Jee_Jee has asked the following question:
You can see a difference between the two situations: most kids can figure out that unlike eagles, elephants don't fly after some time – when they see no flying elephants. Jee_Jee has seen that no counterpart of string theory is being constructed with other objects – but he still believes that this non-existent entity exists. It must have been omitted because string theorists are stupid or they hide some dirty secret or something like that. Why does he believe such a thing instead of making the same straightforward conclusion as the kid makes about the flying elephants?
The short question actually encompasses numerous laymen's delusions whose invalidity has been shown in my texts for decades – with very limited results (the bath of stupidity in the world is almost infinite). Jee_Jee seems to assume every single one of these three fundamentally flawed basic assumptions:
The brain defect known as egalitarianism
The first one is the egalitarianism. Jee_Jee clearly assumes that if some mathematical constructions may be done starting with \(D-1=1\) strings and \(D=2\) world sheets (the strings' history in space), it must be possible to do the same thing with two-dimensional membranes and higher-dimensional branes, too. Everything must work "analogously", right?
That is simply never the case. When two objects \(X,Y\) are different, it is always incorrect to assume that things that may be done with \(X\) may be done with \(Y\) or vice versa. It is wrong to assume that \(f(X)=f(Y)\) – that they have the same properties. When things are different, they also have different properties and different operations may be applied to \(X\) and \(Y\).
Needless to say, it is mainly a well-known large set of the stupid people of the present era – the leftists – who suffer from this misconception. And they seem to assume the egalitarianism about everything. The most important subclass of such assumptions concerns the groups of people. Men must always have the same characteristics as women, races and nations must have the same characteristics as others. Well, they just don't. There is no reason why they should and simple observations show that indeed, when they're different, they have different properties, too.
But it's not just the sexes, races, and nations. People make egalitarian assumptions about mathematical structures and even numbers, too. For example, there exists a simple compact Lie group \(E_8\) whose dimension is \(248\). Surely, there must also exist a simple compact Lie group whose dimension is \(842\) or any other large number, right? Otherwise we would violate the equality of the numbers \(248,842\). Well, let me tell you a secret. The numbers \(248\) and \(842\) are not equal. They differ "quantitatively" but they also differ "qualitatively". Lots of complex operations or structures that exist with \(248\) appearing at a certain place simply don't exist with the number \(842\) substituted instead.
(A real-world example: there are octonions which are isomorphic to \(\RR^8\) as a vector space. So according to some simple people, including people who pretend to be theoretical physicists at top U.K. universities, there must also be \(2^k\)-ions for any integer \(k\), like 64-ions, right? Well, it's not the case. The octonions are the largest division algebra constructed out of reals in a similar way.)
String theory is unique
The second misconception that I listed is really a special example of the myth of egalitarianism – but a special example that is very important and prominent and that lives its own life. Laymen typically assume that "string theory" is just a phrase constructed by adding a random noun representing an object, "string", to the word "theory". Surely we can replace the "string" with any other analogous "object" and we will construct another theory that is on par with string theory.
Well, we can't. There is only one string theory. You know, sometimes it's being said that a whole is more than the sum of the pieces. Well, "string theory" is much more than the sum of "string" and "theory". Only when "strings" play the corresponding role in the construction of a "theory", you end up with one of the most precious set of ideas that are known to the mankind.
The first mathematical expression that was identified to arise from string theory was found by Gabriele Veneziano exactly 50 years ago. Half a century is a pretty long time but it's still true that we don't know any other "competitor" or "counterpart" of the exceptional mathematical structure where strings play a central role. The structure has the properties it has. If you replace one piece by a different one, you are almost guaranteed to end up with an inconsistent set of ideas – unless you're lucky, you replace many things properly, and you end up with a new, equivalent definition of string theory.
Mr Tau, a mute gentle magician from Czech fairy-tales, has a hat on his head: \(\hat\tau\). The hat is what actually allows all the magic. Germans clearly didn't call him "Herr Tau" because "Herr" would indicate he was a Nazi. ;-)
Quantum mechanics is primary, classical physics is derived from it (if the quantum mechanical starting point exists at all)
Finally, Jee_Jee obviously manipulates with the word "quantization" as if it were a straightforward procedure that is always possible. One starts with a classical theory or even a classical picture of an object (such as a string or a brane) and then he simply adds hats (like Mr Tau's hat on the picture above) which make the theory "quantum mechanical". In his world view, "quantization" is just a boring bureaucratic procedure that only requires patience. Unfortunately, that's an idea about the relationship between a classical theory and a quantum mechanical theory that most students believe when they go through some basic courses.
