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Quantum Dynamics of Lorentzian Spacetime Foam

A simple spacetime wormhole, which evolves classically from zero throat radius to a maximum value and recontracts, can be regarded as one possible mode of fluctuation in the microscopic ``spacetime foam'' first suggested by Wheeler. The dynamics of a particularly simple version of such a wormhole can be reduced to that of a single quantity, its throat radius; this wormhole thus provides a ``minisuperspace model'' for a structure in Lorentzian-signature foam. The classical equation of motion for the wormhole throat is obtained from the Einstein field equations and a suitable equation of state for the matter at the throat. Analysis of the quantum behavior of the hole then proceeds from an action corresponding to that equation of motion. The action obtained simply by calculating the scalar curvature of the hole spacetime yields a model with features like those of the relativistic free particle. In particular the Hamiltonian is nonlocal, and for the wormhole cannot even be given as a differential operator in closed form. Nonetheless the general solution of the Schr\"odinger equation for wormhole wave functions, i.e., the wave-function propagator, can be expressed as a path integral. Too complicated to perform exactly, this can yet be evaluated via a WKB approximation. The result indicates that the wormhole, classically stable, is quantum-mechanically unstable: A Feynman-Kac decomposition of the WKB propagator yields no spectrum of bound states. Though an initially localized wormhole wave function may oscillate for many classical expansion/recontraction periods, it must eventually leak to large radius values. The possibility of such a mode unstable against growth, combined with


I. INTRODUCTION
A quantum description of gravitation is one of the most eagerly sought goals of present-day physics. Approaches to such a description may be loosely classified as "grand schemes" and "small schemes." The former are attempts at a comprehensive quantum theory of gravitation, such as supergravity or superstring theory, canonical quantization of general relativity with or without new variables, et cetera.
The latter are attempts to analyze the quantum-gravitational physics of particular, simple systems, in hopes of understanding features expected to emerge from a complete, general theory. These small schemes include the extensive "minisuperspace" quantum-cosmology program [1], the numerous calculations of the effects of Planckscale spacetime wormholes and "baby universes" on the constants of macroscopic physics [2], and the present work, in which we seek to probe the dynamics of such a Planck-scale "foam" structure itself. An outline of this investigation has been published previously [3]; here we present the calculations and results in detail.
The concept of "spacetime foam," first suggested by Wheeler [4] some 35 years ago, has become standard lore in our quantum picture of gravitation: On scales of the Planck length, quantum fluctuations of the spacetime geometry become so dominant that spacetime takes on all manner of nontrivial topological structure, such as wormholes and handles; smooth, simply connected spacetime only emerges as a classical limit on larger scales. Yet in the intervening decades our understanding of quantum gravity still has not advanced sufficiently to prove or disprove this famous conjecture.
The difficulties inherent in such a proof, or in any detailed treatment of spacetime foam, are well known. Topological change in a Lorentzian manifold necessarily entails the complications of singularities or degeneracy of the geometry, or of closed timelike curves [5]. To circumvent these problems one may consider Euclidean manifolds instead [1]. Euclidean quantum gravity is physically distinct from Lorentzian, unlike ordinary quantum field theory in which the two formulations are equivalent via analytic continuation. Indeed on a quantum level it might be that spacetime is fundamentally Euclidean, with familiar Lorentzian spacetime only a classical limit attained far from the Planck regime. But Euclidean quantum gravity has its own formidable problems [6], including the failure of the Euclidean action to be positive definite or bounded below, the problem of interpretation, and the matter of recovering Lorentzian spacetime.
Both versions of spacetime foam continue to be subjects of great interest.
Much recent work on Euclidean foam has explored its possible role in determining the fundamental constants of nature [2]. The suggestion that Lorentzian wormholes, extracted from microscopic foam and suitably enlarged, might conceivably be used for space or time travel [7] has inspired a great deal of recent and ongoing work on issues of causality and consistency in spacetime physics. Both sets of ideas have profound significance for fundamental physics, but both regard the spacetime foam itself as a given, neither probing the questions of its existence or structure substantially.
