String Theory – part 10: Quantum Geometry

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In around a decade Einstein single-handedly smashed Newton’s views on gravity which had been dominant in physics for a few centuries. By doing so he presented to the world a completely new and a very in-depth theory of gravity. Physicists have since been impressed by the elegance of the theory and its numerous implications for the future of physics. Here we have to remember, though, that there were a few factors that undoubtedly helped Einstein a lot in developing his theory. Arguably, the most significant of those factors was the previously established mathematical model for describing the curved space of arbitrary dimensionality, which had been found and explained by the German mathematician Bernhard Riemann in the XIX century. In that elaborated geometrical model Riemann made a giant step from Euclidean flat space to the more general case capable of explaining the geometry of any arbitrarily curved space, Euclidean flat space being just a special case of this general model. Einstein’s genius was in the realisation that Riemann’s model was exactly what he needed to describe his ideas mathematically. After working on the mathematical details of his new theory he was able to make a bold claim about perfect agreement between Riemannian geometry and the physics of gravity.

But now, a century after Einstein developed his theory of gravity, String Theory gives us a quantum mechanical description of gravity that demands a reassessment of the rules of General Relativity on the scales close to the Planck length. And since General Relativity is based on Riemannian geometry, this geometrical model has to be reassessed as well in order to agree with the new physical model. If General Relativity postulates that the properties of curved space is in precise agreement with Riemannian geometry, String Theory advocates the applicability of this assumption only if the region of space being analysed is sufficiently large. On the scale of Planck lengths a new type of geometry, dubbed quantum geometry, should be at play.

In contrast with the Riemann geometry, which aided Einstein in developing his theory, however, there are no developed geometrical models that would be applicable on the ultramicroscopic scales which are typical for String Theory. Which is why these days physicists and mathematicians working with String Theory (and some other theories such as Loop Quantum Gravity) are striving to construct such a new geometrical model piece by piece. This new model has not been developed as yet, but in spite of this a few remarkable properties have already been found. In this chapter we shall be talking about them. As always, a much more detailed description can be found in Brian Greene’s book called “The Elegant Universe”.

The Main Principles of Riemannian Geometry

When a person is trampolining, the trampoline is being deformed under the person’s weight. The closer to the person’s body we inspect the trampoline, the greater the deformation. If there is a picture depicted on the surface of the trampoline, then while being exposed to some noticeable weight the deformation of its form would be quite obvious. If the person stays near the centre of the trampoline you would see that closer to the circumference the deformation becomes lower.

This illustrates an extremely important principle for the description of curved surfaces that Riemann established in his work. Based on the earlier works performed by Carl Friedrich Gauss, Nikolay Lobachevsky and several other mathematicians, Riemann showed that a detailed analysis of the distances between the points on the surface of an object, or inside of that object, gives you a rigorous mathematical way of calculating the curvature of the object. In the example above, the curvature of the trampoline refers to the level of deformation. The greatest curvature corresponds to the point that’s being exposed to the person’s weight. The distances between two points in that region would differ the most from the distances you would expect for a flat Euclidean surface. The closer we get to the edge of the trampoline, assuming there are no more weights on its surface, the lesser the curvature.

Einstein used Riemann’s mathematical results and gave them a physical interpretation. As we discussed in chapter 3, Einstein’s General theory of Relativity showed that the gravitational interaction can be interpreted as caused by the deformation of a space-time region. From the mathematical point of view, this deformation – or curvature – means that the distance between two points is changed relative to a flat surface. From the physical point of view, the gravitational force exerted on an object is caused by the curvature of space-time. There remains a slight problem with this though. As we inspect ever smaller regions of spacetime we should expect a greater agreement between physics and mathematics since the mathematical notion of a point becomes closer and closer to the physical reality, right? But as we saw in chapter 5, this way actually leads to a catastrophe in which the very equations of the General theory of Relativity stop making any sense. String Theory, on the other hand, restricts the smallness of lengths that could be examined even in principle, because strings represent the smallest possible fundamental entities, hence examining regions smaller than the length of a string cannot, again even in principle, have any physical meaning. As long as you have reached the size of a single string, there is no way further. In String Theory there could be no such thing as a dimensionless particle, which is good because that would not allow the realisation of a quantum theory of gravity. That means that Riemannian geometry, which has defined the very way for measuring distances since 1915, has to be modified by String Theory for the ultramicroscopic realm.