But this isn't how Nature works at all.
The "quantization" should be understood as a heuristic procedure that attempts to find a quantum mechanical theory whose "classical limit" is a particular classical theory. For this reason, "quantization" should be understood as a method to solve an "inverse problem". But in reality, the quantum mechanical theory is the generic, full-blown theory with \(\hbar \gt 0\) and the classical theory (like a theory of an oscillating classical string) is just the \(\hbar \to 0\) limit of the quantum mechanical theory.
The quantization is possible – a quantum mechanical theory may be found – if you start with a classical limit from a simple enough class, e.g. with the Hamiltonian of the form \(H=p^2/2m+V(x)\). But the idea that there is a one-to-one correspondence between classical theories and quantum mechanical theories is completely wrong.
Some quantum mechanical theories don't have any classical limits – the proper quantity written as \(\hbar\) may be impossible to define and/or the limit may refuse to exist (limits sometimes don't exist).
On the contrary and more importantly, for some classical theories, there simply exists no quantum mechanical theory whose \(\hbar \to 0\) limit would be equivalent to the classical theory. (The non-existence is completely analogous to the non-existence of a "relativistic counterpart" of a randomly picked non-relativistic theory.) It means that the "addition of the hats" simply isn't guaranteed to exist. These obstructions are analogous to Cumrun Vafa's "Swampland" but at a simpler level. Vafa's "Swampland" includes effective quantum field theories that can't be completed to a string theory vacuum (that also describes quantum gravity consistently). At a more primitive level, there exists a "Classical Swampland" of classical theories that can't be completed to a full-blown quantum mechanical theory.
Again, as implicitly or explicitly stated in all blog posts about the foundations of quantum mechanics, the main problem is that almost all the laymen, beginners, and eternal beginners keep on thinking classically. Classical physics and classical pictures are still at the center of their thinking. They sometimes listen to a sentence or two, write an equation or two that are "quantum mechanical" but those are always assumed to be just some cherries on a pie that really don't play any important role and that may only change some details.
All the important questions, e.g. all the qualitative ones, are still being answered by classical physics by these folks. But this is just completely wrong. Everything in Nature is quantum mechanical and when classical physics is enough to answer some qualitative question, it's just an example of a coincidence or good luck. If your thinking isn't fundamentally quantum mechanical, it is simply incorrect according to modern physics.
Some mathematics: why strings are unique
Fine. What goes wrong when we try to replicate the basic constructions from rudimentary perturbative string theory with branes substituted instead of strings?
Early chapters of string theory textbooks discuss the basic dynamics of the string world sheet, the two-dimensional submanifold of the spacetime that describes the history of a one-dimensional string. The coordinates along the world sheet are often called \(\sigma,\tau\) – the first one is a spatial coordinate along the string and the second one is a time, or closer to a time.
At each point \((\sigma,\tau)\) of the two-dimensional world sheet, we have \(X^\mu(\sigma,\tau)\), the numbers remembering the position of the corresponding point of the world sheet (a point on the string at some moment of time) in the spacetime. The index \(\mu\) is one arising from the spacetime vectors. We ultimately want the spacetime to have some metric tensor\[
g_{\mu\nu}(X^\kappa)
\] at each spacetime point. When this metric tensor field in the spacetime is known, it is possible to calculate the induced metric on the world sheet – we switch to \(\tau,\sigma\to \sigma^\alpha,\alpha=0,1\):\[
h_{\alpha\beta} = \partial_\alpha X^\mu \partial_\beta X^\nu g_{\mu\nu}(\sigma^\gamma).
\] My main point is that there is some metric tensor on the world sheet that is inherited from the spacetime. So the theory on the world sheet is unavoidably a theory of gravity. When we search for a quantum mechanical version of the theory, it is a theory of quantum gravity. And quantum gravity is hard – and when done in the straightforward way, it is inconsistent.
Why? Because we get all the divergences in the quantum multi-loop Feynman diagrams for the scattering of the gravitons, the physical excitations of the metric tensor. The only case when these inconsistencies are avoided is when there are actually no physical excitations of the metric tensor at all.