This work addresses the dynamics of Lorentzian spacetime foam, and the implications of that dynamics for its existence and nature. We begin by envisioning Lorentzian spacetime, filled on Planck scales with all manner of wormholes and other structures-modes of topological fluctuation-continually winking into existence, persisting for microscopic time periods, and pinching off. The moments of creation and disappearance of these structures, i.e., the actual points of topological change, may or may not require a Euclideanized treatment; our analysis does not attempt to deal with these points.
Classical wormhole geometries can be used to model such a picture. In general such wormholes, persisting for arbitrary lengths of time (in particular, sufficient to allow passage of matter or energy), must involve stress-energy distributions which violate the weak energy condition-negative energy density in some reference frames must appear somewhere in the wormholes' throats. Whether this suffices to rule out wormholes on macroscopic scales is unknown at present, but it is not expected to be a problem for microscopic structure, where quantum effects could readily give rise to such stress-energy [7,8]. Moreover it does not appear to present a barrier to the growth of wormholes from Planck scales to large sizes, as the present work indicates.
To make the analysis tractable we employ a brutally simplified wormhole model to represent one mode of spacetime-foam fluctuation. A class of wormholes can be constructed by excising the world tube of some surface from a 3 + 1dimensional spacetime region and joining this to another such region, with like ex-cision, at that surface [9]. The join, the throat of the resulting wormhole, contains a surface layer of stress-energy specified by the Einstein field equations expressed as junction conditions [10]. Here we take the external regions of such a wormhole to be flat, empty Minkowski space, the throat a sphere of time-varying radius [9]. Our model is thus an extreme version of a wormhole in which the spacetime curvature in the throat is much greater than that in the regions surrounding the mouths. Our simplifications reduce the geometric degrees of freedom of the spacetime to just one, the throat radius. Thus our analysis is a minisuperspace model of a wormhole, analogous to the minisuperspace models used in quantum cosmology [1]. In fact our model is simplified even further than those: We treat the matter providing the stress-energy on the wormhole throat via an equation of state, rather than as a separate dynamical field. The quantum/gravitational dynamics of the complete wormhole system becomes then the quantum mechanics of one variable, the throat radius. The result we seek is the evolution, in time as defined in the wormhole's flat exterior regions, of a wave function for that radius.
In the absence of a general theory of quantum gravity, there is no prescription from first principles for quantizing a system like this wormhole. The existence of an external time, in terms of which the evolution of the system is naturally to be described, makes unsuitable the usual time-independent Wheeler-de Witt equation [11]. Instead we choose to "solve the constraint" explicitly, i.e., to impose the Hamiltonian constraint classically, to reduce the phase space of the system.
Combining the classical junction conditions at the wormhole throat-consisting of the Hamiltonian constraint plus a dynamical equation-with an equation of state for the matter on the throat yields an equation of motion for the throat radius alone. We then construct an action for the dynamics in the reduced phase space, i.e., depending only on the radius and its time derivatives, for which that equation of motion is the Euler-Lagrange equation. This can be done in a variety of ways, amenable to various forms of quantization. One such action, associated with gravitational dynamics in a straightforward way, is obtained from the scalar curvature of the wormhole spacetime. The wormhole quantum mechanics described by this action is treated in detail in this paper. Alternative actions and their implications for wormhole dynamics are examined in a subsequent paper [12].