General Relativity has had tremendous success in the last century, and the description of macrosystems in terms of Riemannian geometry had no noticeable obstacles in that domain. In the ultramicroscopic domain of String Theory, however, the Riemann model cannot be applicable because the fundamental entities have some non-zero length. This geometrical model has to be replaced with the next level model – quantum geometry. And this change seems to imply some new extraordinary effects as we shall see shortly.

Applications for Cosmology

As we discussed in the earlier chapters, according to Friedmann’s equation the fate of the Universe depends critically on one parameter – the overall energy density of the Universe. If that density is greater than some critical number, then the Universe would stop expanding at some point in the future and the collapse will begin. If the density is lesser than that critical number, then the Universe will expand forever, and the gravitational tug of matter would not be able to stop the expansion. If the value of energy density is equal to one, i.e. it equals the critical number, then the Universe will still expand forever, but the rate of expansion will asymptotically approach zero.

This reasoning, however, didn’t take into account dark energy that was discovered only in the last decade of the XX century. Now the resolution to the above conundrum seems to inevitably point to the Universe expanding forever at an ever accelerating rate. (There are some models, particularly the one, which is called the Cyclic Universe, developed by Paul Steinhardt and Neil Turok, in which the rate of expansion eventually ceases to increase due to the decay of dark energy, and another big bang occurs, creating the Universe anew. The model is very interesting and worth exploring, but it does not relate to our topic, so we won’t discuss it here.)

But although the collapse of the entire Universe to a single point now seems to be extremely unlikely, the question as to what would such a collapse mean for the geometrical structure of the Universe is extremely important. Moreover, it brings into consideration the topic of this article, quantum geometry. Let us consider such a collapse, but to make things simpler and more understandable we shall omit the complexity of the four dimensional Universe (or 10-dimensional according to String Theory). Instead, we are going to consider the “Wire-Universe”, which we introduced in the 8th chapter for explaining the main principles of the Kaluza-Klein model. That universe contains only 2 dimensions like the surface of a wire, and objects living in that universe could move only in two directions – back and forth and along the circumference of the wire. The ideas that we shall obtain while considering this simplified example will help us to understand the main principles which String Theory draws for the more complicated 10-dimensional universe.

Let’s imagine that the circumferential dimension starts out fully expanded, but then begins to contract. In that case our wire-universe would resemble more and more the “Lineland” – a universe with only one dimension. The question we are interested in here is whether the geometrical structure of our imaginary universe would allow us to distinguish between the model based on strings, and the model based on point-like particles.

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As you can see in the figure 1a above, particles can move in any direction on the 2-dimensional surface of the wire-universe, i.e. they can move along the extended linear dimension, along the circular dimension, or along any other path in those 2 dimensions. The same thing applies to strings, but for strings there is one additional possibility shown in the figure 1b. Closed strings can wrap around the circular dimension. In that case a string would still vibrate and do all sorts of crazy things strings typically do, but it would remain in this configuration, wrapped around the circular dimension. Actually, it could even wrap around that dimension multiple times. String theorists use the term “topological mode” for such strings. It should be clear that pointless particles cannot have this mode since they do not extend in any direction. Let’s try to figure out what this topological mode of motion may imply for both strings and the dimensions around which they wrap.

Physical Characteristics of Wrapped Strings

What is the main difference between wrapped and unwrapped strings? It lies in a minimally possible mass a string would have. This minimal mass of the wrapped strings is defined by the size of the circular dimension and the number of times the string wraps around it. This limit exists due to the length of the string, whose minimal value is defined again by the circumference of the circular dimension. The minimal value of the string’s length, in turn, defines the minimally possible value of its mass. The greater the length, the larger the mass, because with greater length the string, in a sense, grows. Since the circumference of the circular dimension is proportional to its radius, this radius determines the contribution of the topological mode to the mass of the string.

Here a reader might question how String Theory could be compatible with the existence of massless particles if all the strings possess some length, and hence have to possess some mass as well. This question is certainly reasonable. The answer lies in the inevitable presence of quantum fluctuations which can cancel out very small contributions to the string’s mass. In fact, this is exactly what happens with massless particles such as photons and gravitons. However, these quantum fluctuations are not capable of cancelling out the contribution to the mass of a wrapped string, and this draws a very important distinction between these types of strings.

