Why are quantum mechanics and relativity incompatible?

Despite the fact that we have achieved some success in understanding the internal structure of the universe (Higgs boson, aha), we still have gaping gaps in our knowledge. In the end, why do we still not have the Theory of the Great Unification and the Theory of Everything? .. And why Einstein’s General Theory of Relativity cannot make friends with quantum mechanics?

By the way, why do we need to be friends with them at all?
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All our knowledge of the laws of the universe can be divided into two large groups. One will be quantum mechanics, from which the Standard Model has grown, along with all its fundamental particles and three interactions: electromagnetic, strong and weak. Another group will be GTR, developed by Einstein, describing the fourth fundamental interaction - gravity, as well as black holes, the expansion of the universe, and even time travel.

Can they coexist together?

You probably already guessed that we do not exactly know how quantum mechanics and general relativity can unite in quantum gravity. In spite of the greater number of curious theories about how this can be done, I will not dwell on them now, but just try to explain why it is needed at all.

Two Kingdoms

Quantum mechanics and general relativity are usually applied on very different scales. For example, quantum mechanics has long remained a mystery to scientists because its effects become significant only on the scale of individual atoms. If you have a good imagination, you can imagine how with the help of quantum mechanics you can describe the density of, say, a cat, but this can be done only with a big stretch.

The effects of GR, in turn, become noticeable in strong gravitational fields. For example, time near the surface of the Earth flows more slowly, and light bends around clusters of galaxies. These phenomena can, in general, be ignored, but only until we want to understand, for example, what is happening on the surface of neutron stars. In short, GRT works on a large scale, from star systems to the entire Universe.

But there are very interesting places where GTR and quantum mechanics intersect.

For example, in black holes, excellent astrophysical laboratories. With relatively small sizes, they have an extremely strong gravitational field. Moreover, the first attempts to combine gravitational and quantum effects were first undertaken just at the border of black holes. For example, the famous Hokking Radiation , which, by the way, after billions of years should evaporate even the most massive black holes and inevitably lead to the thermal death of the universe.

In general, to describe them outside we have more or less obtained. But the deeper we approach their center, the less we understand what is actually happening there.

Singularities

If you throw something beyond the horizon of a black hole, it will never go back. Moreover, in a world where gravity is the main player, everything that ends up in a black hole will eventually be literally a point - the so-called “singularity”. At the moment of the big bang, there was the same problem: an incredibly large density enclosed in an incredibly small space. At first, probably infinitely small.

We have never seen “pure singularity” directly - and there are good reasons for believing that we will never be. This is rather sad from the point of view of its study, but, nevertheless, not so bad, considering that we will not be torn apart by gravitational forces.

According to GR predictions, black holes have literally zero radius, but in quantum mechanics something completely different happens. In it, there is the principle of uncertainty, which, among other things, asserts that we fundamentally can not determine the absolutely exact position of any particle of matter. In practice, this means that those entities that we call "particles" cannot be arbitrarily small. According to quantum mechanics, no matter how hard we try, a mass equal to the mass of the sun will not be able to be placed in an area smaller than 10 ^ -73 meters. This size is breathtakingly small, but nonetheless non-zero.

If this were the only discrepancy between the quantum world and gravity (which, moreover, probably was already known to readers), one could forgive them for skepticism regarding the scale of the tragedy.

But the real problems between general relativity and quantum mechanics begin much earlier than this scale of 10 ^ -73 meters.

Classical and Quantum Theories.

GRT is a classical field theory that describes the universe as a continuous distribution of numbers — absolutely deterministic numbers — unless, of course, you have sufficiently accurate tools to measure them. These numbers can tell everything about the curvature of space-time, everywhere and always. The curvature itself, in turn, is entirely described by mass and energy. John Wheeler accurately noted:

Mass tells space-time how to bend, space-time tells mass how to move

But quantum theories are completely different! In the quantum world, particles interact with each other with the help of other particles - carriers of interaction. Electromagnetic forces, for example, use photons, strong interaction - gluons, weak - W and Z bosons.

