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First-quantized formulation of string theory is healthy

...and enough to see strings' superiority...

As Kirill reminded us, two weeks ago, a notorious website attracting unpleasant and uintelligent people who just never want to learn string theory published an incoherent rant supplemented by misguided comments assaulting Witten's essay

What every physicist should know about string theory
in Physics Today. Witten presented the approach to string theory that is common in the contemporary textbooks on the subject, the first-quantized approach, and showed why strings eliminate the short-distance (ultraviolet) problems, automatically lead to the gravity in spacetime, and other virtues.

Witten's office as seen under the influence of drugs

This introduction is simple enough and I certainly agree that every physicist should know at least these basic things about string theory but at the end, I think that it isn't the case, anyway. Here I want to clarify certain misunderstandings about the basics of string theory as sketched by Witten; and their relationships, similarities, and differences from quantum mechanics of point-like particles and quantum field theory.

First, let's begin by some light statements that everyone will understand.
These are elephants in the room which are not being addressed.
This latest version takes ignoring the elephants in the room to an extreme, saying absolutely nothing about the problems...
Another huge elephant in the room ignored by Witten’s story motivating string theory as a natural two-dimensional generalization of one-dimensional theories is that the one-dimensional theories he discusses are known to be a bad starting point...
Given the thirty years of heavily oversold publicity for string theory, it is this and the other elephants in the room that every physicist should know about.
So, it can’t have much to do with the real world that we actually live in. These are elephants in the room which are not being addressed.
This makes almost the same argument as the new one, but does also explain one of the elephants in the room (lack of a non-perturbative string theory).
Warren, I think there’s a difference between elephants in the room (we don’t know how to connect string theory to known 4d physics, with or without going to a string field theory) and something much smaller (mice? cockroaches?)...
I kid you not: there are at least 7 colorful sentences in that rant that claim the existence of the elephants in the room. And I didn't count the title and its copies. He must believe in a reduced version of the slogan of his grandfather's close friend (both are co-responsible for the Holocaust), "A lie repeated 7 times becomes the truth."

Sorry but there are no elephants in the room – in Witten's office, in this case. I've seen the office and I know many people who have spent quite some time there and all of them have observed the number of elephants in that room to be zero. It's enough for me to say this fact once.

If that annoying blogger sees the elephants everywhere, he should either reduce the consumption of drugs, increase the dosage of anti-hallucination pills, or both. If his employees had some decency and compassion, they would have already insisted that this particular stuttering computer assistant deals with his psychiatric problems in some way before he can continue with his work.

Fine. Now we can get to the physical issues – to make everyone sure that every single "elephant" is a hallucination.

Before strings were born but after the birth of quantum mechanics, people described the reality by theories that are basically derived from point-like particles and "their" fields. We're talking about the normal models of
  1. quantum mechanics of one or many non-relativistic point-like particles
  2. attempts to make these theories compatible with special relativity
  3. quantum field theory.
You may say that these three classes of theories are increasingly new and increasing more correct and universal. String theory is even newer and more complete so you could think that "it should start" from the category (3) of theories, quantum field theories in the spacetime.

But that's not how it works, at least not in the most natural beginner's way to learn string theory. String theory is not a special kind of the quantum field theories. That's why the construction of string theory has to branch off the hierarchy above earlier than that. String theory already starts with replacing particles with strings in the steps (1) and (2) above and it develops itself analogously to the steps from (1) to (2) to (3) above – but not "quite" identical steps – into a theory that looks like a quantum field theory at long distances but is fundamentally richer and more consistent than that, especially if you care about the behavior at very short distances.

For point-like particles, the Hamiltonians like\[

H = \sum_k \frac{p_k^2}{2m_k} + \sum_{k\lt \ell} V(r_k-r_\ell)

\] work nicely to describe physics of the atoms but they are not compatible with special relativity. The simplest and nice enough generalizations that is relativistic looks like the Klein-Gordon equation\[

(-\square-m^2) \Phi = 0

\] where we imagine that \(\Phi\) is a "wave function of one particle". In quantum mechanics, the wave function ultimately has to be complex because the energy \(E\) pure vector must depend on time as \(\exp(-iEt)\). We may consider \(\Phi\) above to be complex and rewrite the second derivatives with respect to time as a set of equations for \(\Phi\) and \(\partial\Phi/ \partial t\).