Now, the world sheet has \(D=2\) dimensions. The metric tensor is a symmetric one and it has \[
\frac{D(D+1)}{2}
\] components. However, the theory also has a diffeomorphism symmetry. The coordinates \(\sigma^\alpha\) may be replaced with new ones – by picking \(D\) functions of the coordinates \(\sigma^\alpha\). This freedom (a gauge symmetry of a general-relativity-like theory, now used for the world sheet or the world volume) may be used to eliminate \(D\) components out of \(D(D+1)/2\) components of the metric tensor. So the number of (off-shell) components of the metric tensor that are left is\[
\frac{D(D+1)}{2} - D = \frac{D(D-1)}{2}.
\] It is just like the number of components of an antisymmetric tensor. You can check that for \(D\geq 2\), the number above is still positive so there would still be some physical modes of the gravitational field, even after the gauge-fixing (a special choice of the coordinates \(\sigma^\alpha\)). So no consistent theory would exist. All such theories would be on par with a naively quantized Einsteinian gravity which is non-renormalizable.
However, we may find one additional symmetry on the world sheet or the world volume, the scaling symmetry whose transformations (Weyl transformations) are determined by a scalar field \(\varphi\) as a parameter. The metric tensor is just being multiplied by \(\exp(\varphi(\sigma^\alpha))\) at each point. If we manage to rewrite the theory in such a way, the number of components of the world sheet or the world volume metric tensor that are left undetermined will be\[
\frac{D(D-1)}{2} - 1 = \frac{(D+1)(D-2)}{2}.
\] And this number of components may actually vanish for a positive \(D\) but only if \(D=2\). It means that when \(D=2\), we may gauge-fix the diffeomorphism symmetry on the world sheet and (locally on the world sheet) eliminate all the "problematic" components of the metric tensor on the world sheet. We end up with a theory of a free string that oscillates nicely, even in quantum mechanics – because the quantum mechanical oscillation is equivalent to some Klein-Gordon theories on the world sheet (if the spacetime is flat).
We may also allow these strings to join and split – and that results in the interactions of the resulting particles in the spacetime. It is a non-trivial fact that the theory continues to be consistent even after the interactions are allowed. The rest of string theory may be uncovered by analyzing the strings, their joining and splitting, and strings with different boundary conditions. You also find branes in the theory but their properties depend on the "more fundamental" strings – they're not fundamental in the same thing as the strings and you can't use them as the starting point of a construction of the theory.
If you tried to do the same "straightforward derivation" but the \(D=2\) world sheets were replaced with \(D\gt 2\) world volumes, you would end up with a theory on the world volume that has some components of the metric tensor that cannot be eliminated. It would be a theory of gravity similar to Einstein's general relativity – but one on the world volume, not in the spacetime – and there is no known consistent way to "quantize" such a theory. The problems (non-renormalizable loop divergences) is the reason why we considered a theory of extended objects in the first place.
So it is only the strings whose dimensionality is high enough to make the theory smoother at short distances in the spacetime; but low enough to avoid a replication of the spacetime-like problems of general relativity in the world volume.
Obstructions preventing you from finding a quantum mechanical theory with a classical limit
In the explanation above, I finally argued that the "brane theory" of fundamental branes, constructed in analogy with strings, was impossible because of the same problems with "divergent multi-loop Feynman diagrams" that spoil the straightforward quantization of Einstein's equations in the spacetime. You could have written "analogous" things with a higher number of world sheet dimensions up to a certain point but at some moment of an attempted calculation, the expressions would become divergent, meaningless, and incurable.
When we end up with a non-renormalizable theory after we start with a classical field theory as a starting point, we may say that the corresponding quantum mechanical theory doesn't exist. Well, a theory with a given classical limit (such as Fermi's theory of weak interactions or Einstein's equations) may ultimately exist but it must contain "a greater collection of degrees of freedom" than just those assumed in the classical theory that we wanted to quantize (e.g. W-bosons, Z-bosons, and the Higgs bosons in the case of the weak interactions; and excited string states in the case of gravity).
Non-renormalizability isn't the only obstruction that prevents us from "quantizing" certain classical theories. An even nicer and "more fatal" problem that we may often face are anomalies. They are quantum mechanical corrections to symmetries or conservation laws. Typically, a conservation law for a current \(\partial_\mu J^\mu = 0\) gets some loop Feynman diagrams on the right hand side. By some choices of the renormalization procedures, such terms on the right hand side may often be set equal to zero for a single current or a single symmetry, but not for all the symmetries of the classical theory simultaneously.
