Annu. Rev. Astron. Astrophys. 1992. 30:
499-542 Copyright © 1992 by . All rights reserved |
5.1 Wormholes and the Cosmological Constant
One of the most provocative explanations for the small value of the cosmological constant invokes quantum cosmology and fluctuations in the topology of spacetime known as ``wormholes.'' Although we will not give a technical description of the relevant arguments, we will try to give a pedagogical introduction to the essential ideas and the troubles with their realization.
Quantum cosmology is the study of the universe as a quantum gravitational system (Wheeler 1968, DeWitt 1967). Since one does not have a consistent quantum theory of gravity, it is common to use approximation schemes based on Feynman's path integral formulation of quantum mechanics (Feynman & Hibbs 1965). In this picture, we compute the wavefunction for a particle with initial state _{0} to be in state by integrating over all paths that connect the two states:
where p is a path from _{0} to , and S[] is the action for the path. In quantum cosmology, a ``state'' is a three-dimensional slice of a four-dimensional spacetime, and the wave function of a particle is replaced by the ``wave function of the universe'' (), which is the probability amplitude that the universe contains .
Since an oscillating integral such as that of Equation 35 will generally not converge, it is common to analytically continue the time parameter to imaginary values: t -> i. This transformation changes the signature of the metric from (- + + +) to (+ + + +), so the resulting paths are in a Euclidean space rather than a Lorentzian one. At the same time, the action becomes imaginary, so that we may write S -> i S_{E}, where S_{E} is called the Euclidean action. The path integral is then damped by a decaying exponential, and will converge if S_{E} is bounded below. In quantum cosmology, this transformation implies that we should integrate over manifolds of Euclidean signature rather than Lorentzian spacetimes. We therefore compute the wave function of the universe via
where M is a four-dimensional Euclidean space containing a three dimensional slice . (We will not discuss the contentious issue of boundary conditions; see Hartle & Hawking 1983; Vilenkin 1982, 1988; Linde 1984).
Observation tells us that our universe is large and smooth on a global scale; therefore, our next step is to estimate the integral in Equation 36 for three-surfaces which are large and smooth. Although performing the integral is well beyond our capabilities, it is possible to estimate it as the exponential of an ``effective action'' [M]. The effective action may be thought of as an action with all quantum fluctuations integrated out: [dM] exp (- S_{E}[M] / ) = exp (-[M_{c}] / ), where M_{c} is the ``classical'' space, for which is stationary. As Coleman (1988b) points out, it may not seem very useful to define a function () in terms of a path integral over another function (S_{E}) which we do not know; however, an approximate expression for for large spaces is known. The leading terms are simply those of the (Euclidean) action for general relativity:
where g is the determinant of the metric g_{µv} and R is the Ricci scalar (R g^{µv} R_{µv}). This expression may be roughly thought of as a power series expansion in the inverse size of the space; for large manifolds, gravitation is always dominant, and we may neglect terms representing other fields.
Since Equation 37 is simply the action for general relativity, its stationary point is the solution to Einstein's equations with cosmological constant; in Euclidean space, this is a four-dimensional sphere. For such spaces, R = 4 and d^{4} x g = 24^{2} / ^{2}. Inserting these into Equation 37 yields = - 3 / G. Since the path integral in Equation 36 is the exponential of - / , we have
If we consider as an independent parameter, this expression is infinitely peaked at = 0: the cosmological constant problem is solved! The answer is simply that universes in which = 0 dominate the path integral, making it overwhelmingly probable that the cosmological constant vanishes.
However, the cosmological constant may not normally be thought of as a free parameter. Hawking (1984; see also Baum 1984) proposed a field which would contribute to the action in such a way as to mimic a cosmological constant; this field would be varied in the path integral, turning into a free parameter and making Equation 38 the wave function of the universe. However, there is no compelling reason to believe in the existence of such a field (except for solving the cosmological constant problem).