When we do so, we find out that the probability density – the time component of the natural Klein-Gordon current \(j^\mu\) – isn't positive definite. It would be a catastrophe if \(j^0\) could have both signs: probabilities would sometimes be positive and sometimes negative. But probabilities cannot be negative. (The "wrong" states have both a negative norm and negative energy so that the ratio is positive but the negative sign of the energy and especially the probability is a problem, anyway.) That's why the "class" of point-like theories (2) is inconsistent.

The disease is automatically solved once we second-quantize \(\Phi\) and interpret it as a quantum field – a set of operators (operator distributions if you are picky) – that act on the Hilbert space (of wave functions) and that are able to create and annihilate particles. We get\[

\Phi(x,y,z,t) = \sum_{k} \zav{ c_k\cdot e^{ik\cdot x} + c_k^\dagger \cdot e^{-ik\cdot x} }

\] Add the vector sign for \(\vec k\), hats everywhere, correct the signs, and add the normalization factors or integrals instead of sums. None of those issues matter in our conceptual discussion. What matters is that \(c_k,c^\dagger_k\) are operators and so is \(\Phi\). Therefore, \(\Phi\) no longer has to be "complex". It may be real – because it's an operator, we actually require it is Hermitian. And I have assumed that \(\Phi\) is Hermitian in the expansion above.

There must exist the ground state – by which we always mean the energy eigenstate \(\ket 0\) corresponding to the lowest eigenvalue of the Hamiltonian \(H\) – and one may prove that\[

c_k \ket 0 = 0.

\] The annihilation operators annihilate the vacuum completely. For this reason, the only one-particle states are the linear superpositions of various vectors \(c^\dagger_k \ket 0\). This is a linear subspace of the full, multi-particle Fock space produced by the quantum field theory. But both the Fock space and this one-particle subspace are positively definite Hilbert spaces. The probabilities are never zero.

You may say that the "dangerous" states that have led to the negative probabilities in the "bad theories of the type (2)" are the states of the type \(c_k\ket 0\) which may have been naively expected to be nonzero vectors in the theories (2) but they are automatically seen to be zero in quantum field theory. Quantum field theories pretty much erases the negative-probability states by hand, automatically.

Now, if you take bosonic strings in \(D=26\), for the sake of simplicity, and ban any internal excitations of the string, the physics of this string will reduce to that of the tachyonic particle. The tachyonic mass is a problem (tachyons disappear once you study the \(D=10\) superstring instead of bosonic string theory; but since 1999, we know that tachyons are not "quite" a hopelessly incurable inconsistency, just signs of some different, unstable physical evolution).

But otherwise the string's physics becomes identical to that of the spinless Klein-Gordon particle. In quantum field theory, the negative-probability polarizations of the spinless particle "disappear" from the spectrum because \(c_k \ket 0=0\) and be sure that exactly the same elimination takes place in string theory.

The correctly treated string theory, much like quantum field theory, simply picks the positive-definite part of the one-particle Hilbert space only. At the end, much like quantum field theory, string theory allows the multi-particle Fock space with arbitrary combinations of arbitrary numbers of arbitrarily excited strings in the spacetime. And this Hilbert space is positive definite.

First-quantized approach to QFT is just fine

Many unpleasant people at that blog believe that for the negative-probability states to disappear, we must mindlessly write down the exact rules and slogans that are taught in quantum field theory courses and no other treatment is possible. They believe that quantum field theory is the only framework that eliminates the wrong states.

But that's obviously wrong. We don't need to talk about quantum fields at all. At the end, we are doing science – at least string theorists are doing science – so what matters are the physical predictions such as cross sections or, let's say, scattering amplitudes.