The wormhole quantum mechanics which follows from the "curvature action" is in some respects similar to the Newton-Wigner first-quantized description of a relativistic free particle [13,14]. For both the Hamiltonian is non-polynomial in the momentum, i.e., is a nonlocal operator in the coordinate representation. Indeed, for the wormhole the Hamiltonian is a rational function of "velocity," which is related to the canonical momentum via a transcendental equation which cannot be inverted in closed form. This unconvential form of the Hamiltonian makes it impossible to deduce the behavior of the system merely by examining the "potential" term; explicit solution for the evolution of the wave function is needed. Moreover the corresponding Schrödinger equation for wormhole wave functions cannot be written explicitly as a differential equation. It can, however, be solved, at least formally: The wave-function propagator is obtained by using the action in a Feynman path integral. Exact evaluation of this integral is problematic; the action is complicated, and questions of the appropriate measure and skeletonization (factor-ordering problems), as well as the issue of the correct class of paths over which to integrate, arise.
But the integral can be evaluated in a WKB approximation, as a sum of contributions from classical paths and small fluctuations about them. The measure and factor-ordering problems are thus avoided. It is crucial, though, to include all the appropriate paths. The analogy to the relativistic particle suggests that spacelike as well as timelike and lightlike paths must be included in the integral [14,15]. Also, since negative throat radii are undefined, a suitable boundary condition must be imposed on the wave functions, i.e., on the propagator, at zero radius. Zero radius acts as a reflecting barrier, giving rise to classical paths-extrema of the actionwhich "bounce" there one or more times. (Such a boundary condition incorporates our neglect of any topological change at zero radius.) The WKB version of the propagator, then, consists of a sum of several terms, contributions of paths with different numbers of bounces; cancellation of terms effects the boundary condition at zero, just as for a particle in a half space. Explicit expressions for all the terms are obtained here. With these the evolution of a wave function is calculated via a convolution integral, evaluated numerically.
The results suggest that even with a matter equation of state chosen so that the classical evolution of the throat radius is bounded, the wormhole is quantummechanically unstable. This is not unprecedented-classically stable black holes, for example, are unstable to Hawking evaporation. In this case a Feynman-Kac decomposition of the wormhole propagator yields no spectrum of bound states.
The calculated evolution of an initially localized wave function follows a classical trajectory for many expansion/recollapse/bounce cycles, but the wave function must eventually leak to arbitrarily large throat-radius values. Its behavior thus resembles that for a particle initially confined to a well with finite, though perhaps high and wide, walls, or an α particle in a heavy nucleus. This simple wormhole geometry appears to represent a mode of spacetime-foam structure unstable against growth to macroscopic size.
Though subject to the limitations of our model, viz., the simplifications and approximations we have applied, and the uncertainties associated with our quantization scheme, such a conclusion is potentially very significant. It indicates that Planck-scale gravitational physics might be constrained by observations on much larger scales: The apparent absence of any topological structure to spacetime on scales accessible to observations, ranging from particle physics to astronomy, could imply either that not all modes of topological fluctuation are possible or that microscopic (Lorentzian) spacetime foam does not exist altogether.
The wormhole model and its associated classical and quantum dynamics are presented in Sec. II below. Evaluation of the wormhole propagator in the WKB approximation is shown in detail in Sec. III. The implications of the result for wormhole wave-function evolution are discussed in Sec. IV. Conclusions and caveats follow in Sec. V. Units with G = c =h = 1 are used throughout. Sign conventions and general notation follow those of Misner, Thorne, and Wheeler [16].

A. Classical dynamics
A spherically symmetric "Minkowski wormhole" [9] provides a very simple where S i j is the stress-energy tensor of a surface layer on the throat, and the righthand side is the jump discontinuity in the throat extrinsic curvature K i j , minus its trace K m m . Here the nontrivial components of these equations are give the classical equation of motion for the wormhole. As is to be expected [7,8], Eq. (2.2a) indicates that negative energy densities appear.
For simplicity we specify the equation of state as a property of the model, rather than derive it from some more fundamental description of the surface layer.
It could be chosen to make the equation of motion extremely simple. The choice p = −σ/2, for example, would implyR = 0. The dynamics of such a model is trivial.