How does the existence of these topological configurations influence the geometrical characteristics of the dimensions the strings are wrapped around? Let’s think about what happens in the last moments of the circular dimension’s collapse in the wire-universe. Here String Theory draws a completely different prediction from Einstein’s General theory of Relativity. According to String Theory, when the size of the dimension in question reaches the Planck length and continues to collapse, all the processes thereafter are completely identical to those in which the dimension expands out! This implies that when a collapsing universe reaches the Planck size, further collapse is forbidden by the String Theory mathematical apparatus, and the universe, instead, starts to expand. The rules of geometry in this situation are turned upside down. The circular dimension can contract to the Planck size, but the topological modes make its further collapse impossible, and instead changes it to expansion. Let us see why such a bizarre phenomenon takes place in String Theory.

The States of a String

The existence of the configurations considered above implies that the energy of a string has two sources: vibrations and topology. If we are to consider the wire-universe, we can see that both types of energy (or both contributions) depend on the radius of the curled up additional dimension. This dependency, however, would play no role in the Universe with dimensionless particles, because such particles, obviously, cannot wrap around our dimension. Let us first define the dependency of vibrational and topological contributions to the size of the additional dimension. It would be useful to know that the vibrational component itself can be divided further into two parts: vibrational patterns that were considered in chapter 6, and a simple motion of a string as a whole from one point to another. The entire vibrational component is a superposition of a string’s vibrations and its simple motion, but the first of these plays only a minor role for the current considerations, so we shall omit it for the time being and focus our attention on the ordinary motion of a string through space.

There are two important factors at play here. Firstly, the energy of this motion is inversely proportional to the radius of our rolled up dimension, which is a direct consequence of Heisenberg’s uncertainty principle. The smaller the radius, the more localized a string is, and hence its momentum (i.e. the energy of its motion) increases. This explains the aforementioned inverse proportionality. Secondly, as we saw earlier in this article, the topological contributions to string’s energy is linearly proportional to the radius of the additional dimension. This implies that a large radius corresponds to the large contributions of topological energy and small contributions of vibrational energy, and vice versa.

As a result of these considerations, we are left with a very important consequence. Any large radius of the circumferential dimension corresponds to a certain small radius, where the sum of topological and vibrational energies in the universe with a large radius is exactly equal to the corresponding sum in the universe with a small radius. But despite a huge difference in size, the physical characteristics depend only on the total amount of energy (the sum of topological and vibrational energy), which implies that there is no physical difference between these two states of the wire-universe!

Let us take an example to make things clearer. Suppose that our wire-universe under consideration has a circumferential dimension with a radius 10 times the Planck length. A string could be wrapped around this dimension any number of times (i.e. any positive integer). The number of times a string is wrapped around that dimension is called the winding number. The topological energy of our string, in turn, is defined by its length and is proportional to the radius times the winding number. In our case that would amount to 10 * ℓ * winding number (where ℓ denotes the Planck length). Additionally, any string can vibrate which provides it with additional energy. As we mentioned earlier, we are now interested only in the vibration that corresponds to a simple motion of a string through space, and that the energy provided by this motion is inversely proportional to the radius of the circumferential dimension, i.e. 1/r or 1/10*ℓ. This parameter is called the vibrational number, which can take up only integer values due to the main principle of quantum mechanics – the discreteness of energy.

If we know the radius of the curled up dimension, the winding number and the vibrational number, finding the total energy of a string is a trivial operation. In the table 1 below we can see the results for a number of different configurations in the wire-universe of radius 10ℓ. An entire table would be infinitely large since the winding and vibrational numbers can take up any arbitrary values, but the given results are sufficient for our understanding of what’s going on. In this table we plot the configurations with large topological contributions to energy (multiples of 10), and small vibrational contributions (multiples of 1/10). In the table 2 we have the opposite situation: the radius of the circumferential dimension here is equal to the 1/10 of the Planck length (ℓ/10), thus making the topological contributions multiples of 1/10, and the vibrational contributions multiples of 10.