Even in quantum field theory, we may avoid lots of the talking and mindless formalism if we just want the results – the physical predictions. We may write down the Feynman rules and draw the Feynman diagrams needed for a given process or question directly. We don't need to repeat all the history clarifying how Feynman derived the Feynman rules for QED; we can consider these rules as "the candidate laws of physics". When we calculate all these amplitudes, we may check that they obey all the consistency rules. In fact, they match the observations, too. And that's enough for science to be victorious.

The fun is that even in the ordinary physics of point-like particles, the Feynman diagrams – which may be derived from "quantum fields" – may be interpreted in the first-quantized language, too. The propagators represent the path integral over all histories i.e. trajectories of one point-like particle from one spacetime point to another. The particle literally tries all histories – paths – and we sum over them. When we do so, the relevant amplitude is \(G(x^\mu,y^\mu)\).

However, the Feynman diagrams have many propagators that are meeting at the vertices – singular places of the diagrams. These may be interpreted as "special moments of the histories". The point-like particles are literally allowed to split and join. The prefactors that the Feynman rules force you to add for every vertex represent the probability amplitudes for the splitting/joining event, something that may depend on the internal spin/color or other quantum numbers of all the particles at the vertex.

The stringy Feynman diagrams may be interpreted totally analogously, in the one-string or first-quantized way. Strings may propagate from one place or another – this propagation of one string also includes the general evolution of its internal shape (a history is an embedding of the world sheet into the spacetime) – and they may split and join, too (the world sheet may have branches and loops etc.). In this way, we may imagine that we're Feynman summing over possible histories of oscillating, splitting, and joining strings. The sum may be converted to a formula according to Feynman-like rules relevant for string theory. And the result may be checked to obey all the consistency rules and agree with an effective quantum field theory at long distances.

And because the effective quantum field theories that string theory agrees with may be (for some solutions/compactifications of string theory) those extending the Standard Model (by the addition of SUSY and/or some extra nice new physics) and this is known to be compatible with all the observations, string theory is as compatible with all the observations as quantum field theory. You don't really need anything else in science.

Strings' superiority in comparison with particles

So all the calculations of the scattering amplitudes etc. may be interpreted in the first-quantized language, both in the case of point-like particles and strings. For strings, however, the whole formalism automatically brings us several amazing surprises, by which I mean advantages over the case of point-like particles, including
  1. the automatic appearance of "spin" of the particles from the internal motion of the strings
  2. unification of all particle species into different vibrations of the string
  3. the automatic inclusion of interactions; no special rules for "Feynman vertices" need to be supplemented
  4. automatic removal of short-distance (ultraviolet) divergences
  5. unavoidable inclusion of strings with oscillation eigenstates that are able to perturb the spacetime geometry: Einstein's gravity inevitably follows from string theory
It's a matter of pedagogy that I have identified five advantages. Some people could include others, perhaps more technical ones, or unify some of the entries above into bigger entries, and so on. But I think that this "list of five advantages" is rather wisely chosen.

I am going to discuss the advantages one-by-one. Before I do so, however, I want to emphasize that too many people are obsessed with a particular formalism but that's not what the scientific method – and string theorists are those who most staunchly insist on this method – demands. The scientific method is about the predictions, like the calculation of the amplitudes for all the scattering processes. And string theory has well-defined rules for those. Once you have these universal rules, you don't need to repeat all the details "how you found them" – this justification or motivation or history may be said to be "unnecessary excess baggage" or "knowledge to be studied by historians and social pseudoscientists".

Someone could protest that this method only generalizes the Feynman's approach to quantum field theory. However, this protest is silly for two reasons: it isn't true; and even if it were true, it would be irrelevant. It isn't true because the dynamics of string theory may be described in various types of the operator formalism (quantum field theory on the world sheet with different approaches to the gauge symmetries; string field theory; AdS/CFT; matrix theory, and so on). It's simply not true that the "integrals over histories" become "absolutely" unavoidable in string theory. Second, even if the path integrals were the only way to make physical predictions, there would be nothing wrong about it.

Fine. Let me now discuss the advantages of the strings relatively to particles, one by one.