When the anomalous symmetry is a gauge symmetry, the theory is rendered completely inconsistent because the gauge symmetry has to survive in order to "kill" (well, "decouple") the time-like mode of the photon or another gauge boson (whose existence would lead to predictions of negative probabilities for every other process). Anomalies usually require left-right-asymmetric interactions.
Even the ordinary Standard Model – whose left-right asymmetry is concentrated in the sector of the weak interactions – would be anomalous and therefore inconsistent if you omitted all leptons or if you omitted all quarks. You might think that the leptons (electrons etc.) and the quarks (\(u,c,t;d,s,b\)) are being added as "matter fields" to the Standard Model in the form of two independent packages, the leptons and the quarks, and you could add these packages separately. Classically, this statement looks obvious. When the Standard Model is consistent with both leptons and quarks, it should be consistent if you remove one of them, you could think.
But this is not the case. Leptons actually require quarks and vice versa. Only when leptons and quarks are added simultaneously, all the anomalies cancel and the classically expected gauge symmetries may be reproduced at the quantum level.
This is an example of a concept that is omnipresent in quantum field theories – including the quantum field theories that define the world sheet dynamics of strings (or, more problematically, the world volume dynamics of branes). Also, aside from anomalies and other inconsistencies, the quantum mechanical effects may qualitatively change the behavior of the classical theory that you assumed as an inspiration to construct the quantum mechanical theories. An example of that are the monodromies in the Seiberg-Witten \(\NNN=2\) gauge theory in four dimensions. Without quantum mechanics, you wouldn't even think of such monodromies, let alone about their being unavoidable.
The anomalies, monodromies, and other effects are proportional to \(\hbar\) and/or its higher power. They are invisible classically because \(\hbar=0\) in the classical theory but they are important in the full-blown, quantum mechanical theory and may render many proposed theories inconsistent. Note that if we focus our research on the characteristic properties of a quantum mechanical theory, we often set \(\hbar=1\) so the quantum effects are not "small" (or close to zero) in any sense – they're at least as important as everything else. It may be a good idea for me to give you the following explanation why quantum mechanics is capable of rendering many theories inconsistent even though they would be consistent classically. A result in the quantum mechanical theory may be computed in two different ways. The expressions could be \(C+2\hbar\) and \(C-3\hbar\). In the \(\hbar\to 0\) limit, both expressions are equal to \(C\) and everything is fine. But in quantum mechanics, \(\hbar\neq 0\) so the two expressions are different which is a straight inconsistency – an inconsistency that is simply invisible in the classical limit.
So objects that aren't equal must be assumed to have different, occasionally qualitatively different characteristics – and they are often capable or incapable of different treatments. String theory is a very special and unique mathematical structure. There is only one theory in theoretical physics that has this degree of depth and relevance to the explanations of the most fundamental features of the Universe – and it's string theory. The uniqueness is analogous to the uniqueness of \(E_8\) among the exceptional compact simple Lie groups or the uniqueness of the monster group among the sporadic groups. The uniqueness of \(E_8\) and the monster group among the isolated groups of the two kinds has been rigorously proven. The uniqueness of string theory hasn't been proven quite rigorously but the evidence supporting the uniqueness is extremely strong and, within "patches" where extra assumptions are made about the form of the theory or description, the restricted proofs are rigorous, too.
And quantum mechanics isn't just a cherry on a pie. It's the primary theory, classical physics is derived from quantum mechanics as a limit (not the other way around), and if there's no quantum mechanical theory with a certain classical limit, it means that this classical limit is prohibited in Nature. Fundamental, weakly coupled branes are an example of that ban.
Dear Lubos, thanks for this article. Please excuse my ignorance but could you please briefly explain why we don't find quantization of higher dimensional objects (such as branes) discussed in various standard references? Shouldn't the rules of QM determine the dynamics of branes too? Or may be it is done somewhere but I have not been able to find it.Well, the reason why this "quantization" isn't discussed in any of these texts is that it is not possible. The question is analogous to the question: "Why don't textbooks of zoology feature photographs of flying elephants that would resemble the flying eagles?" You know, elephants don't fly. In the same sense, there is no "brane theory" that would be fully analogous to "string theory".