Classically the wormhole throat either sits at fixed radius or expands or contracts linearly. Its quantum behavior is the same as that of a free particle; a wave function initially concentrated about some R value will disperse to infinity. But this is not a however, is not suited to the action construction we use below. (It can be treated by other methods [12].) Instead we choose which yields the equation of motion The corresponding classical trajectories are parabolic: This choice of reduced-phase-space action is motivated by the fact that it coincides with the Einstein-Hilbert action for the wormhole spacetime. The wormhole's scalar curvature R is nonzero only on the throat: where ρ is r−R(t) in one exterior region and R(t)−r in the other. The corresponding gravitational action-the integral of R plus a surface term [17] eliminating the totalderivative term-is just Eq. (2.6).
The Hamiltonian dynamics of the wormhole, on its reduced phase space, follows immediately from the action. With Lagrangian L given by the integrand in Eq. (2.7), the canonical momentum is The Hamiltonian is (2.9) The Hamiltonian cannot be expressed in terms of R and P in closed form, since the transcendental relation (2.9) betweenṘ and P/R cannot be inverted explicitly.
It should be noted that the "energy" corresponding to this Hamiltonian is not to be identified with the ADM mass of the wormhole, which is zero by construction.
To thus fix the "energy" would reduce the equation of motion to first order, and the classical initial-parameter space to one dimension, making canonical quantization impossible. In fact the Hamiltonian (2.9) corresponds to the classically conserved "mass" 2(2πR 3/2 σ) 2  and their implications for wormhole quantum mechanics are examined in another paper [12]. The rest of this paper treats the quantum dynamics obtained from the geometric action.

B. Quantum dynamics
In this minisuperspace model, then, the quantum wormhole is to be described by a wave function ψ(R, t). With the Hamiltonian constraint (2.2a) imposed classically, i.e., incorporated into the reduction of the wormhole's phase space, no Wheeler-de Witt equation imposes "time independence" of ψ [11]; instead its evolution in wormhole-exterior time t is generated by the Hamiltonian (2.9).
There is no hope of treating the corresponding Schrödinger equation directly: The transcendental nature of relation (2.8) and the form of H imply that the Hamiltonian contains all powers of the momentum P , or as an operator on ψ, of ∂/∂R.
It is therefore a nonlocal operator, similar to that for a relativistic particle, but rather more unwieldy [14]. Operator ordering in H is problematic as well. But the solution of the Schrödinger equation can be treated. The general solution, i.e., the time evolution of any wave function, can be given in terms of a propagator: The propagator G is given formally by a Feynman path integral with C denoting the class of paths over which the integral is defined.
The appropriate class of paths is an important issue. The example of the relativistic particle [14], and more general quantum-gravitational considerations [15], suggest that all paths linking the initial and final points should be includedspacelike as well as timelike and null. [Spacelike paths, with |Ṙ| > 1, correspond to "negative energy" contributions, in terms of the Hamiltonian (2.9).] Including spacelike paths implies that G need not vanish outside the "light cone" |R−R 0 | = t, i.e., that it may admit "acausal propagation." Moreover, the wormhole geometry is only defined for nonnegative R, hence only paths with R(t) ≥ 0 should be included. This restriction can be implemented as for a particle confined to a half space, i.e., as if there were an infinite potential wall at R = 0. This implies the boundary condition ψ(0, t) = 0, hence that G[R, t; R 0 , 0] should vanish for R or R 0 zero. (More precisely, ψ must entail no current in the −R direction at R = 0, so boundary conditions more general than this are possible. For simplicity we do not treat such here.) By imposing this condition we exclude consideration of topology-changing processes, i.e., wormhole creation or disappearance, at R = 0. The treatment of such processes might require a "second quantized" framework (actually "third quantized," outside the restrictions of a minisuperspace model), rather than the "first quantized" (actually "second quantized") description used here. Their bearing on our results is discussed further in Sec. V.