The spin is included

One of the objection by the "Not Even Wrong" community, if I use a euphemism for that dirty scum, is:
Another huge elephant in the room ignored by Witten’s story motivating string theory as a natural two-dimensional generalization of one-dimensional theories is that the one-dimensional theories he discusses are known to be a bad starting point, for reasons that go far beyond UV problems. A much better starting point is provided by quantized gauge fields and spinor fields coupled to them, which have a very different fundamental structure than that of the terms of a perturbation series of a scalar field theory.
It's probably the elephant with the blue hat. Needless to say, these comments are totally wrong. It is simply not true that the point-like particles and their trajectories described in the quantum formalism – with the Feynman sum or the operators \(\hat x,\hat p\) – are a "bad starting point". They're a perfectly fine starting point. They're how quantum mechanics of electrons and other particles actually started in 1925.

Quantum field theory is one of the later points of the evolution of these theories, not a starting point, and it is not the final word in physics, either.

For point-like particles, the first-quantized approach building on the motion of one particle may look like a formalism restricted to the spinless, scalar, Klein-Gordon particles. But again, this objection is no good because of two reasons. It is false; and even if it were true, it is completely irrelevant for the status of string theory.

The comment that one may only get spinless particles in the first-quantized treatment of point-like particles is wrong e.g. because one can study the propagation of point-like particles in the superspace, a spacetime with additional fermionic spinorial coordinates. And the dynamics of particles in such spaces is equivalent to the dynamics of a superfield which is a conglomerate of fields with different spins. One gets the whole multiplet. More generally and less prettily, one could describe particles with arbitrary spins by adding discrete quantum numbers to the world lines of the Klein-Gordon particles.

But the second problem with the objection is that it is irrelevant because
the stringy generalization of the Klein-Gordon particle is actually enough to describe elementary particles of all allowed values of the spin.
Why? You know why, right? It's because the string has internal dynamics. It may be decomposed to creation and annihilation operators of waves along the string, \(\alpha_{\pm k}^\mu\). The spectrum of the operator \(m^2\) is discrete and the convention is that negative subscripts are creation operators; positive ones are annihilation operators. The ground state of the bosonic string \(\ket 0\) is a tachyon that carries center-of-mass degrees of freedom remembering the location or the total momentum of the string behaving as a particle. And the internal degrees of freedom, thanks to the \(\mu\) superscript (which tells you which scalar field on the string was excited), add spin to the string.

Bosonic strings only carry the spin with \(j\in \ZZ^{0,+}\). If you study the superstring, basically the physics of strings propagating in a superspace, you will find out that all \(j\in \ZZ^{0,+}/2\) appear in the spectrum.

It should be obvious but once again, the conclusion is that
the first-quantized Klein-Gordon particle physics is actually a totally sufficient starting point because once we replace the particles with strings propagating in a superspace, we get particles (and corresponding fields) of all the required spins in the spectrum.
The claim that there's something wrong with this "starting point" or strategy is just wrong. It's pure crackpottery. If you asked the author of that incorrect statement what's wrong about this starting point, he could only mumble some incoherent rubbish that would ultimately reduce to the fact that the first lecture in a low-brow quantum field theory course is the only thing he's been capable of learning and he just doesn't want to learn anything beyond that because his brain is too small and fucked-up for that.

But that doesn't mean that there's something wrong with other ways to construct physical theories. Some of the other ways actually get us much further.

Unification of all particle species into one string

One string can vibrate in different ways. Different energy or mass eigenstates of the internal string oscillations correspond to different particle species such as the graviton, photon, electron, muon, \(u\)-quark, and so on. This is of course a characteristic example of the string theory's unification power.

This idea wasn't quite new in string theory, however. Already in the early 1960s, people managed to realize that the proton and the neutron (or the \(u\)-quark and the \(d\)-quark; or the left-handed electron and the electron neutrino) naturally combine into a doublet. They may be considered a single particle species, the nucleon (if I continue with the proton-neutron case only), which may be found in two quantum states. These two states are analogous to the spin-up and spin-down states. The \(SU(2)\) mathematics is really isomorphic which is why the quantum number distinguishing the proton and the neutron was called the "isospin".