You can see a difference between the two situations: most kids can figure out that unlike eagles, elephants don't fly after some time – when they see no flying elephants. Jee_Jee has seen that no counterpart of string theory is being constructed with other objects – but he still believes that this non-existent entity exists. It must have been omitted because string theorists are stupid or they hide some dirty secret or something like that. Why does he believe such a thing instead of making the same straightforward conclusion as the kid makes about the flying elephants?
The short question actually encompasses numerous laymen's delusions whose invalidity has been shown in my texts for decades – with very limited results (the bath of stupidity in the world is almost infinite). Jee_Jee seems to assume every single one of these three fundamentally flawed basic assumptions:
- Some kind of egalitarianism. In this case, if something can be "done" with strings, it may be done with any other objects such as branes.
- String theory is a collection of cheap operations and lots of other analogous things may be constructed. The myth about the cheapness or genericity of string theory is encouraged by self-described critics of string theory i.e. crackpots such as Sabine Hossenfelder.
- Classical physics is the starting point while quantum mechanical theories are derived – and the quantization is just a formality or a cherry on a pie that is always possible to add.
The brain defect known as egalitarianism
The first one is the egalitarianism. Jee_Jee clearly assumes that if some mathematical constructions may be done starting with \(D-1=1\) strings and \(D=2\) world sheets (the strings' history in space), it must be possible to do the same thing with two-dimensional membranes and higher-dimensional branes, too. Everything must work "analogously", right?
That is simply never the case. When two objects \(X,Y\) are different, it is always incorrect to assume that things that may be done with \(X\) may be done with \(Y\) or vice versa. It is wrong to assume that \(f(X)=f(Y)\) – that they have the same properties. When things are different, they also have different properties and different operations may be applied to \(X\) and \(Y\).
Needless to say, it is mainly a well-known large set of the stupid people of the present era – the leftists – who suffer from this misconception. And they seem to assume the egalitarianism about everything. The most important subclass of such assumptions concerns the groups of people. Men must always have the same characteristics as women, races and nations must have the same characteristics as others. Well, they just don't. There is no reason why they should and simple observations show that indeed, when they're different, they have different properties, too.
But it's not just the sexes, races, and nations. People make egalitarian assumptions about mathematical structures and even numbers, too. For example, there exists a simple compact Lie group \(E_8\) whose dimension is \(248\). Surely, there must also exist a simple compact Lie group whose dimension is \(842\) or any other large number, right? Otherwise we would violate the equality of the numbers \(248,842\). Well, let me tell you a secret. The numbers \(248\) and \(842\) are not equal. They differ "quantitatively" but they also differ "qualitatively". Lots of complex operations or structures that exist with \(248\) appearing at a certain place simply don't exist with the number \(842\) substituted instead.
(A real-world example: there are octonions which are isomorphic to \(\RR^8\) as a vector space. So according to some simple people, including people who pretend to be theoretical physicists at top U.K. universities, there must also be \(2^k\)-ions for any integer \(k\), like 64-ions, right? Well, it's not the case. The octonions are the largest division algebra constructed out of reals in a similar way.)
String theory is unique
The second misconception that I listed is really a special example of the myth of egalitarianism – but a special example that is very important and prominent and that lives its own life. Laymen typically assume that "string theory" is just a phrase constructed by adding a random noun representing an object, "string", to the word "theory". Surely we can replace the "string" with any other analogous "object" and we will construct another theory that is on par with string theory.
Well, we can't. There is only one string theory. You know, sometimes it's being said that a whole is more than the sum of the pieces. Well, "string theory" is much more than the sum of "string" and "theory". Only when "strings" play the corresponding role in the construction of a "theory", you end up with one of the most precious set of ideas that are known to the mankind.
The first mathematical expression that was identified to arise from string theory was found by Gabriele Veneziano exactly 50 years ago. Half a century is a pretty long time but it's still true that we don't know any other "competitor" or "counterpart" of the exceptional mathematical structure where strings play a central role. The structure has the properties it has. If you replace one piece by a different one, you are almost guaranteed to end up with an inconsistent set of ideas – unless you're lucky, you replace many things properly, and you end up with a new, equivalent definition of string theory.