A more natural mechanism for making a free parameter is provided by wormholes - topologically nontrivial spacetime geometries. Roughly, a wormhole may be thought of as a thin tube which connects two separated regions of a Euclidean space (see Figure 11). (The Euclidean wormholes we consider are distinct from wormholes which connect spatial regions in a Lorentzian geometry; see Wheeler 1964 and Morris et al 1988.) Since the action for an infinitesimally small wormhole is negligible, manifolds consisting of large spheres connected by wormholes are approximate stationary points of the effective action (Equation 37), and therefore contribute to the path integral; the effect of these configurations has been the object of some debate (Hawking 1979, 1982, 1988; Teitelboim 1982; Strominger 1984; Gross 1984; Lavrelashvili et al 1987). A resolution was provided by Coleman (1988a) and Giddings & Strominger (1988), who found that wormholes induced a distribution of values for all the constants of nature-precisely what is necessary to solve the cosmological constant problem. In other words, the interaction of our ``universe'' with other universes through wormholes allows the cosmological constant to attain a range of values; since the effective action is stationary at = 0, this value is singled out. (One must not take this concept too literally - the universes being spoken of are fictional Euclidean spaces used to calculate a path integral, not alternate worlds that coexist with our own.)
Figure 11. Example of a Euclidean space that contributes to the Feynman path integral for quantum cosmology. This manifold consists of large spheres connected by wormholes. |
This is essentially the argument assembled by Coleman in his celebrated paper (Coleman 1988b). (See also Banks 1988: for later variations, see Accetta et al 1989, Adler 1989, Elizalde & Gaztanaga 1990, Hosoya 1989, Kosower 1989, Rubakov 1988, Unruh 1989b, and Veneziano 1989.) One subtlety arises because there are many connected spheres contributing to the path integral; the associated combinatorics makes the wave function of the universe a double exponential, ~ exp (e^{3 / G}) this displays the infinite peak at = 0 in an even more impressive way. As a solution to the cosmological constant problem, this proposal has at least two very favorable features. First, although highly speculative physics is essential to the argument, there was no need to introduce any new laws or invent new phenomena; all that was necessary was to include in the path integral wormhole configurations which should be there anyway. Second, the communication with other large universes explains how our universe ``knows ahead of time'' to set = 0 at low temperature, rather than at early times. In Coleman's phrase, ``prearrangement is replaced by precognition'' (Coleman 1988b).
At the same time, there are many unanswered questions relating to Coleman's proposal; we will mention just a few. Before looking at worm-holes specifically, it is worth noting a long-standing problem of quantum cosmology: The Euclidean action for gravitation is not bounded below, and therefore the path integral of Equation 36 does not converge. Many remedies to this problem have been proposed, including allowing to vary along a complex contour (Gibbons et al 1978), adding additional terms to the action such that it becomes bounded below (Horowitz 1985), or staying in Lorentzian-signature space all along (Farhi 1989, Strominger 1989).
Unfortunately, the solution to the cosmological constant problem seems to depend intimately on the ``wrong'' sign for the action (Giddings & Strominger 1989), and attempts to base analogous calculations in Lorentzian space do not find a peak at = 0 (Fischler et al 1989, Cline 1989).
If we accept for the moment the viability of Euclidean quantum gravity, there is still some question of the reliability of approximating the path integral by large spheres connected by small wormholes. V. Kaplunovsky (unpublished) and Fischler & Susskind (1989) have suggested that large wormholes may dominate small ones; since then a debate has raged back and forth with no clear winner (Preskill 1989; Coleman & Lee 1989, 1990; Polchinski 1989a; Iwazaki 1989). Equally troubling is an argument by Polchinski (1989b) that the integration over spheres connected by worm-holes induces a phase (-i)^{d + 2} in the wave function, where d is the dimension of spacetime. Thus, in a four-dimensional universe ~ exp (-e^{3 / G}), which exhibits no peak at = 0. Lastly, several authors have explored a suggestion by Coleman (1988b) that wormholes may determine all of the constants of nature. To date, attempts to implement this plan have not met with great success (Preskill 1989, Hawking 1990, Klebanov et al 1989, Preskill et al 1989).
These results serve to emphasize that quantum cosmology is an ambitious but unsettled subject, insufficiently developed for crucial questions to be definitively answered. The solution to the cosmological constant problem offered by wormholes is certainly elegant as well as provocative; only further work will allow us to judge its physical relevance.