Table 1: the wire-universe with large radius

Vibrational number

Winding number

Total energy

1

1

1/10 + 10 = 10.1

1

2

1/10 + 20 = 20.1

1

3

1/10 + 30 = 30.1

2

1

2/10 + 10 = 10.2

2

2

2/10 + 20 = 20.2

2

3

2/10 + 30 = 30.2

3

1

3/10 + 10 = 10.3

3

2

3/10 + 20 = 20.3

3

3

3/10 + 30 = 30.3

Table 2: the wire-universe with small radius

1

1

10 + 1/10 = 10.1

1

2

10 + 2/10 = 10.2

1

3

10 + 3/10 = 10.3

2

1

20 + 1/10 = 20.1

2

2

20 + 2/10 = 20.2

2

3

20 + 3/10 = 20.3

3

1

30 + 1/10 = 30.1

3

2

30 + 2/10 = 30.2

3

3

30 + 3/10 = 30.3

It may seem that these two tables are quite different, but upon closer inspection it becomes clear that the two contain the same elements in the total energy column, albeit they show up on different rows. To find an element in the table 2 that corresponds to that same element in the table 1 we just need to swap the vibrational and winding numbers. In other words, vibrational and topological contributions to the total energy take up each other’s values with changing the radius to its reciprocal (10 to 1/10 in our case). That is why there is no difference between these two radii when we take the total energy of a string into consideration. The change of radius from 10 to 1/10 is compensated precisely by swapping the vibrational and winding numbers. Furthermore, our choice of radius was made just for the sake of calculation simplicity, but a similar result would be obtained for any other radius!

Now, as you might remember, it was mentioned that we were previously considering the vibrational contribution only in terms of the ordinary motion of a string through space, leaving its own vibrations behind the scene. Those vibrations, however, also make their contribution to the total energy of a string as well as to their electric charge, and hence need to be included in calculations. What’s important to us here, though, is that these contributions do not depend on the radius. Thus even if these contributions were included in the tables 1 and 2, they would provide identical amounts to the total energy in both tables. Which means that they can be neglected if we want to compare the strings’ energies in the wire-universes of radius 10 and of radius 1/10, and need to be added only if we need to compute the precise values of the total energy. Consequently, we may conclude that the masses and electric charges of strings in the universe with radius 10 are equal to those in the universe with radius 1/10. And since these masses and charges are what determines the characteristics and behaviour of strings, and what governs the very laws of physics, there is no physical difference between these two universes! The results of any experiment in one universe would exactly match the results of a similar experiment performed in the other universe.

Three Important Questions

Given all of the above, there are three quite important questions remaining to be considered. Firstly, if we were creatures living inside the wire-universe, wouldn’t we just be able to measure the length of the circumferential dimension by some kind of a ruler? So why do we even speak about the impossibility of differentiating between those two radii? Secondly, weren’t we previously told that String Theory got rid of the scales that are below the Planck length? Why do we even need to talk about a circumferential dimension with a radius of ℓ/10 (i.e. 10 times smaller than the Planck length) if it is forbidden by String Theory? And finally, who cares about that two-dimensional wire-universe if we live and are interested in the universe with 9 (or, according to the M-theory, 10) spatial dimensions?

Let us start with the last of these three questions since it would provide a good background for dealing with the other two. Although we were exploring the realms of the wire-universe in the previous sections, that choice was made just for the sake of relative simplicity. If we took into consideration a universe with three extended and six compactified dimensions, the result would remain the same. Any circumference has a radius, and if we change it to its inverse, we would get a physically identical universe.

We currently don’t know whether the three unfolded dimensions in our universe expand out to infinity or close in on themselves eventually. The best estimates to date suggest that the Universe is flat, meaning that it’s not of a spheroidal shape like the Earth, but it could be the case that it is just so astonishingly vast that even our best telescopes do not have enough sensitivity to catch up the closed curvature of the Universe. Or, to put it in other words, when you stand somewhere on the surface of the Earth you cannot discern its round shape because you are not capable of seeing far enough. But if you could, you would be able to determine its actual shape. Likewise, our most advanced telescopes may see not sufficiently far for determining the round shape of the Universe. In which case the three familiar dimensions would take up a spherical shape and fall into the category of indistinguishable 10*ℓ and ℓ/10 radii.

For example, suppose the radius of the Universe is 1 trillion light years, which would be roughly equal to 1063 * ℓ. If String Theory is correct, then a completely identical universe would have a radius of ℓ/1063. That is the Planck length divided by 10 to the 63rd power! And since the Universe grows with time, its inverse will continue to shrink. But how is this even possible? How a 2 metres tall human being could even fit into such an infinitesimal universe? How such a piece could be identical to the vastness of the Universe we live in? And here we get to our second question: we were told in the previous chapters that String Theory forbids anything to have a size smaller than the Planck length. But if our radius r is greater than the Planck length, then 1/r is inevitably smaller. So what does actually happen? The answer to this question has to do with some non-trivial characteristics of space, and it will also provide the answer to the first of our three questions.