Mr Tau, a mute gentle magician from Czech fairy-tales, has a hat on his head: \(\hat\tau\). The hat is what actually allows all the magic. Germans clearly didn't call him "Herr Tau" because "Herr" would indicate he was a Nazi. ;-)
Quantum mechanics is primary, classical physics is derived from it (if the quantum mechanical starting point exists at all)
Finally, Jee_Jee obviously manipulates with the word "quantization" as if it were a straightforward procedure that is always possible. One starts with a classical theory or even a classical picture of an object (such as a string or a brane) and then he simply adds hats (like Mr Tau's hat on the picture above) which make the theory "quantum mechanical". In his world view, "quantization" is just a boring bureaucratic procedure that only requires patience. Unfortunately, that's an idea about the relationship between a classical theory and a quantum mechanical theory that most students believe when they go through some basic courses.
But this isn't how Nature works at all.
The "quantization" should be understood as a heuristic procedure that attempts to find a quantum mechanical theory whose "classical limit" is a particular classical theory. For this reason, "quantization" should be understood as a method to solve an "inverse problem". But in reality, the quantum mechanical theory is the generic, full-blown theory with \(\hbar \gt 0\) and the classical theory (like a theory of an oscillating classical string) is just the \(\hbar \to 0\) limit of the quantum mechanical theory.
The quantization is possible – a quantum mechanical theory may be found – if you start with a classical limit from a simple enough class, e.g. with the Hamiltonian of the form \(H=p^2/2m+V(x)\). But the idea that there is a one-to-one correspondence between classical theories and quantum mechanical theories is completely wrong.
Some quantum mechanical theories don't have any classical limits – the proper quantity written as \(\hbar\) may be impossible to define and/or the limit may refuse to exist (limits sometimes don't exist).
On the contrary and more importantly, for some classical theories, there simply exists no quantum mechanical theory whose \(\hbar \to 0\) limit would be equivalent to the classical theory. (The non-existence is completely analogous to the non-existence of a "relativistic counterpart" of a randomly picked non-relativistic theory.) It means that the "addition of the hats" simply isn't guaranteed to exist. These obstructions are analogous to Cumrun Vafa's "Swampland" but at a simpler level. Vafa's "Swampland" includes effective quantum field theories that can't be completed to a string theory vacuum (that also describes quantum gravity consistently). At a more primitive level, there exists a "Classical Swampland" of classical theories that can't be completed to a full-blown quantum mechanical theory.
Again, as implicitly or explicitly stated in all blog posts about the foundations of quantum mechanics, the main problem is that almost all the laymen, beginners, and eternal beginners keep on thinking classically. Classical physics and classical pictures are still at the center of their thinking. They sometimes listen to a sentence or two, write an equation or two that are "quantum mechanical" but those are always assumed to be just some cherries on a pie that really don't play any important role and that may only change some details.
All the important questions, e.g. all the qualitative ones, are still being answered by classical physics by these folks. But this is just completely wrong. Everything in Nature is quantum mechanical and when classical physics is enough to answer some qualitative question, it's just an example of a coincidence or good luck. If your thinking isn't fundamentally quantum mechanical, it is simply incorrect according to modern physics.
Some mathematics: why strings are unique
Fine. What goes wrong when we try to replicate the basic constructions from rudimentary perturbative string theory with branes substituted instead of strings?
Early chapters of string theory textbooks discuss the basic dynamics of the string world sheet, the two-dimensional submanifold of the spacetime that describes the history of a one-dimensional string. The coordinates along the world sheet are often called \(\sigma,\tau\) – the first one is a spatial coordinate along the string and the second one is a time, or closer to a time.
At each point \((\sigma,\tau)\) of the two-dimensional world sheet, we have \(X^\mu(\sigma,\tau)\), the numbers remembering the position of the corresponding point of the world sheet (a point on the string at some moment of time) in the spacetime. The index \(\mu\) is one arising from the spacetime vectors. We ultimately want the spacetime to have some metric tensor\[
g_{\mu\nu}(X^\kappa)
\] at each spacetime point. When this metric tensor field in the spacetime is known, it is possible to calculate the induced metric on the world sheet – we switch to \(\tau,\sigma\to \sigma^\alpha,\alpha=0,1\):\[
h_{\alpha\beta} = \partial_\alpha X^\mu \partial_\beta X^\nu g_{\mu\nu}(\sigma^\gamma).