What distinguishes the proton and the neutron is some "detailed information (a qubit) inside this nucleon". It's still the same nucleon that can be switched to be a proton, or a neutron. And the same is true for string theory. What is switched are the vibrations of the internal waves propagating along the string. And there are many more ways to switch them. In fact, we can get all the known particle species – plus infinitely many new, too heavy particle species – by playing with these switches, with the stringy oscillations.

Interactions are automatically included

In the Feynman diagrams for point-like particles, you have to define the rules for the internal lines, the propagators, plus the rules for the vertices where the propagators meet. These are choices that are "almost" independent from each other.

Recall that from a quantum field theory Lagrangian, the propagators are derived from the "free", quadratic terms in the Lagrangian. The vertices are derived from the cubic and higher-order, "interaction" terms in the Lagrangian. Even when the free theory is simple or unique, there may be many choices and parameters to be made when you decide what the allowed vertices should be and do.

The situation is different in string theory. When you replace the interaction vertex by the splitting string, by the pants diagram, it becomes much smoother, as we discuss later. But one huge advantage of the smoother shape is that locally, the pants always look the same. Every square micron (I mean square micro-Planck-length) of the clothes looks the same.

So once you decide what is the "field theory Lagrangian" per area of the pants – the world sheet – you will have rules for the interactions, too. There is no special "interaction vertex" where the rules for the single free string break down. Once you allow the topology of the world sheet to be arbitrary, interactions of the strings are automatically allowed. You produce the stringy counterparts of all Feynman diagrams you get in quantum field theory.

In the simplest cases, one finds out that there is still a topological invariant, the genus \(h\) of the world sheet, and the amplitude from a given diagram may be weighted by \(g_s^{2h}\), a power of the string coupling. But it may be seen that \(g_s\) isn't really a parameter labeling different theories. Instead, its value is related to the expectation value of a string field that results from a particular vibration of the closed string, the dilaton (greetings to Dilaton). This "modulus" may get stabilized – a dynamically generated potential will pick the right minimum, the dilaton vev, and therefore the "relevant value" of the coupling constant, too.

So the choice of the precise "free dynamics of a string" already determines all the interactions in a unique way. This is a way to see why string theory ends up being much more robust and undeformable than quantum field theories.

Ultraviolet divergences are always gone

One old well-known big reason why string theory is so attractive is that the ultraviolet i.e. short-distance divergences in the spacetime don't arise, not even at intermediate stages of the calculation. That's why we don't even need any renormalization that is otherwise a part of the calculations in quantum field theory.

I must point out that it doesn't mean that there's never any renormalization in string theory. If we describe strings using a (conformal) quantum field theory on the two-dimensional world sheet, this quantum field theory requires analogous steps to the quantum field theories that old particle physicists used for the spacetime. There are UV divergences and renormalization etc.

But in string theory, no such divergences may be tied to short distances in the spacetime. And the renormalization on the world sheet works smoothly – the conformal field theory on the world sheet is consistent and, whenever the calculational procedure makes this adjective meaningful, renormalizable. (Conformal theories are scale-invariant so they obviously can't have short-distance problems; the scale invariance means that a problem at one length scale is the same problem at any other length scale.)

This UV health of string theory may be seen in many ways. For example, if you compactify string theory on a circle of radius \(R\), too short a value of \(R\) doesn't produce a potentially problematic dynamics with short-distance problems because the compactification on radius \(R\) is exactly equivalent to a compactification on the radius \(\alpha' / R\), basically \(1/R\) in the "string units", because of the so-called T-duality.

Also, if you consider one-loop diagrams, the simplest diagrams where UV divergences normally occur in quantum field theories, you will find out that the relevant integral in string theory is over a complex \(\tau\) whose region is\[

{\rm Im}(\tau)\gt 0, \,\, |\tau| \gt 1, \,\, |{\rm Re}(\tau)| \lt \frac 12.

