Conifolds and Tunneling in the String Landscape: Part 4

The ever-lengthening series continues!

Part 1 gave an overview of the two main themes of our investigation of tunneling in the String Landscape—the collection of possible configurations that string theory’s extra dimensions can find themselves in, including certain flux fields through the extra dimensions.

In part 2 I told a more detailed story of what it means to curl up extra dimensions and I explained the notion of a flux that can keep the sizes and shapes of these dimensions stable.

In part 3 I explained a very special configuration called the “conifold.” When the extra dimensions are wrapped up in the shape of a Calabi-Yau manifold, there are extra parameters that control certain aspects of that shape. The conifold configuration of the Calabi-Yau manifold represents a specific choice of parameter values at which some part of the space collapses to zero size. If you had a little six dimensional ant that lived in the Calabi-Yau, it would experience the part of the space near the collapsed region as appearing to be like a six dimensional cone.

We now have all the ingredients for understanding what is meant by “String Landscape.” The extra dimensions are curled up into a Calabi-Yau shape. Generalized electric and magnetic fields pierce through parts of this shape to give you fluxes. For a given Calabi-Yau there are huge numbers of possible fluxes to put on the shape—that large number of flux configurations is usually what is referred to by a string landscape. It’s a landscape when you consider the energy function which will look very bumpy with lots of peaks, ridges, valleys, and wells. The model universe will want to settle down in a well of low energy much as a ball will roll into the valley between two peaks rather than stay teetering at the tip of one of the peaks.

So if we are able to map out some of this landscape, we should be able to see which flux configurations are favored by string theory for that particular Calabi-Yau. We simply search for the lowest energy wells in the landscape of energy that describes the various flux configurations. Interestingly, the landscape for the simple models that we investigated in our paper has certain patterns. Energy wells that the universe would like to settle into tend to appear periodically as one travels around and around the conifold configurations of the extra dimensions. Thus, if you find an energy well somewhere in the landscape, and then you—the intrepid explorer—wind around the conifold point, then you will often find another energy well after making roughly a 360 degree turn about the conifold point.

This pattern presents us with an opportunity to explore the details of the following scenario applied to string theory. In a quantum system, objects will not usually just stay forever in a given configuration, even if it has a low energy and is at the bottom of an energy well. Quantum mechanics implies that systems will jitter and jiggle around such configurations. Usually these jitters are very small, but very very rarely, such a random jitter can drive you out of the low energy configuration you are in and into a neighboring one over the mountain ridge dividing the two.

This situation is often illustrated using the example of a particle in a box. In quantum theory, a particle isn’t described simply as a point in space. Instead it has associated with it a wavefunction that, for our purposes, tells us the probability of finding it at any given point. In this way, quantum theory tells us that even though when we observe them in space, particles look like little points, in order to understand how they move through space, we must model them as somehow extending out over all space in this wave-like manner.

So imagine placing a particle into a box and enclosing it tight. If you make the box out of some sort of impenetrable material—impermium let’s call it—then you know that the wave describing the particle is completely contained inside your impermium box. This means that sometime later, when you open the box up and try to locate the particle, you will inevitably find it somewhere inside the box.

But impermium is not a real material. Real materials always have some degree of permeability. So, if you build an almost impenetrable box and put the particle in, what happens? Well, sometime later you are likely to find that the particle is still in there when you open it up again. However, there is a tiny chance that the particle could have slipped out or “tunneled” through the box’s walls to escape. This is because a penetrable box allows the wavefunction of the particle to leak out a very slight amount. So despite having been initially placed in the box, there is a tiny but non-zero probability that it will get out after some time has passed.

Well, energy landscapes for general quantum systems behave in an analogous manner. If a system finds itself in a valley, but one of the mountain ridges is adjacent to another valley, then there is a tiny but non-zero probability that the system can tunnel out from its low-energy configuration described by the first valley and hop into the other low-energy configuration described by the adjacent valley. Note that this would not be possible if there was no quantum mechanics since there would be no quantum jitters that could drive the system up the mountain side and down into the adjacent valley.