\] My main point is that there is some metric tensor on the world sheet that is inherited from the spacetime. So the theory on the world sheet is unavoidably a theory of gravity. When we search for a quantum mechanical version of the theory, it is a theory of quantum gravity. And quantum gravity is hard – and when done in the straightforward way, it is inconsistent.
Why? Because we get all the divergences in the quantum multi-loop Feynman diagrams for the scattering of the gravitons, the physical excitations of the metric tensor. The only case when these inconsistencies are avoided is when there are actually no physical excitations of the metric tensor at all.
Now, the world sheet has \(D=2\) dimensions. The metric tensor is a symmetric one and it has \[
\frac{D(D+1)}{2}
\] components. However, the theory also has a diffeomorphism symmetry. The coordinates \(\sigma^\alpha\) may be replaced with new ones – by picking \(D\) functions of the coordinates \(\sigma^\alpha\). This freedom (a gauge symmetry of a general-relativity-like theory, now used for the world sheet or the world volume) may be used to eliminate \(D\) components out of \(D(D+1)/2\) components of the metric tensor. So the number of (off-shell) components of the metric tensor that are left is\[
\frac{D(D+1)}{2} - D = \frac{D(D-1)}{2}.
\] It is just like the number of components of an antisymmetric tensor. You can check that for \(D\geq 2\), the number above is still positive so there would still be some physical modes of the gravitational field, even after the gauge-fixing (a special choice of the coordinates \(\sigma^\alpha\)). So no consistent theory would exist. All such theories would be on par with a naively quantized Einsteinian gravity which is non-renormalizable.
However, we may find one additional symmetry on the world sheet or the world volume, the scaling symmetry whose transformations (Weyl transformations) are determined by a scalar field \(\varphi\) as a parameter. The metric tensor is just being multiplied by \(\exp(\varphi(\sigma^\alpha))\) at each point. If we manage to rewrite the theory in such a way, the number of components of the world sheet or the world volume metric tensor that are left undetermined will be\[
\frac{D(D-1)}{2} - 1 = \frac{(D+1)(D-2)}{2}.
\] And this number of components may actually vanish for a positive \(D\) but only if \(D=2\). It means that when \(D=2\), we may gauge-fix the diffeomorphism symmetry on the world sheet and (locally on the world sheet) eliminate all the "problematic" components of the metric tensor on the world sheet. We end up with a theory of a free string that oscillates nicely, even in quantum mechanics – because the quantum mechanical oscillation is equivalent to some Klein-Gordon theories on the world sheet (if the spacetime is flat).
We may also allow these strings to join and split – and that results in the interactions of the resulting particles in the spacetime. It is a non-trivial fact that the theory continues to be consistent even after the interactions are allowed. The rest of string theory may be uncovered by analyzing the strings, their joining and splitting, and strings with different boundary conditions. You also find branes in the theory but their properties depend on the "more fundamental" strings – they're not fundamental in the same thing as the strings and you can't use them as the starting point of a construction of the theory.
If you tried to do the same "straightforward derivation" but the \(D=2\) world sheets were replaced with \(D\gt 2\) world volumes, you would end up with a theory on the world volume that has some components of the metric tensor that cannot be eliminated. It would be a theory of gravity similar to Einstein's general relativity – but one on the world volume, not in the spacetime – and there is no known consistent way to "quantize" such a theory. The problems (non-renormalizable loop divergences) is the reason why we considered a theory of extended objects in the first place.
So it is only the strings whose dimensionality is high enough to make the theory smoother at short distances in the spacetime; but low enough to avoid a replication of the spacetime-like problems of general relativity in the world volume.
Obstructions preventing you from finding a quantum mechanical theory with a classical limit
In the explanation above, I finally argued that the "brane theory" of fundamental branes, constructed in analogy with strings, was impossible because of the same problems with "divergent multi-loop Feynman diagrams" that spoil the straightforward quantization of Einstein's equations in the spacetime. You could have written "analogous" things with a higher number of world sheet dimensions up to a certain point but at some moment of an attempted calculation, the expressions would become divergent, meaningless, and incurable.
When we end up with a non-renormalizable theory after we start with a classical field theory as a starting point, we may say that the corresponding quantum mechanical theory doesn't exist. Well, a theory with a given classical limit (such as Fermi's theory of weak interactions or Einstein's equations) may ultimately exist but it must contain "a greater collection of degrees of freedom" than just those assumed in the classical theory that we wanted to quantize (e.g. W-bosons, Z-bosons, and the Higgs bosons in the case of the weak interactions; and excited string states in the case of gravity).
