9) *) The use of annihilation and creation operators for baby universes originated in Hawking's work. 7 ) Recently, the author has explored the possibility of the second (third?) quantization of universes. B) Downloaded from https://academic.oup.com/ptp/article-abstract/81/6/1248/1888183 by guest on 10 April 2019