We do not need to dive into black holes to see the conflict between classical and quantum theories. Remember the famous "experiment with two slots." In it, a beam with electrons (or photons, or any other particles) passes through a screen with two narrow slits. Due to quantum uncertainty, there is no way to determine the specific slot through which an electron flies. It literally passes through both slots at the same time. Even in itself, this phenomenon is rather strange, but in the context of gravity it becomes completely incomprehensible. If an electron passes through one hole, it must create a slightly different gravitational field than if it passes through another.

Even more strange is that, according to Wheeler’s postponed experiment, it is possible to create conditions under which the electron will select a gap in the past , after retrospective observations at the end of the experiment. Go crazy, right?

In other words, the world of gravity must be absolutely deterministic; in quantum mechanics this is not happening.

Gravity is special

There is an even deeper problem. Unlike, say, electricity, which only interacts with charged particles, gravity seems to interact with everything. All kinds of masses and energies are influenced by gravity and create gravitational fields. Also, unlike electricity, there are no negative masses that could neutralize the positive ones.

We can imagine a quantum theory of gravity, at least in principle. Just like the main forces, we will have a particle-carrier of interaction, called the graviton in absentia, which will transmit the signal.

We can even imagine experiments carried out on a smaller and smaller scale, in which we will observe more and more virtual gravitons between particles. The problem is that on a small scale energy becomes more and more. For example, the nucleus of an atom is much more difficult to destroy than to tear an electron from it.

At very short distances, a swarm of gravitons with tremendous energy must create an incredible density of energy, and this is where the problems begin. In theory, gravity should interact with all forms of energy, and since we generate infinitely more high-charged particles, they must create the strongest gravitational field. You probably already see the problem. In the end, all the calculations end with a fan of infinity climbing from everywhere.

In electromagnetism and other quantum interactions, when going to very small scales, the results of calculations become extremely discouraging. This distance, also known as plank length, is many times smaller than an atom - only 10 ^ -35m. Once again, I note that now it is absolutely incomprehensible how the laws of nature should work on scales smaller than this distance. Quantum mechanics says that in this microcosm tiny black holes can appear and disappear absolutely randomly, thus assuming that space-time itself is far from evenly distributed if you take a closer look at it.

We try to avoid these inconsistencies of theories through a process called renormalization. Renormalization is simply an intricate way to say that we only do calculations to a certain limit. It allows you to get rid of the infinities in most theories and live peacefully on. Since most interactions include only the difference of two energies, it does not matter if you add or subtract a constant from all data (even, apparently, if this constant is infinity), the result is still satisfactory.

Not everyone, of course, agrees with this. The great Richard Feynman said:

This trick we are doing ... Technically, it is called renormalization. But no matter how clever he is called, I would call it insanity! Appeal to such hocus pocus does not give us the right to say that the theory of quantum electrodynamics is mathematically consistent. It is surprising that so far it has not really been able to prove it; I think that renormalization from the point of view of mathematics cannot be considered true in the full sense of the word.

Even despite these objections, things are even sadder with gravity. Since gravity acts on all particles (as opposed to electromagnetism), these infinite energies pull the infinite curvature of spacetime behind them. And even the renormalization does not allow us to get rid of it.

What do we know?

Despite the fact that we do not have a theory of quantum gravity, we have some idea of ​​how it should look. For example, it should definitely have a graviton, and since gravity seems to propagate everywhere, graviton (like a photon) must have zero mass, because heavy interaction carriers (such as W and Z bosons) can interact only at very small distances.

There are also interesting connections between classical and quantum theories. For example, electromagnetism is generated by electrical charges and currents. In the mathematical model, these sources must produce particles - carriers of interaction with spin -1. Such particles with an odd spin must create repulsive forces - and indeed, the two electrons will repel each other.

It is worth mentioning that GTR is also known as the “tensor theory”, since it describes all kinds of sources in combination with pressure, flow, and density of energy distribution. Quantum versions of tensor theories describe particle carriers of interaction with spin -2; therefore, the graviton must also have such a spin. And - surprise - carriers of interaction with an even spin attract the same particles, which is in excellent agreement with the way gravity works.

Well, hooray us. We still know something about how a graviton should look. But regarding all these infinities - the devil, we have no idea what all the same is really happening!