The idea of scalar fields and potentials is approximate once we leave the supermoduli-space, as is the notion of a stable de Sitter vacuum. The problem is familiar. How do we make precise sense of an unstable state in quantum mechanics? In ordinary quantum mechanics the clearest situation is when we can think of the unstable state as a resonance in a set of scattering amplitudes. The parameters of a resonance, i.e. its width and mass, are well defined and do not depend on the exact way the resonance was formed. Thus, even black holes have precise meaning as resonant poles in the S-matrix. Normally we cannot compute the scattering amplitudes that describe the formation and evaporation of a black hole, but it is comforting that an exact criterion exists.
In the case of a black hole the density of levels is enormous, being proportional to the exponential of the entropy. The spacing between levels is therefore exponentially small. On the other hand, the width of each level is not very small. The lifetime of a state is the time it takes to emit a single quantum of radiation, and this is proportional to the Schwarzschild radius. Therefore the levels are broadened by much more than their spacing. The usual resonance formulae are not applicable, but the precise definition of the unstable state as a pole in the scattering amplitude is. I think the same things can be said about the unstable de Sitter vacua, but it can only be understood by returning to the 'causal patch' way of thinking. Therefore let us focus on the causal patch of one observer. We have discussed the observer's future history and found that it always ends in an infinite expanding supersymmetric open FRW universe. Such a universe has the usual kind of asymptotic future, consisting of time-like and light-like infinities. There is no temperature in the remote future and the geometry permits particles to separate and propagate freely, just as in flat spacetime.
Now let us consider the observer's past history. The same argument which says that the observer will eventually make a transition to A = 0 in the far future can be run backward. The observer could only have reached the de Sitter vacuum by the time-reversed history and so must have originated from a collapsing open universe. The entire history is shown in Fig. 16.5. The history may seem paradoxical, since it requires the second law of thermodynamics to be violated in the past. A similar paradox arises in a more familiar setting. Let us return to the sealed room filled with gas molecules, except that now one of the walls has a small hole that lets the gas escape to unbounded space. Suppose we find the gas filling the room in thermal equilibrium at some time. If we run the system forward, we will eventually
find that all the molecules have escaped and are on their way out, never to return. But it is also true that, if we run the equations of motion backwards, we will eventually find all the molecules outside the room moving away. Thus the only way the starting configuration could have occurred is if the original molecules were converging from infinity toward the small hole in the wall.
If we are studying the system quantum mechanically, the metastable configuration with all the molecules in the room would be an unstable resonance in a scattering matrix describing the many-body scatterings of a system of molecules with the walls of the room. Indeed, the energy levels describing the molecules trapped inside the room are complex due to the finite lifetime of the configuration.
This suggests a view of the intermediate de Sitter space in Fig. 16.5 as an unstable resonance in the scattering matrix connecting states in the asymptotic A = 0 vacua. In fact, we can estimate the width of the states. Since the lifetime of de Sitter space is always longer than the recurrence time, generally by a huge factor, the width y satisfies 7 » exp(-S). On the other hand, the spacing between levels, AE, is of order exp(-S). Therefore 7 » AE, so that the levels are very broad and overlapping, as for the black hole. No perfectly precise definition exists in string theory for the moduli fields or their potential when we go away from the supermoduli-space. The only precise definition of the de Sitter vacua seems to be as complex poles in some new sector of the scattering matrix between states on the supermoduli-space.
Knowing that a black hole is a resonance in a scattering amplitude does not tell us much about the way real black holes form. Most of the possibilities for black hole formation are just the time-reverse of the ways in which it evaporates. In other words, the overwhelming number of initial states that can lead to a black hole consist of thermal radiation. Real black holes in our universe form from stellar collapse, which is just one channel in a huge collection of S-matrix 'in states'. In the same way, the fact that cosmological states may be thought of in a scattering framework does not itself shed much light on the original creation process.
Vacua come in two varieties: supersymmetric and non-supersymmetric. Most likely the latter do not have vanishing cosmological constant, but it is plausible that there are so many of them that they practically form a continuum. Some tiny fraction have a cosmological constant in the observed range. With nothing favouring one vacuum over another, the Anthropic Principle comes to the fore, whether or not we like the idea. String theory provides a framework in which this can be studied in a rigorous way. Progress can certainly be made in exploring the landscape. The project is in its infancy, but in time we should know just how rich it is. We can argue the philosophical merits of the Anthropic Principle, but we cannot argue with quantitative information about the number of vacua with each particular property, such as the cosmological constant, Higgs mass or fine structure constant. That information is there for us to extract.
Counting the vacua is important but not sufficient. A greater understanding of cosmological evolution is essential to determining if the large number of possibilities are realized as actualities. The vacua in string theory with A> 0 are not stable and decay on a time-scale smaller than the recurrence time. This is very general and also very fortunate, since there are serious problems with stable de Sitter space. The instability also allows the Universe to sample all or a large part of the landscape by means of bubble formation. In such a world the probability that some region of space has suitable conditions for life to exist can be large.
