Why do physicists believe that string theory can be the "theory of everything"

Original author: Ethan Siegel
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The theory of strings is based on the idea that instead of zero-dimensional elementary particles, the Universe consists of one-dimensional strings. String

theory is one of the most ingenious, contradictory and unproved ideas of physics. It is based on a physical trend that has lived for many centuries - that at some fundamental level, all the various forces, particles, interactions and manifestations of reality are connected together as different parts of the same platform. Instead of four independent fundamental interactions — strong, electromagnetic, weak, and gravitational — there is one unified theory covering all of them.

In many ways, string theory is the best candidate for a quantum theory of gravity, combining interactions at the highest energy levels. And although there is no experimental evidence for this, there are convincing theoretical reasons for believing that this is the case. In 2015, the largest living string theory specialist, Edward Witten, wrote a paper that every physicist should know about string theory. And this is what it means - even if you are not a physicist.

The difference between standard interactions of quantum field theory (left) for point particles and interactions in string theory (right) for closed strings.

It's amazing how sometimes a lot in common is found in the laws of nature, concerning seemingly unrelated phenomena. The mathematical structures of such phenomena are often very similar, and sometimes even identical. The attraction of two massive bodies according to Newton's laws is almost identical to the attraction / repulsion of electrically charged particles. The oscillations of the pendulum are completely analogous to the movement of mass on a spring or a planet around a star. Gravitational waves, waves on water, light waves - they all have surprisingly similar properties, despite the fact that it comes from fundamentally different physical sources. And in the same vein, although many are not aware of this, the quantum theory of a single particle and the approach to the quantum theory of gravity are also similar to each other.

A Feynman diagram representing the scattering of two electrons — this requires summing up all possible histories of particle interactions.

Quantum field theory works like this: take a particle and produce a mathematical “summation of all its histories”. It is impossible to simply calculate where the particle was, and where it is now, and how it got there - since there is an internal and fundamental quantum uncertainty in nature. Instead, we summarize all the possible ways in which it could arrive at the current state (“past history”) with the corresponding probabilistic weights, and then we calculate the quantum state of one particle.

To work with gravity, and not with quantum particles, you need to change a little something. Since Einstein's General Theory of Relativity is not connected with particles, but with the curvature of space-time, we will not average all possible particle histories. Instead, we average all possible space-time geometries.

Gravity by the rules of Einstein and everything else (strong, weak and electromagnetic interactions) by the rules of quantum physics are two different sets of laws that govern everything in the Universe.

It is very difficult to work in three spatial dimensions, and when we encounter a complex physical problem, we often try to solve its simpler version first. If you go down one dimension, everything will become easier. The only possible one-dimensional surfaces are an open string, with two separate ends not connected to each other, or a closed string, the ends of which are connected and form a loop. In addition, the curvature of space — very complex in three dimensions — becomes a trivial matter. Therefore, if we want to add matter, we use a set of scalar fields (just like for a certain kind of particles) and a cosmological constant (working exactly as a member of the equation responsible for mass): a perfect analogy.

The additional degrees of freedom that a particle receives in several dimensions do not play a special role; as long as we can determine the impulse vector, this remains the main measurement. Therefore, in one dimension, quantum gravity looks just like a free quantum particle in any arbitrary number of dimensions.

A graph with vertices where three edges meet - the key component of constructing an integral along a path related to one-dimensional quantum gravity.

The next step is to include interactions and move from a free particle without scattering amplitudes or effective cross sectionsto the one that can have a physical role associated with the universe. Graphs similar to the one above allow us to describe the physical concept of action in quantum gravity. If we write down all possible combinations of such graphs and sum up them — applying the same laws as usual, for example, the law of conservation of momentum — we can complete the analogy. Quantum gravity in one dimension is very similar to the interaction of a single particle in any number of dimensions.

The probability of detecting a quantum particle in a particular place never equals 100%; probability is distributed in space and in time.

The next step is to move from one spatial dimension to 3 + 1 dimensions: where the Universe has three spatial and one time dimension. But this theoretical “upgrade” for gravity can be very difficult. You can find a different approach if we decide to work in the opposite direction.

Instead of counting the behavior of a single particle (a zero-dimensional entity) in any number of dimensions, perhaps we could calculate the behavior of a string, an open or closed (one-dimensional entity). And based on this, look for analogies to a more complete theory of quantum gravity in a more realistic number of dimensions.

Feynman diagrams (top) are based on point particles and their interactions. By transforming them into analogs for string theory (below), we obtain surfaces capable of possessing nontrivial curvature.

Instead of points and interactions, we immediately start working with surfaces, membranes, and so on. Having obtained a real multidimensional surface, we can bend it in non-trivial ways. We begin to observe her very interesting behavior; such that may be the basis of the curvature of space-time observed in the Universe in the framework of GR.

But although one-dimensional quantum gravity gives us quantum field theory for particles in a possibly curved space-time, by itself it does not describe gravity. What is missing in this puzzle? There is no correspondence between operators, or functions representing quantum-mechanical interactions and properties, as well as states, that is, how particles and their properties change over time. This “state-of-operator” correspondence was a necessary but missing ingredient.

But if we go from point particles to string entities, this correspondence manifests itself.

The deformation of the space-time metric can be represented by a fluctuation ('p'), and if applied to the string analogy, it will describe the fluctuation of space-time and correspond to the quantum state of the string.

In the transition from particles to strings, a real correspondence of operator-states appears. Fluctuations in the spacetime metric (that is, the operator) automatically represent the state in the quantum mechanical description of string properties. Therefore, the quantum theory of gravity in space-time can be created on the basis of string theory.

But this is not all that we will get: we will also obtain quantum gravity combined with other particles and space-time interactions, with those that correspond to other string operators in field theory. There is also an operator describing the fluctuations of the space-time geometry, and another one for the quantum states of the string. The most interesting thing in string theory is that it is capable of giving us a working quantum theory of gravity.

Brian Green makes a string theory presentation.

All this does not mean that the problem is solved, and that string theory is the path to quantum gravity. The great hope of string theory is that these analogies can hold on to all scales, and that there will be an unequivocal correspondence of the type of "one to one" string picture of the world and the Universe that we see around us.

So far, the picture of the world with strings and superstrings is consistent only in a few sets of dimensions, and the most promising of them does not give us the 4-dimensional Einstein gravity, which describes our Universe. Instead, we discover the Brans-Dicke 10-dimensional theory of gravity . To restore the gravity present in our Universe, it is necessary to “get rid” of six dimensions and direct the coupling constant ω to infinity.

If you have heard the term “compactification” as applied to string theory, this is simply a word for us to solve these puzzles. So far, many people assume the existence of a complete and convincing solution suitable for compactification. But the question of how to obtain Einstein gravity and 3 + 1 dimensions from a 10-dimensional theory remains open.

Two-dimensional projection of the Calabi-Yau manifold , one of the popular methods of compactification of additional, unnecessary measurements of string theory

String theory offers a path to quantum gravity, which few alternatives can match. If we draw reasonable conclusions about how mathematics works, we will be able to derive from it both GTR and the Standard Model. Today it is the only idea that gives us this - that’s why they are so desperately chasing it. It doesn’t matter whether you advocate for the success of string theory or failure, or how you feel about the absence of verifiable predictions, it undoubtedly remains one of the most active areas of theoretical physics research. In essence, string theory stands out as the leading idea among physicists' dreams of a final theory.

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