The condition on ψ and G gives R = 0 here the role of a reflecting boundary for wormhole wave functions.

III. EVALUATION OF THE PROPAGATOR: WKB APPROXIMATION
The exponential factors arise from the classical paths R cl between the initial and final values. The prefactors come from Gaussian integrals over small deviations from these, giving semiclassical corrections [19]. To that order G (WKB) is independent of the choice of path-integral measure. Specifying the regime in which this approximation is accurate is problematic, since we have neither an exact solution for comparison nor even an explicit form of the Schrödinger equation. This question is considered further in Sec. V below.
The first step in evaluating G (WKB) , then, is finding the appropriate classical paths. More than one type of path always contributes. There is exactly one parabolic trajectory of form (2.5), and taking only nonnegative radius values, between any initial and final points (R 0 , 0) and (R, t). Specifically, this trajectory has and action Because paths are restricted to the half-space of positive radius values, there are also classical trajectories which are piecewise of form (2.5), but which "bounce" one or more times at zero radius; extremizing the action determines the bounce times.
Such a path which bounces just once, at time t 1 , has action extremal with respect to variation of t 1 . That condition is t there is always one such solution, and may be three. A classical path bouncing more than once, at times t 1 , . . . , t n , has action That this be extremal with respect to all bounce times is equivalent to the conditions Any real solution to the coupled quadratic equations (3.7a) and (3.7c) with 0 ≤ t 1 < t n ≤ t yields an n-bounce path contributing to G (WKB) . For a given n there may be zero, two, or four such. Conditions (3.5) and (3.7a-c) are equivalent to requiring that the parameters α for all the parabolic segments of a bouncing path be equal, which is to say that the wormhole "bounces elastically."

8c)
and t (d) The evaluation of G (WKB) proceeds from these results. Expression (3.1) eventually takes the form with n the number of bounces, as above, and k labelling the n-bounce paths, the range of k depending of course on n. The classical actions S with α and t 0 from Eqs. (3.2); with t 1 from Eq. (3.5); For example, as R approaches zero, the contribution of the direct (zero-bounce) classical path becomes equal in magnitude and opposite in sign to that of a onebounce path 1 with bounce time approaching t. This phase prescription accords with that used, for example, for the simple problem of a particle confined to a half space [20].
Form (3.9) and the accompanying expressions show that even the approximate propagator for this simplified system is a complicated quantity. Some of its features are illustrated in Fig. 1. The propagator has singularities on the outgoing and ingoing or "bounced" light cones R = R 0 +t, R = R 0 −t, and R = t−R 0 , respectively.
(It has other singular points as well, described below.) As expected, given the inclusion of spacelike trajectories in the path integral as discussed in Sec. II, the propagator is nonvanishing outside the light cone. Thus it admits acausal, i.e., 1 The absolute value of the argument of the logarithm is taken in the action (2.6) and subsequent expressions to ensure that this cancellation still occurs when the direct path is just timelike and the one-bounce path just spacelike, i.e., for R 0 ≈ t and R approaching zero.
superluminal, propagation. Overlying these gross features the propagator exhibits a great deal of high-frequency structure.

IV. QUANTUM EVOLUTION OF THE WORMHOLE
In interval. This is determined from an expansion in R 0 of the appropriate term about the light cone. In particular we obtain on the outgoing light cone, on the ingoing light cone, and is intractable-the general expressions for the bounce times t 1 and t n appearing in the S (n) and F (n) are too unwieldy even for computer manipulation! But for R 0 = R the solutions (3.8) yield manageable results, and nearby R 0 and R values can be handled perturbatively. The terms in the sum (3.9) are then labeled a, b, c, and d, corresponding to those solutions. In the limit t ≫ R, R 0 , with R = R 0 + ε and |ε| ≪ R 0 , the functional determinants and classical actions in those terms are: for n less than t/(4R 0 ); and for n less than (t 2 + 4R 2 0 )/(4R 0 t). The n = 0 term is a c term, given by Eqs. (4.2d) and (4.2e), and the n = 1 terms are a, b, and c terms given by Eqs. (4.2a-e).
