In the last century our understanding of the Universe has gotten an unimaginable boost. Theoretical concepts of both General Relativity and Quantum Mechanics have allowed us to comprehend and, in some cases, even predict physical phenomena taking place on atomic and subatomic scales, and on the scales of the entire Universe, so to speak. This is truly a fundamental achievement. The fact that the civilization inhabiting a planet revolving about an ordinary star in quite an ordinary galaxy have managed to figure out such astonishing aspects of the physical world really impresses. But scientific institutions are built in such a way that scientists, and physicists in particular, are not going to stop until they have come to the deepest understanding and have figured out all the aspects of the physical world.

There is a good bit of evidence that GR and QM do not allow us to achieve this deepest understanding. This is what we touched upon in the first article of this series, and here we are going to consider this in a bit of detail. Since the application fields of these theories are so different, it is usually the case that *either* GR’s concepts *or* QM’s concepts need to be involved in solving a particular problem. But as we saw earlier, there are situations where both these theories are necessary to get a picture of what is going on. The centre of a black hole and the Universe as a whole in the moment of the Big Bang are a couple of such examples. But our attempts to combine the two lead to nothing but a catastrophe. For example, when we combine the equations of these theories together, a reasonable question leads to an answer which makes no sense at all, such as a probability equalling not 20 or 75 or 100 percent but *infinity*! But what does a probability greater than one, let alone infinity, even mean? In this case we have to conclude that there is a flaw in our understanding of physics. This inconsistency, which Brian Greene explains in a distinct chapter in his book “The Elegant Universe”, is what we shall focus on in this article.

**The Uncertainty Principle**

When Heisenberg derived his uncertainty principle physics concepts hugely shifted in a way which had never been imagined before. Probabilities, wave functions, quanta and all that demanded a radical change from a previously deterministic point of view. The uncertainty principle unambiguously brings about an indeterministic aspect into the physical framework. We considered this principle in the latest article, but for those who have not seen it, I should briefly mention what it leads to. According to the uncertainty principle, the Universe becomes extremely outrageous when we investigate space and time on micro scales. In the previous article we showed that there are some pairs of characteristics of a particle, whose exact values could not be known at the same time. One of such pairs is particle’s position and velocity. In order to define the exact position of a particle you have to light it up (bring a photon in contact with it). But the photon carries energy, hence it brings a high uncertainty in the value of particle’s velocity. And if we decrease the energy of a photon, its wavelength increases, which brings a high uncertainty in the position of a particle. What this tells us is that the world is essentially chaotic on its tiniest of scales.

This short explanation could bring up a natural question: does this uncertainty show up only when we – tactless observers – poke our nose into the microworld? No, this is *not the case*! Another example from the previous article, which considered the behaviour of a particle in a box whose edges are contracting, shows the fundamentality of the uncertainty principle clearer, since in this case we don’t bring a photon in contact with the particle in question. But this example does not uncover all the stunning aspects of the uncertainty principle either. What this principle shows is that even in the calmest situation which we can imagine – a completely empty space – there is miraculous activity on the subatomic scales. And this activity increases as we investigate the space-time fabric on smaller and smaller scales.

This conclusion is based on the fact that another pair of characteristics – energy and time – is also tied by the uncertainty principle. If you have a financial problem you could borrow some money to solve it, and then return the quantity back. Similarly, a particle could borrow energy from the Universe and then return it back. In this case though, the energy can be borrowed for a very short period of time. And the amount of this energy depends on how fast it will be returned. Thus, if a particle borrows energy for an infinitesimally small period of time, the amount of energy could be quite large.