The bubble universe based on Linde's eternal inflation seems promising, but it is unclear how to think about it with precision. There are real conceptual problems having to do with the global view of spacetime. The main problem is to reconcile two pictures: the causal patch picture and the global picture. String theory has provided a testing ground for some important relevant ideas, such as black hole complementarity [7,8] and the
Holographic Principle [11,12]. Complementarity requires the observer's side of the horizon to have a self-contained conventional quantum description. It also prohibits a conventional quantum description that covers the interior and exterior simultaneously. Any attempt to describe both sides as a single quantum system will come into conflict with one of three sacred principles . The first is the Equivalence Principle, which says that a freely falling observer passes the horizon without incident. The second says that experiments performed outside a black hole should be consistent with the rules of quantum mechanics as set down in Dirac's textbook. No loss of quantum information should take place and the time evolution should be unitary. Finally, the rules of quantum mechanics forbid information duplication. This means that we cannot resolve the so-called information paradox by creating two copies (quantum xeroxing) of every bit as it falls through the horizon - at least not within the formalism of conventional quantum mechanics. The Complementarity and Holographic Principles have been convincingly confirmed by the modern methods of string theory (see, for example, ref. ). The inevitable conclusion is that a global description of geometries with horizons, if it exists at all, will not be based on the standard quantum rules.
Why is this important for cosmology? The point is that the eternal inflationary production of an infinity of bubbles takes place behind the horizon of any given observer. It is not something that has a description within one causal patch. If it makes sense, a global description is needed. However, if cosmic event horizons are at all like black hole horizons, then any global description will involve wholly new elements. If I were to make a wild guess about which rule of quantum mechanics has to be given up in a global description of either black holes or cosmology, I would guess it is the Quantum Xerox Principle . I would look for a theory which formally allows quantum duplication but cleverly prevents any observer from witnessing it. Perhaps then the replication of bubbles can be sensibly described.
Progress may also be possible in sharpening the exact mathematical meaning of the de Sitter vacua. Away from the supermoduli-space, the concept of a local field and the effective potential is at best approximate in string theory. The fact that the vacua are false meta-stable states makes it even more problematic to be precise. In ordinary quantum mechanics the best mathematical definition of an unstable state is as a resonance in the amplitude for scattering between very precisely defined asymptotic states. Each meta-stable state corresponds to a pole whose real and imaginary parts define the energy and inverse lifetime of the state.
I have argued that each causal patch begins and ends with an asymptotic 'roll' toward the supermoduli-space. The final states have the boundary conditions of an FRW open universe and the initial states are time-reversals of these. This means we may be able to define some kind of S-matrix connecting initial and final asymptotic states. The various intermediate meta-stable de Sitter phases would be exactly defined as resonances in this amplitude.
At first this proposal sounds foolish. In General Relativity, initial and final states are very different. Black holes make sense. White holes do not. Ordinary things fall into black holes and thermal radiation comes out. The opposite never happens. But this is deceptive. Our experience with string theory has made it clear that the fundamental microphysical input is completely reversible and that black holes are most rigorously defined in terms of resonances in scattering amplitudes.6 Of course, knowing that a black hole is an intermediate state in a tremendously complicated scattering amplitude does not really tell us much about how real black holes form. For that, we need to know about stellar collapse and the like. But it does provide an exact mathematical definition of the states that comprise the black hole ensemble.
To further illustrate the point, let me tell a story. Two future astronauts in the deep empty reaches of outer space discover a sealed capsule. On further inspection, they find a tiny pinhole in the capsule from which air is slowly leaking out. One says to the other, 'Aha! We have discovered an eternal air tank. It must have been here forever.' The other says, 'No, you fool. If it were here forever, the air would have leaked out an infinitely long time ago.' So, the first one thinks and says, 'Yes, you are right. Let's think. If we wait long enough, all the air will be streaming outward in an asymptotic final state. That is clear. But because of microreversibility, it is equally clear that - if we go far into the past - all the air must have been doing the reverse. In fact, the quantum states with air in the capsule are just intermediate resonances in the scattering of a collection of air molecules with the empty capsule.' The second astronaut looks at the first as if he were nuts. 'Don't be a dope,' he says. 'That's just too unlikely. I guess someone else was here not so long ago and filled it up.'
Both of them can be right. The quantum states of air in a tank are mathematical resonances in a scattering matrix. And it may also be true that the laws of an isolated system of gas and tank may have been temporarily interfered with by another presence. Or we might say that the scattering
6 The one exception is a black hole in anti-de Sitter space, which is stable.
states need to include not only air and tank but also astronauts and their apparatus. It is in this sense that I propose that de Sitter space can be mathematically defined in terms of singularities in some kind of generalized S matrix. But in so doing, I am not really telling you much about how it all started.
From the causal patch view-point, the evolutionary endpoint seems to be an approach to some point on the supermoduli-space. After the last tunnelling, the Universe enters a final open FRW expansion toward some flat supersymmetric solution. This is not to be thought of as a unique quantum state but as a large set of states with similar evolution. Running the argument backward (assuming microscopic reversibility), we expect the initial state to be the time-reversal of one of the many future endpoints. We might even hope for a scattering matrix connecting initial and final states. de Sitter minima would be an enormously large density of complex poles in the amplitude.
One last point: the final and initial states do not have to be 4-dimensional. In fact, in the example given in ref. , the modulus describing the overall size of the compact space rolls to infinity, thus creating a 10- or possibly 11-dimensional universe.
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