The order-ε terms in the actions are obtained from the relation ∂S cl /∂R = P (t) and Eq. (2.8) for the momentum P , evaluated on the appropriate classical path.
Terms of order ε and higher in the functional determinants give rise to subdominant contributions and are not needed here.
The above expansions in inverse powers of t do not converge for n = t/(4R 0 ).
Indeed, the a, b, and c terms in the propagator are singular for these parameter values, at which the roots (3.8a-c) coincide. But the contribution of these singular terms to the evolution of a wave function can still be estimated. At fixed R 0 , such a singularity or "caustic point" occurs at intervals in t of 4R 0 , the period of a classical bouncing trajectory with maximum radius value R 0 . The singular contribution is thus associated with the classically bouncing peak seen in Fig. 2.
Unlike the similar singularity in a harmonic-oscillator propagator [21], however, this singularity in G (WKB) does not correspond to a δ function. Rather, in the immediate vicinity of the caustic, i.e., for ε small compared to t −2 (in Planck units), the c term has functional determinant and action The contribution to a wave function from this vicinity is therefore of order t −3/2 , not unity, as a δ function would yield. However, the form of the action (4.3b) suggests that a significant contribution from this term may arise from a region of width t −1/2 in ε. Such a contribution might be of order unity, as Fig. 2  But telling features of that behavior can be extracted from the propagator as approximated by Eqs. (4.2). Did the Hamiltonian (2.9) describe a bound system, its ground-state energy (or, for a free system, the bottom of its continuous spectrum) would be given by the Feynman-Kac [22] limit of the propagator in imaginary time. This can be evaluated via Eqs. (4.2). The propagator is still a sum of terms with rapidly varying phases with respect to increasing τ , owing to the t 2 terms in the actions. Consequently it oscillates forever; the limit on the right side of Eq. (4.4) does not exist. This is illustrated in Fig. 4.
This means that the quantum wormhole possesses no spectrum of bound states.
The expectation value of R cannot be confined in the late-time limit. As for a system with an inverted potential diverging to negative infinity, or a metastable system such as a particle confined by finite walls, the wormhole wave function must eventually run or "leak" to arbitrarily large radius values-the wormhole is unstable against eventual growth to large size. The previous results suggest that it is the slow leaking behavior which characterizes these wormholes. They appear reminiscent of α particles in a nucleus, oscillating perhaps millions of times before escaping to infinity.

V. IMPLICATIONS AND LIMITATIONS
Spherically symmetric Minkowksi wormholes [9]  i.e., they are classically stable, they will nonetheless grow to large size by quantum "diffusion." Many systems exhibit similar behavior. For a particle with the familiar quadratic kinetic term in the action, the form of the potential determines whether such diffusion or spreading occurs: A potential well with walls or barriers which fall off at large distances will allow a classically bound particle to leak out via quantum tunneling (as in the case of α decay), while one which increases monotonically with distance will not. For these wormholes, with more complicated action (2.6), so simple an analysis is not possible. The more involved examination of the wormholes' quantum dynamics described here is needed to see that spreading of the wave function to large radii will take place.
Our result suggests that dynamics, and stability considerations in particular, may be of great importance in understanding the quantum nature of gravitation.
A definitive demonstration of the existence of an unstable mode of fluctuation in spacetime foam would have profound implications. Since a macroscopic structure of wormholes is not observed, i.e., spacetime appears to be smooth and topologically trivial on all scales accessible to laboratory physics, it would imply the existence of a mechanism for suppressing such a mode, or even the absence of (Lorentziansignature) spacetime foam altogether. Of course the present work is far from such a definitve demonstration; it serves to point up lines along which these matters should be studied further.