The uncertainty principle shows that the exact energy and momentum values are *uncertain* on subatomic scales. These values fluctuate from one to another in a completely spontaneous manner. It seems like *nothing* (an empty region of space) borrows energy and momentum from the Universe and gives them back all the time. And here is a twist. Einstein’s formula **E = mc^2** tells us that energy can be transformed into matter and vice versa. For example, if the fluctuation of energy is sufficiently large, the borrowed energy could be transformed into matter, and a pair of particles with opposite electric charges (e.g. an electron and a positron) emerges out of what seemed as nothing. But because the energy must be returned very shortly, these particles immediately meet and annihilate each other, giving the energy off, hence they are called *virtual* particles. To speak a little bit more precisely, we can say that a region of space is empty when the intensity of *all* fields in that region is zero. But according to the uncertainty principle the amplitude of a wave and its rate of change are inversely proportional to each other. And since the intensity equalling zero implies zero amplitude, there is a high uncertainty in the rate of change of that amplitude, which means that at the next moment the amplitude will *not* be equal to zero! *On average* though, the amplitude remains undisturbed since in some places it takes a positive value whereas in others the value is negative. Quantum Mechanical uncertainty clearly shows that the Universe is highly blusterous and chaotic on its micro scales. But because the amount of borrowed energy on average equals the amount of returned energy, this tempestuous activity is never observed on normal scales. And as we shall see later, this chaos is the main obstacle for GR and QT to be merged together.

**Quantum Field Theory**

Throughout the 1930s and 1940s a huge number of physicists were working on finding a robust mathematical apparatus that would have helped to becalm the microscopic chaos. It was figured out then that Schrödinger’s equation represents just an approximation to the quantum mechanical realm since it does not take into account Einstein’s theory of relativity. In fact, Schrödinger initially tried to include Special Relativity in his equations, but this attempt was unsuccessful since the predictions based on it were inconsistent with experimental data. So he decided to make the first step towards the unified theory leaving Special Relativity out of consideration but providing a mathematical apparatus that was consistent with the idea of wave-particle duality and other experimental data. But later, physicists realised that Special Relativity is essential for the description of the microworld, since otherwise they did not take into account the interchangeability of matter, energy and momentum.

Initially physicists focused on the unification of Special Relativity and the part of Quantum Mechanics which describes the electromagnetic field and its interaction with matter. As a result, the theory known as *Quantum Electrodynamics* (QED) was developed. This was one of the theories dubbed *relativistic quantum field theory*. This theory is *relativistic* because it takes into account Special Relativity; *quantum *because it is formulated based on the principles of Quantum Mechanics; and it is a *field theory* because it combines the quantum theory with the notion of a classical force field, in this case Maxwell’s electromagnetic field.

QED is, without doubt, the most precise theory which has ever been developed. Physicists had been using it to derive the predictions (using the most powerful computers at a time) which then were experimentally confirmed to the precision of more than one billionth! What this means is that the results of theoretical considerations match experimental results up to nine decimal places and even more. This accordance of abstract mathematics with real-world experimental data is simply astonishing to say the least. The details of QED are so subtle and its role in physics is so vast that there are entire books written about it, for example this brilliant book by Richard Feynman, who was one of the main contributors to the development of QED.

The success of QED has led other physicists to try to describe other forces – strong, weak and gravitational – in a similar way, through a quantum field theory. This approach has proved very successful for strong and weak nuclear forces. Physicists have been able to describe these forces by the means of a quantum field theory; as a result *Quantum Chromodynamics* and *Electroweak* theories emerged. The former describes strong force with a fantastic precision, while the latter shows that both electromagnetic and weak nuclear forces have the same origin! With the conditions of unimaginably high temperatures and energies – which the Universe had a fraction of a second after the Big Bang – these two forces manifest themselves as one unified force. In a work, for which Sheldon Glashow, Abdus Salam and Steven Weinberg were jointly awarded with a Nobel Prize in physics in 1979, they showed that those two forces naturally merge together into one force in quantum field description even though they seem to have no commonalities in our cold Universe. In a fraction of a second after the Big Bang the temperatures dropped enough for these two forces to be separated out due to the process known as the *spontaneous symmetry breaking* which we shall consider later. Then the Universe continued to cool down so that we now have both these forces having very distinct properties.

So by 1970s physicists had got a very accurate description of three out of four fundamental forces of Nature – strong, weak and electromagnetic – and also showed that at least two of them can be unified in our physical framework. There have been many attempts to put strong nuclear force into this picture, in which case this would become a *Grand Unified Theory*, but so far no one has been able to accomplish this. However, the predictions of both Electroweak theory and Quantum Chromodynamics have been thoroughly tested with all sorts of adjustments, and so far this model has been proven correct countless times. Because of that, we call it the *Standard Model* of particle physics.