Notion of Distance in String Theory

The notion of distance is among such things that we familiarise ourselves with in infancy, and which we experience throughout our entire lives. In fact, it seems to be so clear to us that it’s easy to miss some very interesting concepts lying beneath the surface. In previous chapters we have already seen how Einstein’s theories of Relativity completely changed our understanding of space and time. Now it is time to look more closely at the notion of distance taking into account the insights provided by String Theory. In physics we often try to come up with and use such definitions that allow us to measure the quantity being defined. Sometimes we can make sense out of even the most obscure and abstract concepts by simply measuring the quantities they describe.

How do we give such a definition to the concept of distance? String Theory gives quite a peculiar answer to this question. In 1988 physicists Kumrun Vafa and Robert Brandenberger, then working at Harvard University, showed that there are two different but related definitions of distance in String Theory (assuming the form of an analysed dimension is circular). Both definitions have their own procedures for measuring distance experimentally, each being based on identifying the time it takes a probe moving with a fixed velocity to traverse a given distance. The difference between these procedures is in choosing the probe. In the first case the probe is a string moving within a given dimension whereas in the other it is represented by a string wound around that dimension like we saw in the figure 1b. The length of the probe (a string) used for measuring the distance explains the existence of two different definitions of that notion in String Theory. Conversely, pointless particles cannot wrap around our dimension (or cannot wrap around anything for that matter), and hence there is only one unique definition of distance in a theory based on particles.

What’s the difference between the two procedures? The answer found by Vafa and Brandenberger is anything but trivial. The main idea can be depicted with the use of uncertainty principle. Ordinary (not wound around the circular dimension) strings can freely move through space, thus they can be used to measure the circumference of the dimension under consideration. The circumference in this case is proportional to the radius r, where r is given in terms of the Planck length, ℓ. According to the uncertainty principle, the energy of the strings used for measurement is proportional to 1/r (if this seems obscure you might want to return to the chapter 6). On the other hand, earlier in this chapter we saw that the energy of wound strings is proportional to r. By the same argument it follows that if such strings are used as our probes they can discern the distances proportional to 1/r! It then follows that if both probes are used for the measurement of the circular dimension radius, ordinary strings would measure the radius r, while the wound strings would show the reciprocal, 1/r. And both of these measurements are completely valid! String Theory demonstrates that different probes would measure different radii, the results being inversely proportional to one another. And, what’s even more important, this applies to any measurement of distance, not only to the radius of the circular dimension.

But if String Theory claims there are two different definitions of distance, why have we never noticed the presence of the second one, neither in our everyday life nor by the means of sophisticated physical experiments? Doesn’t this fact disprove String Theory? Well, actually not. The answer lies in the value of r in our universe. When this value largely differs from 1 (again, one Planck unit) one of the definitions is astronomically hard to realise in experiment, whereas the other one simply shows up, so that experimenters do not have to bother themselves with finding it. And, as we know, the value of r in our universe is enormously higher than 1 Planck unit, so that the second definition of distance lies beyond what we are capable of achieving with current technology. But perhaps, it is not impossible, at least in principle.

The different complexity in the realisation of two procedures lies in the masses of the probes needed to perform an experiment (i.e. the different masses of high-energy topological mode and low-energy vibrational mode or vice versa). When the radius r is very far from the Planck length, low-energy probes would have very small masses, whereas the high-energy ones would be no less than one billion billion times the mass of proton. The complexity of the procedure which uses those high-energy probes is beyond what is currently achievable because we cannot even create such heavy configurations of strings nowadays, but it might be possible in the future. In practice we have always used the low-energy configurations for measuring distances. Thus our intuition is based entirely on the definition provided by this configuration and hence we had no idea about the possibility that there might be another definition. All the previous chapters in this series used this familiar notion of distance, but now you might start to feel how our intuition about even the most familiar things might change if String Theory turns out to describe the world we live in.

