*Click the yellow picture twice to see an animation (2 MB). Sorry, the picture is kind of rotating because I forgot 1/2 of the modes. But all the other qualitative features are OK. Update: Fine, people couldn't live without it, so here you have the animation with all modes which is not rotating.*

This simple picture represents a typical shape of a ground state closed string in perturbative string theory. The periodic animation has 300 pictures in it. However, I was only summing the Fourier modes from -10 to +10. If I were summing a greater number, the picture would be much more complicated.

For infinitely many (M) modes, the string would have a huge number of twists and turns and its typical size would diverge logarithmically, as sqrt(log(M)). Although I was just presented as "The Buster of Infinities" in a Czech magazine, I must tell you that this particular infinity is completely harmless and creates no divergences in cross sections and other observables.

The apparent complexity of the string is pretty much unmeasurable because of the uncertainty principle. As you move towards more sophisticated, more quantum theories, you find a larger amount of chaotic microscopic structure but this structure remains unobservable because of the uncertainty principle.

Imagine that you want to prove the divergent size of the string, i.e. that a piece of string is actually very far from the origin. You will find out that the convergent contributions arise from high-frequency modes around the string and their effects average out very quickly.

It may sound paradoxical but the situation is the following: if you send the UV cutoff on the world sheet to infinity, the typical closed string in the ground state occupies the whole Universe and it has infinitely many twists and turns. Nevertheless, the particle may behave as a nearly perfectly localized graviton or another particle.

The reason why it sounds paradoxical is that we are not intuitively used to quantum mechanics. In fact, the same thing occurs already in ordinary quantum mechanics of one particle. In Feynman's path-integral approach, the typical trajectory "x(t)" resembles the Brownian motion. However, in the classical limit, when you only probe low frequencies, the path integral is localized near the "center of mass" of these chaotic curves.

In the same way, the chaotic string that can extend to large distances secretly knows about its center-of-mass position. All the additional structure with infinitely many chaotic degrees of freedom is needed for locality of the quantum theory but most of this structure remains below the finest resolution allowed by the uncertainty principle.

These observations reappear in many situations in string theory. For example, the graviton in the BFSS matrix model is a complicated bound state of many D0-branes (some "elementary particles" in the model) whose internal size also diverges as a power law in N (which is faster than for the string!), where N is the number of D0-branes that should be sent to infinity in the physical limit.

Still, this "huge cloud" of D0-branes interacts like a point. Even if you imagine that two one-mile clouds of D0-branes are penetrating through each other, they behave as vastly separated particles if the centers of mass are just a few fermis away from one another.

The internal structure of black holes is probably analogous. A typical black hole microstate looks like a very chaotic configuration of exotic matter - let me say the word: a fuzzball - but low-frequency probes are unable to see this microscopic layer of reality. Instead, they lead you to believe that the black hole interior looks like the nearly empty smooth space envisioned by general relativity.

The main difference between the closed string in its ground state or the graviton in M(atrix) theory on one side - and a fuzzball on the other side - is that in the former case, the low-lying pure states are pretty much uniquely determined. On the other hand, there are many microstates corresponding to a black hole with given macroscopic parameters (size, mass, charge).

However, I believe that this difference doesn't quite kill the analogy. The thermal mixed state describing the black hole is almost as unique as the pure state of a closed string (or BFSS) graviton. String theory - as the most advanced quantum theory we have - where the "quantum" phenomena appear most radically - leads the physicists to study new hidden layers of reality that can't be directly observed in simple, doable experiments.

The progress goes in both directions: objects look increasingly more discrete but the internal description of these discrete states is increasingly continuous. There is no paradox here. The contrast between string field theory and holographic bounds is another example of this phenomenon.

String field theory is a method to describe string theory by objects that are as similar to quantum field theory as possible: when you require this outcome, the theory looks like a particular quantum field theory with infinitely many fields. All of their masses and interactions are determined by string theory but once they are determined, you can use calculational methods that pretty much resemble quantum field theory.

String fields have a huge amount of numbers in them. Classical mechanics was based on functions of time such as x(t). Classical fields expanded them to functions of many variables, E(x,y,z,t). Wave functions in quantum mechanics have a different interpretation but they are equally complex as classical fields.

However, wave functionals in quantum field theory are more complex: the wave functional is a gadget that gives you a complex number for every function E(x,y,z,t), roughly speaking. When you get to string field theory, you need to make one more step. The wave functional in string field theory returns a complex number for each functional which is a prescription to calculate a number out of a function X(sigma).

In this sense, the objects are as complicated as you would expect from a "third quantization". This situation may look gigantically complex: we deal with a kind of meta-meta-meta-mechanics. :-) The number of degrees of freedom looks more infinite than ever before.

However, when you measure the information that can actually be stored by objects in a region of space and operationally measured, you will find out that this information is actually smaller than ever before. In all frameworks describing the world - classical mechanics, classical fields, and quantum fields - it could have been argued that the amount of information that can be stored in volume V scales like V.

However, in quantum gravity, the maximum information only scales like the surface A: that's what the holographic entropy bounds guarantee. Emotionally speaking, such a conclusion about a radically dropping number of degrees of freedom "contradicts" the previous trend that was making the relevant mathematical objects ever more complex.