According to the Standard Model photons represent the smallest ‘packets’ of the electromagnetic field. Likewise, as we saw in the first article of this series, gluons and weak gauge bosons (W- and Z- bosons) represent the smallest components of strong and weak interactions respectively. Standard Model says that each of these particles is elementary, which means that they do not have any internal structure, just like quarks, electrons and neutrinos.

Photons, gluons and weak gauge bosons provide a microscopic mechanism for the transmission of interactions between matter particles. For example, two electrically charged particles with the same electric charge repel each other because they are surrounded by the swarm of photons whose interaction, in a sense, transfers the information to the particles that they must be repelled of each other. Similarly, two particles with opposite electric charge receive the ‘message’ according to which they must converge. Likewise, strong interaction is transmitted by gluons and weak interaction by weak gauge bosons.

**Symmetries**

You might have noticed that the quantum field theory leaves gravitational interaction behind the scene. But since we know that physicists successfully used this theory for the description of other forces, you might expect that such attempts were made. In such a theory, a particle carrying gravitational interaction would be *graviton*; and the connection of gravity with other forces becomes even clearer if we look at the examples of what is known as *gauge symmetries*.

First let us recall that according to Einstein’s theories of relativity, any observer – irrespective of their motion – could say that it is him, who is at rest, and all other observers are moving, so that all of their points of view have equal weight. Even those who move with acceleration could reconcile it by putting the appropriate gravitational interaction in. Thus, gravity provides a symmetry: it guarantees that all points of view, irrespective of their reference frame, have equal weight. Likewise, strong, weak and electromagnetic interactions are connected to other symmetries, even though these symmetries are far more abstract than that of gravitation.

In order to get an idea of these subtle sorts of symmetry, let us consider strong nuclear force. Every quark could be coloured into one of three ‘colours’ (bizarrely called red, green and blue, although these do not have any resemblance to the real colours). These colours determine quarks’ behaviour in response to the strong interaction, just like electric charge determines particles’ behaviour under electromagnetic interaction. Symmetry steps in when we consider the interaction of quarks with a particular colour. All interactions between quarks with the same colour (red-red, green-green and blue-blue) are identical. Similarly, all interactions between quarks with different colours (red-green, green-blue, blue-red) are also identical. But what’s even more surprising is that if we shifted three colours (three different strong charges, we could say) in a certain way, i.e. if we changed our red, green and blue to, say, magenta, lime and cyan, then even if the shifting parameters were to change from one point in space to another, and from one moment to another, the interaction between quarks would not change at all!

A good analogy could be drawn with a perfect sphere. A sphere is an example of a body having rotational symmetry: it will look the same irrespective of a point from which you are looking at it. In such a sense, we could say that our Universe has the *strong interaction symmetry*: physical phenomena do not change if the charges of strong interaction are shifted. This symmetry is the example of gauge symmetry, as was mentioned earlier.

And what’s particularly important here is that Hermann Weyl in 1920s and Chen-Ning Yang with Robert Mills in 1950s showed that gauge symmetry *demands* the existence of strong, weak and electromagnetic forces, just like the symmetry of all reference frames demands the existence of gravity! According to Yang and Mills, certain types of force fields *compensate* for the charges’ shift, keeping the interaction between particles changeless. In the case of gauge symmetry connected to the changes of quarks’ colours, the demanded force is nothing but strong nuclear force. This means that if there was no strong interaction, physical framework would change with such a colour shift. This shows that even though strong and gravitational interactions are so different, they are connected, in the sense that each of them is essential for maintaining certain kinds of symmetry. Moreover, the existence of electromagnetic and weak nuclear force is also connected to a certain kind of gauge symmetry. Therefore, all four known fundamental forces are directly connected to the principles of symmetry.

As we’ve just seen, we have long known that the four fundamental forces have quite a lot in common, which means that we should probably search for a quantum theory of gravitational interaction within the quantum field theory framework. This searching has been continuously kept up by many physicists for decades, but so far nobody has been able to accomplish it. In the last part of this article we shall try to figure out why this route has been so complex.

**Why Can We Not Live Together**

The laws of GR are usually applied on large scales: from everyday objects all the way up to planets, stars, galaxies and the Universe as a whole. According to Einstein’s picture, the absence of mass implies the flatness of the space-time structure in that region. In order to combine the laws of GR with those of QM we have to investigate the properties of space and time on the microscopic scales. Let us see what happens in this case.