The most fundamental limitation of our calculation is the restriction of the gravitational degrees of freedom to those of the spherically symmetric Minkowski wormhole, i.e., the use of a "minisuperspace model" for topological structure. In fact our model is even more restricted than the usual minisuperspace models [1], since the matter in the hole is treated not as a dynamical field but by the use of We analyze our constrained model by quantization in the reduced phase space, as described above. In the absence of a general framework for quantum-gravity calculations, this method seems best suited to the problem. It does differ markedly, though, from the Wheeler-de Witt approach [11].
Furthermore, we use the particular reduced action (2.6). Other forms corresponding to the classical equation (2.4) are possible; the effect of this choice on the results will be examined elsewhere [12].
Moreover our choice of action implies a choice of Hamiltonian fundamentally different from that used in similar calculations [18], corresponding here to a classi-cally conserved "proper mass" rather than the ADM mass of the wormhole. This choice raises some interesting questions. For example, since the Hamiltonian (2.9) is time independent, its expectation value is conserved. Its form appears to suggest, then, that the expectation value of the throat radius R should remain bounded. It might be expected, however, that any initially localized wave function must contain both positive-and negative-"energy" (|Ṙ| > 1) components; this is a well-known feature of, e.g., the quantum mechanics of a relativistic particle [13,14]. The interplay of these components could account for the eventual spreading to large radius values. More detailed calculations of the wormholes' quantum dynamics should clarify this.
We employ the WKB approximation to evaluate the wormhole propagator.
WKB calculations of quantum instabilities in classically stable systems, such as tunnelling, are well known. Here it is more difficult to be precise about the accuracy of the approximation. It is certainly to be expected to be valid in the late-time limit t ≫ R, R 0 , in which the instability is manifest. But the accuracy of the numerical evolution of a wave function, as shown in Fig. 2, is harder to establish.
Lacking any exact solution for comparison, we have tested the accuracy of the calculation via the composition relation with 0 < t 1 < t. It can be shown analytically that this should hold if the WKB approximation is strictly valid at the intermediate time t 1 . We found, however, that for some t 1 values relation (5.1) is not well satisfied. This may indicate inaccuracy of the WKB approximation, or may be due to numerical difficulties associated with the integration, given the rapidly varying phase of the propagator.
A final limitation, of fundamental significance, is our implementation of the restriction that throat radii are nonnegative. Here we do this as for a particle in a half space, leading to the boundary condition ψ(0, t) = 0. Other implementations might be used, the most general condition being only that the wave function ψ entail no current in the −R direction at R = 0. Our condition eliminates from consideration any processes such as wormhole creation or disconnection at R = 0. Including these processes would drastically alter the physics of the modelessentially, from the quantum mechanics of one variable to quantum field theoryand would require a formalism for describing the topology changes. It should not, however, alter the instability suggested by our results. At issue is the stability of the spacetime foam, of which an individual wormhole is just one fluctuation.
Certainly for any particular wormhole, the probability of growth to large size is affected strongly by the inclusion or exclusion of topology change. Indeed, since the time scale implied by our results for the wave function to leak to large radii is much longer that that for a classical bounce, the chance that a specific wormhole grows large should be much smaller than that it pinches off and disappears at zero radius, if that is allowed with more than an extremely small probability. But given the possibility of topology change, the foam should contain an equilibrium population of holes fluctuating into and out of existence. If it is possible for a hole to grow large, this population will eventually give rise to some large holes. Again the analogy may be drawn to the α decay of a heavy nucleus: α particles continually form and disperse within the nucleus, on a time scale typically much shorter that that of the decay; the instability represented by the tunnelling of an α particle out of the nucleus remains.
The more sophisticated analyses needed to probe the quantum dynamics of spacetime beyond the restrictions and limitations of these calculations present a considerable challenge. Our results suggest, however, that this is an aspect of quantum gravity theory well worth such consideration.