In principle, however, we may say that in a universe described by String Theory one can choose any of the two procedures, whatever is more suitable. When astronomers measure the size of the observable Universe they speak about the radius roughly 46 billion light years, which corresponds to about 2.7*1061 Planck units. This measurement, however, is provided by photons that reach our telescopes. Those photons are low-energy string configurations, so this value provides one of two possible solutions. If the three dimensions in our universe are truly circular, and if String Theory is correct, then we could measure this radius to be equal to the reciprocal of that enormous number by using the topological string configurations. In effect, the low-energy configurations tell us that the Universe is huge and expands, while the high-energy ones would show that it is miniscule and contracts! This is not a discrepancy, but just two different – but equally meaningful – definitions of distance in String Theory.

Now it is time to answer the question as to how a 2-metre human can fit into a miniscule universe much smaller than the Planck length in size. Some bright sparks reading this may have already guessed the answer. The height of 2 metres is measured by the same low-energy probes which give the radius of the observable universe to be equal to around 46 billion light years. Much, much larger than 2 metres. Hence the very question is itself meaningless, because it uses two different definitions of distance to compare the results. What we need is use the same definition for both the person’s height and the radius of the Universe to ask any sort of questions and to make any statements.

The Minimal Size

Now we have the essential toolkit at our disposal to give an answer to the second important question as to what do we mean by distances smaller than the Planck unit if we were told that String Theory claims that no such distances exist. By the time we started this chapter, we had always considered the notion of distance provided by the low-energy string configurations. If we measure distance by this kind of configuration, we always obtain the result greater than the Planck length. To understand this we shall envisage what would happen in a hypothetical situation where the Universe were to collapse in on itself in a Big Crunch. Suppose our low-energy (light) string configurations tell us that the radius of the Universe is huge, but it becomes smaller, so that the Universe collapses. While the Universe is collapsing, the light string configurations become heavier whereas the high-energy topological configurations become lighter. When the radius of the Universe reaches one Planck unit (i.e. r = ℓ) the masses of vibrational and topological modes become identical. Two previously different ways of measuring distances now lead to the same results because 1 is equal to its reciprocal.

While the radius decreases further, the topological modes become lighter than the vibrational modes, and hence since that time they will be used for measuring distances because of a much simpler access. And since the results of any distance measurements obtained by this method are always inversely proportional to what we see using the other method, the radius of our hypothetical universe from now on would again be greater than the Planck length, and it will be increasing! This can be explained by quite a simple consideration: when radius r becomes smaller, its reciprocal 1/r increases in magnitude, and after r continues to collapse after reaching the Planck unit, 1/r continues to increase. Consequently, if we always use the lighter string configurations (which are much easier to access) when we measure distances, the minimal size would never be less than the Planck unit. Which implies that in a process of Big Crunch the radius of the Universe never gets to zero, and hence there would be no singularities, which have troubled physicists for many decades. From the point of view of an experimenter who always uses the easily accessible method to measure distance, the radius of the Universe shrinks to the Planck length but increases thereafter. Collapse is changed to expansion in this scenario.

The use of light string modes conforms to our familiar notion of distance which had been known long before String Theory came out. This very notion is responsible for those seemingly unsolvable problems with quantum fluctuations which were described in chapter 5. If the measured distance gets below the Planck unit, General Relativity and Quantum Mechanics do not go hand in hand. But in String Theory such distances are never accessible if we use that familiar notion of distance. In the physical formulation of GR and in the mathematical formulation of Riemannian geometry there is only one notion of distance, which allows it to be arbitrarily small. On the other hand, in the physical formulation of String Theory and in the mathematical formulation of still being developed quantum geometry there are two notions of distance. Their careful use conforms to our intuition and to General Relativity if the distance being described is sufficiently large, but it radically differs from those once we get to extremely small distances.

Based on these ideas Brandenberger, Vafa and other physicists offered a solution in which the laws of cosmology are rewritten in such a way that in either a Big Bang or a Big Crunch the size of the Universe is never equal to zero, but instead the minimal size equals the Planck length. This is a very interesting proposal for getting rid of the physical, mathematical and logical inconsistencies in our understanding of the Universe. If this proposal is somehow confirmed, the very start of the Universe would no longer require a point with infinite density and zero size. It has surely led to a new field of study for string theorists – string cosmology, which we shall discuss in later articles.

What if Dimensions are not Circular?