However, when you analyze the situation rationally, there's no contradiction here at all. The uncertainty principle prevents all the new numbers hiding in the meta-functionals to carry some actual physical information. Because those observables don't commute with each other, you can't assign them with independent numbers. Black hole complementarity is just another natural step in Nature's successful efforts to make Her truly microscopic and essential organs unobservable.

For infinitely many (M) modes, the string would have a huge number of twists and turns and its typical size would diverge logarithmically, as sqrt(log(M)). Although I was just presented as "The Buster of Infinities" in a Czech magazine, I must tell you that this particular infinity is completely harmless and creates no divergences in cross sections and other observables.

The apparent complexity of the string is pretty much unmeasurable because of the uncertainty principle. As you move towards more sophisticated, more quantum theories, you find a larger amount of chaotic microscopic structure but this structure remains unobservable because of the uncertainty principle.

Imagine that you want to prove the divergent size of the string, i.e. that a piece of string is actually very far from the origin. You will find out that the convergent contributions arise from high-frequency modes around the string and their effects average out very quickly.

It may sound paradoxical but the situation is the following: if you send the UV cutoff on the world sheet to infinity, the typical closed string in the ground state occupies the whole Universe and it has infinitely many twists and turns. Nevertheless, the particle may behave as a nearly perfectly localized graviton or another particle.

The reason why it sounds paradoxical is that we are not intuitively used to quantum mechanics. In fact, the same thing occurs already in ordinary quantum mechanics of one particle. In Feynman's path-integral approach, the typical trajectory "x(t)" resembles the Brownian motion. However, in the classical limit, when you only probe low frequencies, the path integral is localized near the "center of mass" of these chaotic curves.

In the same way, the chaotic string that can extend to large distances secretly knows about its center-of-mass position. All the additional structure with infinitely many chaotic degrees of freedom is needed for locality of the quantum theory but most of this structure remains below the finest resolution allowed by the uncertainty principle.

These observations reappear in many situations in string theory. For example, the graviton in the BFSS matrix model is a complicated bound state of many D0-branes (some "elementary particles" in the model) whose internal size also diverges as a power law in N (which is faster than for the string!), where N is the number of D0-branes that should be sent to infinity in the physical limit.

Still, this "huge cloud" of D0-branes interacts like a point. Even if you imagine that two one-mile clouds of D0-branes are penetrating through each other, they behave as vastly separated particles if the centers of mass are just a few fermis away from one another.

The internal structure of black holes is probably analogous. A typical black hole microstate looks like a very chaotic configuration of exotic matter - let me say the word: a fuzzball - but low-frequency probes are unable to see this microscopic layer of reality. Instead, they lead you to believe that the black hole interior looks like the nearly empty smooth space envisioned by general relativity.

The main difference between the closed string in its ground state or the graviton in M(atrix) theory on one side - and a fuzzball on the other side - is that in the former case, the low-lying pure states are pretty much uniquely determined. On the other hand, there are many microstates corresponding to a black hole with given macroscopic parameters (size, mass, charge).

However, I believe that this difference doesn't quite kill the analogy. The thermal mixed state describing the black hole is almost as unique as the pure state of a closed string (or BFSS) graviton. String theory - as the most advanced quantum theory we have - where the "quantum" phenomena appear most radically - leads the physicists to study new hidden layers of reality that can't be directly observed in simple, doable experiments.

The progress goes in both directions: objects look increasingly more discrete but the internal description of these discrete states is increasingly continuous. There is no paradox here. The contrast between string field theory and holographic bounds is another example of this phenomenon.

String field theory is a method to describe string theory by objects that are as similar to quantum field theory as possible: when you require this outcome, the theory looks like a particular quantum field theory with infinitely many fields. All of their masses and interactions are determined by string theory but once they are determined, you can use calculational methods that pretty much resemble quantum field theory.

String fields have a huge amount of numbers in them. Classical mechanics was based on functions of time such as x(t). Classical fields expanded them to functions of many variables, E(x,y,z,t). Wave functions in quantum mechanics have a different interpretation but they are equally complex as classical fields.

However, wave functionals in quantum field theory are more complex: the wave functional is a gadget that gives you a complex number for every function E(x,y,z,t), roughly speaking. When you get to string field theory, you need to make one more step. The wave functional in string field theory returns a complex number for each functional which is a prescription to calculate a number out of a function X(sigma).

In this sense, the objects are as complicated as you would expect from a "third quantization". This situation may look gigantically complex: we deal with a kind of meta-meta-meta-mechanics. :-) The number of degrees of freedom looks more infinite than ever before.

However, when you measure the information that can actually be stored by objects in a region of space and operationally measured, you will find out that this information is actually smaller than ever before. In all frameworks describing the world - classical mechanics, classical fields, and quantum fields - it could have been argued that the amount of information that can be stored in volume V scales like V.

However, in quantum gravity, the maximum information only scales like the surface A: that's what the holographic entropy bounds guarantee. Emotionally speaking, such a conclusion about a radically dropping number of degrees of freedom "contradicts" the previous trend that was making the relevant mathematical objects ever more complex.

However, when you analyze the situation rationally, there's no contradiction here at all. The uncertainty principle prevents all the new numbers hiding in the meta-functionals to carry some actual physical information. Because those observables don't commute with each other, you can't assign them with independent numbers. Black hole complementarity is just another natural step in Nature's successful efforts to make Her truly microscopic and essential organs unobservable.

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