Why are quantum mechanics and theory of relativity incompatible?

Original author: Dr. Dave goldberg
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Despite the fact that we have achieved some success in understanding the internal structure of the universe (Higgs boson, yeah), there are still gaping gaps in our knowledge. After all, why do we still not have the Theory of Great Unification and Theory of Everything? .. And why can't Einstein's General Theory of Relativity make friends with quantum mechanics?

By the way, why do we even need to be friends with them?

All our knowledge of the laws of the universe can be divided into two large groups. One will include quantum mechanics, from which the Standard Model grew up, along with all its fundamental particles and three interactions: electromagnetic, strong and weak. Another group includes the general theory of relativity developed by Einstein, which describes 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 know exactly how quantum mechanics and general relativity can unite in quantum gravity. Despite the number of curious theories about how to do this, I will not dwell on them now, but simply try to explain why this is necessary 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, using quantum mechanics, you can describe the density of, say, a cat, but this can only be done with a stretch of the imagination.

The effects of general relativity, 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 happens on the surface of neutron stars. In a word, GTR works on a large scale, starting from stellar systems and ending with the entire Universe.

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

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

In general, we are more or less able to describe them from the outside. But the deeper we get closer to their center, the less we understand what really happens there.


If you throw something beyond the event horizon of a black hole, it will never go back. Moreover, in a world where gravity is the main player, everything that enters a black hole will ultimately be a literally concluded point - the so-called “singularity”. At the time of the big bang, there was the same problem: an incredibly high density enclosed in an incredibly small space. In the first instant, probably infinitesimal.

We have never observed a “pure singularity” directly - and there are serious reasons to believe that we will never. 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 the predictions of general relativity, black holes have a literally zero radius, but something else happens in quantum mechanics. There is the principle of uncertainty in it, which, among other things, claims that we cannot fundamentally 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 cannot be enclosed in an area smaller than 10 ^ -73 meters. This size is breathtakingly small, but nonetheless non-zero.

If this were the only inconsistency between the quantum world and gravity (which, moreover, was probably already known to readers), they could be forgiven for skepticism about the scale of the tragedy.

But the real problems between GR and quantum mechanics begin much earlier than these scales of 10 ^ -73 meters.

Classical and Quantum Theories.

GR is a classical field theory that describes the universe as a continuous distribution of numbers - absolutely determined 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 remarked:
Mass tells spacetime how to bend, spacetime tells mass how to move

But quantum theories are completely different! In the quantum world, particles interact with each other using 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 “Two-Slit Experiment”. In it, a beam with electrons (or photons, or any other particles) passes through the screen with two narrow slots. 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 this phenomenon in itself is rather strange, but in the context of gravity it becomes absolutely incomprehensible. If an electron passes through one hole, it should create a slightly different gravitational field than if it passed through another.

Even stranger is that, according to a Wheeler experiment with a deferred choice, it is possible to create conditions under which the electron will choose 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 just does not happen.

Gravity is special.

There is an even deeper problem. Unlike, say, electricity, which interacts only with charged particles, gravity seems to interact with everything. All types 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 the quantum theory of gravity, at least in principle. Just like the main forces, we will have a particle-carrier of interaction, in absentia called the graviton, which will transmit the signal.

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

At the smallest distances, a swarm of gravitons with huge energy should create an incredible energy density, and this is where the problems begin. Gravity in theory should interact with all forms of energy, and since we generate an infinitely greater number of highly charged particles, they must create a strong gravitational field. You probably already see what the problem is. In the end, all the calculations end with a fan of infinities, climbing from everywhere.

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

We are trying to avoid these inconsistencies of theories with a process called renormalization. Renormalization is just a tricky way of saying that we do calculations only to a certain limit. It allows you to get rid of infinities in most theories and live quietly on. Because most interactions include only the difference of the two energies, it does not matter if you add or subtract a constant from all the data (even, apparently, if this constant is infinity), the result is still satisfactory.

Not everyone, of course, agrees with this. Great Richard Feynman said:
This trick that we do ... Technically, it is called renormalization. But no matter how clever a word he calls, I would call him crazy! Appeal to such focus hocus-pocus does not give us the right to assert that the theory of quantum electrodynamics is mathematically consistent. It is surprising that so far it has not been able to prove this; 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 affects all particles (as opposed to electromagnetism), these infinite energies pull the infinite curvature of space-time. And even 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 definitely should have a graviton, and since gravity seems to be spreading everywhere, a graviton (like a photon) should have zero mass, because heavy interaction carriers (such as W and Z bosons) can interact only at very small distances.

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

It is worth mentioning that GR is also known as the "tensor theory", because it describes all kinds of sources in combination with pressure, flow and energy density. Quantum versions of tensor theories describe particles-carriers of interaction with a spin of -2, so a graviton should also have such a spin. And - a surprise - even-spin interaction carriers attract identical particles, which is in excellent agreement with how gravity works.

Well, cheers to us. We still know something about how a graviton should look. But regarding all these infinities - hell, we have no idea what really happens!

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