So what does it look like when you take your quantum system to be the universe and you think about it as tunneling between two adjacent energy wells separated by mountain ridges in an energy landscape? In the simplest cases the transition would occur in some local region of the universe and it would look like a little bubble appearing. Within the bubble you have a state of the universe that is in the adjacent energy well. Outside the bubble is the state of the universe in the original energy well. Naturally this implies that the bubble has a wall at which the two states interface with each other.

If the energy state inside the bubble is higher than the one outside the bubble, then pressure from the outside will collapse the bubble and nothing much will change. But if the energy inside the bubble is lower, this will put pressure on the bubble wall to expand. The constant acceleration pushes the bubble wall to expand close to the speed of light, so eventually what you get is an ever expanding region of the low energy state eating up the surrounding parts of the universe that are in the original energy state.

Note that the wall of interface between the inside and outside of the bubble is itself physically interesting. It is a spherical membrane that expands outwards and it could very well have its own dynamics. In other words, a more careful analysis that allows for complications would involve understanding how the membrane wall ripples and fluctuates as it interacts with things outside of the bubble (and also just due to its own quantum mechanical jitters). The creation of the bubble can thus be understood as equivalent to the spontaneous creation of the bubble wall as a dynamical, physical object in its own right.

This may sound rather farfetched, but there are real life examples of such processes. If you take two plates of opposite charge and oppose them so that they are parallel to each other you have formed a system called a capacitor. There is an electric field that goes from the positively charged plate to the negatively charged plate. If you crank up the strength of this field, you will eventually get to a point where quantum mechanical fluctuations become important. In particular, quantum mechanics tells us that pairs of electrons and anti-electrons are constantly being created and then annihilate each other everywhere in space. Usually these pairs are “virtual” since they come into and out of existence so quickly nobody can directly observe them. However, in a strong enough electric field, the pair can spontaneously come into existence—similarly to the bubble wall from before—and then be accelerated in opposite directions—the negatively charged electron will fly toward the positively charged plate while the positively charged anti-electron will fly into the negatively charged plate. This will actually reduce the strength of the electric field since some if its energy will have been converted into the mass and energy necessary for these electron/anti-electron pairs to come into existence and accelerate apart from each other.

So, if the universe is described by an energy landscape that determines the fluxes through the extra dimensions, then it likely has many hills, ridges, valleys, and wells. In our simple models, we can indeed find wells that are adjacent to each other by going around the conifold configuration. This allows us to explore what it would be like for a bubble to appear, representing a transition from one energy well to another one across the conifold configuration. As it turns out, the dynamics of this transition are somewhat intricate. The universe cannot spontaneously create a bubble membrane that separates the two configurations. Instead, the extra dimensions have to be able to deform as well. In other words, the tunneling transitions taking our model universes from one flux configuration to another must be accompanied by a sort of dance of the extra dimensions. In particular, they deform so that a portion of them shrink down to very tiny sizes—that is, they approach the conifold configuration—and it is essentially (but not exactly) at that point that the membrane or bubble wall is able to be spontaneously generated. This bubble wall acts a lot like the electron/anti-electron pair in that it absorbs some of the flux from the universe’s original energy well and thus leads to a lower energy configuration inside the bubble with less flux.

This has broader potential implications for understanding how configurations may tunnel between each other in the string landscape. We cannot blithely assume that if two configurations can be connected by some appropriately charged bubble wall then they can simply hop into each other. Instead, the shape and size of the extra dimensions needs to also be taken into account. They will likely have to be able to deform in the appropriate ways so that it becomes energetically favorable to spontaneously generate these bubbles.

This work is part of a continuing effort to understand what we string theorists actually mean when we talk about a string landscape and the way a universe—ours perhaps—could evolve when it is described by such a theory. There are lots of details that I’ve left out and there are many more that need to be included to get a completely satisfying picture. That said, the work highlights some of the immense richness of string theory and suggests some very natural directions for further exploration. In particular, these sorts of tunneling processes ought to also be able to describe the tearing and patching together of the universe into yet more radically different configurations—or at least provide an explanation for why such things cannot happen in string theory.

Phew! Okay! I think that that wraps up my “summary” of recent work. I’m half-inclined to produce a summary of this summary! (But I think I should resist that impulse).

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