You might have noticed that in the above considerations we mentioned a few times that the dimensions in our hypothetical universe were circular. What if, you might ask, this is not the case, and they have some other form? Would our claims about the minimal possible size still hold in this case? No one has an answer to this question. A very important property of a circular dimension is that a string can wrap around it. If our dimensions had some other form, but strings could still wrap around them, all of the things we came up with would be applicable. But what if they had such a shape that strings had no way of having the topological configuration? Does String Theory still rule out the possibility of distances having arbitrarily small values in this case?

According to the theoretical research on String Theory, the answer to this question depends on whether an entire dimension or only a region of space is collapsing. The majority of researchers believe that there is a minimal allowable size independent of the form of a given dimension, and the mechanism of its occurrence is similar to the one we described for circular dimensions.

Mirror Symmetry

When physicists had first come up with the above idea some string theorists tried to push it even further. What if, they speculated, not only the shapes having reciprocally proportional sizes, but also the ones differing from each other in their form, led to physically similar universe? In the late 1980s physicists Lance Dixon, Cumrun Vafa, Wolfgang Lerche, and Nicholas Warner made a proposal according to which two very different Calabi-Yau manifolds might lead to identical physical characteristics.

This seemingly unrealistic idea might turn out to be correct due to the following considerations. First of all, recall that in chapter 9 we saw that the number of generations of particles in String Theory corresponds to the number of “openings” in the additional compactified dimensions (look up the figures 1 and 2 on that page for reference). And although those openings can have different dimensionalities, the number of particle generations is dependent upon only their overall number. That is, we do not care about the number of openings with particular dimensionality, and are focussing our attention only on their sum. Now assume that we have two Calabi-Yau manifolds where the number of openings with particular dimensionality is different, but the overall number of those openings is similar. The two are clearly different Calabi-Yau shapes, but they lead to the same number of particle generations! Of course, we are now talking only about one physical parameter being similar, the equivalence of all the parameters would require a much stronger justification. However, even this small step already gives some credence to the hypothesis.

The next major step in this direction was made by Brian Greene alongside Ronen Plesser, who was studying, at Harvard at that time. They used the so-called method of Orbifolds found in the mid 1980s by Dixon, Vafa, Jeffrey Harvey and Edward Witten. This method, quite loosely speaking, represents ‘sticking together’ some points on a given Calabi-Yau shape in accordance to a mathematical scheme which guarantees that the final shape is going to be a Calabi-Yau manifold as well. After a few months of gruelling work Greene and Plesser obtained a very surprising result. Given certain conditions, if you are to stick together some particular points on an initial manifold, then the resulting manifold differs from it, but that difference is very subtle. The number of openings with even dimensionality on the resulting manifold is equal to the number of openings with odd dimensionality on the initial one, and vice versa. This implies that the two manifolds are very different from each other, but on the other hand the overall number of openings is the same, hence the number of particle generations is also the same!

Inspired by the obvious connection between their results and those obtained by the Dixon-Vafa-Lerche-Warner group, Greene and Plesser started to investigate the central question as to whether or not two such manifolds with the same number of openings would be identical regarding other physical characteristics. And after a few months of scrupulous mathematical analysis the two came to a positive answer. Such pairs of Calabi-Yau shapes have since been known as Mirror Manifolds. Although these manifolds cannot be considered as being, literally, ‘mirror reflections’, if they are used as the additional dimensions in String Theory, they lead to physically indistinguishable universes.

At the same time another group, led by Philip Candelas, independently found out that a far larger number of Calabi-Yau shapes go in such pairs where members of each have the same property: even-odd dimensionality correspondence. It so happened that both of these results go hand in hand: Greene and Plesser found that both members of such pairs lead to identical physical characteristics, while Candelas et al. showed that this correspondence shows up for a very large number of Calabi-Yau manifold pairs.

Next time we shall be talking about some other intriguing ideas found in String Theory. Particularly, about the possibility of the very space-time fabric being torn apart and getting reglued anew.

Previous articles of the series can be found at:

Part 1: Following Einstein’s Dream

Part 2: Special Relativity – The Picture of Space and Time

Part 3: General Relativity – The Heart of Gravity

Part 4: Quantum Mechanics – The World of Weirdness

Part 5: General Relativity vs Quantum Mechanics

Part 6: The Basic Principles of String Theory

Part 7: Supersymmetry

Part 8: Hidden Dimensions

Part 9: Road to Experimental Proof

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