Non-renormalizability isn't the only obstruction that prevents us from "quantizing" certain classical theories. An even nicer and "more fatal" problem that we may often face are anomalies. They are quantum mechanical corrections to symmetries or conservation laws. Typically, a conservation law for a current \(\partial_\mu J^\mu = 0\) gets some loop Feynman diagrams on the right hand side. By some choices of the renormalization procedures, such terms on the right hand side may often be set equal to zero for a single current or a single symmetry, but not for all the symmetries of the classical theory simultaneously.
When the anomalous symmetry is a gauge symmetry, the theory is rendered completely inconsistent because the gauge symmetry has to survive in order to "kill" (well, "decouple") the time-like mode of the photon or another gauge boson (whose existence would lead to predictions of negative probabilities for every other process). Anomalies usually require left-right-asymmetric interactions.
Even the ordinary Standard Model – whose left-right asymmetry is concentrated in the sector of the weak interactions – would be anomalous and therefore inconsistent if you omitted all leptons or if you omitted all quarks. You might think that the leptons (electrons etc.) and the quarks (\(u,c,t;d,s,b\)) are being added as "matter fields" to the Standard Model in the form of two independent packages, the leptons and the quarks, and you could add these packages separately. Classically, this statement looks obvious. When the Standard Model is consistent with both leptons and quarks, it should be consistent if you remove one of them, you could think.
But this is not the case. Leptons actually require quarks and vice versa. Only when leptons and quarks are added simultaneously, all the anomalies cancel and the classically expected gauge symmetries may be reproduced at the quantum level.
This is an example of a concept that is omnipresent in quantum field theories – including the quantum field theories that define the world sheet dynamics of strings (or, more problematically, the world volume dynamics of branes). Also, aside from anomalies and other inconsistencies, the quantum mechanical effects may qualitatively change the behavior of the classical theory that you assumed as an inspiration to construct the quantum mechanical theories. An example of that are the monodromies in the Seiberg-Witten \(\NNN=2\) gauge theory in four dimensions. Without quantum mechanics, you wouldn't even think of such monodromies, let alone about their being unavoidable.
The anomalies, monodromies, and other effects are proportional to \(\hbar\) and/or its higher power. They are invisible classically because \(\hbar=0\) in the classical theory but they are important in the full-blown, quantum mechanical theory and may render many proposed theories inconsistent. Note that if we focus our research on the characteristic properties of a quantum mechanical theory, we often set \(\hbar=1\) so the quantum effects are not "small" (or close to zero) in any sense – they're at least as important as everything else. It may be a good idea for me to give you the following explanation why quantum mechanics is capable of rendering many theories inconsistent even though they would be consistent classically. A result in the quantum mechanical theory may be computed in two different ways. The expressions could be \(C+2\hbar\) and \(C-3\hbar\). In the \(\hbar\to 0\) limit, both expressions are equal to \(C\) and everything is fine. But in quantum mechanics, \(\hbar\neq 0\) so the two expressions are different which is a straight inconsistency – an inconsistency that is simply invisible in the classical limit.
So objects that aren't equal must be assumed to have different, occasionally qualitatively different characteristics – and they are often capable or incapable of different treatments. String theory is a very special and unique mathematical structure. There is only one theory in theoretical physics that has this degree of depth and relevance to the explanations of the most fundamental features of the Universe – and it's string theory. The uniqueness is analogous to the uniqueness of \(E_8\) among the exceptional compact simple Lie groups or the uniqueness of the monster group among the sporadic groups. The uniqueness of \(E_8\) and the monster group among the isolated groups of the two kinds has been rigorously proven. The uniqueness of string theory hasn't been proven quite rigorously but the evidence supporting the uniqueness is extremely strong and, within "patches" where extra assumptions are made about the form of the theory or description, the restricted proofs are rigorous, too.
And quantum mechanics isn't just a cherry on a pie. It's the primary theory, classical physics is derived from quantum mechanics as a limit (not the other way around), and if there's no quantum mechanical theory with a certain classical limit, it means that this classical limit is prohibited in Nature. Fundamental, weakly coupled branes are an example of that ban.
Only strings, not branes, can be consistently quantized in the same way
![Only strings, not branes, can be consistently quantized in the same way]()
Reviewed by MCH
on
September 14, 2018
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