You can see the successive diminishing of scales in the figure 1. The bottom in this figure represents an empty region of space on our everyday scales, and each next level shows the tiny areas of the same region investigated on smaller and smaller scales. As you can see, initially – on the first three steps of magnification – nothing happens at all, the structure of space keeps its initial guise. If we continued to magnify this structure taking into account only classical physics, we would expect to see the same picture with every successive magnification, irrespective of how small the investigated scales are. Quantum Mechanics, however, radically changes this picture. According to QM, *everything*, including gravitational field, experiences quantum fluctuations, which are caused by the uncertainty principle. Although classical picture says that gravitational field in empty space equals zero, quantum picture claims that it fluctuates from one value to another and equals zero only on average. Moreover, the uncertainty principle tells us that the magnitude of fluctuations grows with every successive magnification.

Since the presence of the gravitational field implies the curvature of space-time, those quantum fluctuations lead to dramatic deformation of space as shown in the figure 1. On the fourth level of magnification these deformations start showing themselves, but the overall structure remains quite smooth. On the fifth level, however, the deformations become incredibly strong, so that the space does not look flat and smooth at all – it becomes curved to an unimaginable extent. The investigated region of space takes on a turbulent and curled form. This is known as *quantum foam*, the term first suggested by John Wheeler. This is where the notions such as *left and right*, *up and down* and even *before and after* lose any meaning! It is here where we encounter the fundamental discrepancy between General Relativity and Quantum Mechanics. One of the basic principles of GR, the flat and smooth geometry of space-time, breaks down under furious conditions of microscopic realm. On subatomic scales the inherent property of quantum theory – the uncertainty principle – comes into contradiction with the basic principle of General Relativity – the flat and smooth geometric model of space and time.

This conflict manifests itself in a certain way. The calculations based on both GR and QM usually give a similar nonsensical answer – infinity. This implies that there is a fundamental flaw in our physical framework and that General Relativity just cannot cope with the furious aspects of quantum foam. Here I should mention that infinity also appeared in the results of the calculations based on other types of quantum field theory as well. However, physicists were able to calm down these infinities, using the process known as *renormalization*.

If we reversed our magnification process and went back to ordinary scales, however, the fluctuations of gravitational field would cancel each other out, hence we would see a flat region of space again. This is like an image you see on the web. If you look at an image it seems like the changes of colour occur in a continuous manner. If you sufficiently magnified the image, however, you would be able to see that it consists of discrete individual points – or pixels, as we call them. But in order to obtain the information about the pixellation of an image you had to magnify it; if we did not do that, the image would look smooth. Likewise, an empty space-time region would look flat and smooth on regular scales (and even on the scales of atoms), until we inspect those regions on extraordinarily tiny scales.

The principles of GR and QM allow us to calculate the approximate distances on which the devastating nature of quantum fluctuations would make space-time structure look like what is shown at the last level of magnification of the figure 1. The incredible smallness of both Planck’s constant and universal gravitational constant leads to the value of Planck’s length – which involves both of these constants – being so tiny that it is beyond imagining. Its value is approximately equal to *10 to the negative 35 metres*, which is one hundred-millionth of one billionth of one billionth of one billionth of a metre! *If we were to expand an atom to the size of observable universe, Planck’s length would correspond to merely the height of an average tree*!

If this conflict shows itself only on such fantastically tiny scales, should we care, you might ask. Some physicists would argue that such scales need not be considered in order to make and test physically meaningful predictions. Others, however, are highly concerned by the fact that two pillars on which the framework of theoretical physics currently holds are in a direct contradiction with each other.

As I’ve already mentioned, there have been lots of attempts to overcome this contradiction. But despite the fact that some of these attempts were very elaborate, they all have failed. This was the case until String Theory stepped in. (There are some other interesting approaches on combining the concepts of GR and QM, e.g. Roger Penrose’s Twistor Theory and Abhay Ashtekar’s, Ted Jakobson’s and Lee Smolin’s Loop Quantum Gravity which both are not of our concern in this series, but you can find plenty of information on them on the web.) Next time we shall finally start considering String Theory step by step.

Thanks everyone for your time!

## 4 thoughts on “String Theory – part 5: General Relativity vs Quantum Mechanics”