\] The most stringy condition defining this "fundamental domain" is \(|\tau|\gt 1\) which eliminates the region of a very small \({\rm Im}(\tau)\). But this is precisely the region where ultraviolet divergences would arise if we integrated over it. In quantum field theory, we would have to integrate over a corresponding region. In string theory, however, we don't because these small values of \({\rm Im}(\tau)\) correspond to "copies" of the same torus that we already described by a much higher value of \(\tau\).

In the Feynman sum over histories, we only sum each shape of the torus once so including the "small \(\tau\) points aside from the fundamental region" would mean to double-count (or to count infinitely many times) and that's not what Feynman tells you to do.

For this reason, if there are some divergences, they may always be interpreted as infrared divergences. It is always possible for every similar divergence in string theory to be interpreted as "the same" divergence that would arise in the effective field theory approximating your string theory vacuum as an "infrared divergence", and no additional divergences occur. In this sense, any kind of string theory – even bosonic string theory – explicitly removes all potential ultraviolet divergences. And it does so without breaking the gauge symmetries or doing similar nasty things that would be unavoidable if you imposed a similar cutoff in quantum field theory.

String theory is extremely clever in the way how it eliminates the UV divergences.

The crackpot-in-chief on the anti-string blog wrote:
From the talks of his I’ve seen, Witten likes to claim that in string perturbation theory the only problems are infrared problems, not UV problems. That’s never seemed completely convincing, since conformal invariance can swap UV and IR. My attempts to understand exactly what the situation is by asking experts have just left me thinking, “it’s complicated”.
I am pretty sure that they meant "it's complicated for an imbecile like you". There is nothing complicated about it from the viewpoint of an intelligent person and string theory grad students understand these things when they study the first or second chapter of the string textbooks. Indeed, modular transformations swap the UV and IR regions and that's exactly why the would-be UV divergences may always be seen as something that we have already counted as IR divergences and we shouldn't count them again.

Grad students understand why there are no UV divergences in string theory but smart 9-year-old girls may already explain to their fellow kids why string theory is right and how compactifications work. According to soon-to-be Prof Peo Webster, who's "personally on Team String Theory", the case of \({\mathcal T}^* S^3\) requires some extra work relatively to an \(S^5\). She explains non-renormalizability and other basic issues that Dr Tim Blais has only sketched.

If you first identify all divergences that may be attributed to long-distance dynamics, i.e. identified as IR divergences, there will be no other divergences left in the string-theoretical integral. Isn't this statement really simple? It's surely too complicated for the crackpot but I hope it won't be too complicated for the readers of this blog.

Now, you may ask about the IR divergences. Aren't they a problem?

Well, infrared divergences are a problem but they are never an inconsistency of the theory. Instead, they may always be eliminated if you ask a more physically meaningful question. When you ask about a scattering process, you may get an IR-divergent cross section. But that's because you neglected the fact that experimentally, you won't be able to distinguish a given idealized process from the processes where some super-low-energy photons or gravitons were emitted along with the final particles you did notice. If you compute the inclusive cross section where the soft particles under a detection threshold \(E_{\rm min}\) – which may be as low as you want but nonzero – are allowed and included, the infrared divergences in the simple processes (without soft photons) exactly cancel against the cross section coming from the more complicated processes with the extra soft photons.

This wisdom isn't tied to quantum field theory per se. The same wisdom operates in any theory that agrees with quantum field theory at long distances – and string theory does. So even in string theory, it's true that IR divergences are not an inconsistency. If you ask a better, more realistically "inclusive" question, the divergences cancel.

In practice, bosonic string theory has infrared divergences that are exponentially harsh and connected with the instability of the vacuum – any vacuum – that allows tachyonic excitations. Tachyons are filtered out of the spectrum in superstring theory but massless particles such as the dilaton may be – and generically, are – sources of "power law" IR divergences, too. However, in type II string theory etc., all the infrared divergences that arise from the massless excitations cancel due to supersymmetry. So ten-dimensional superstring theories avoid both UV (string theory always does) and IR (thanks to SUSY) divergences.

But one must emphasize that in some more complicated compactifications, some IR divergences will refuse to cancel – we know that they don't cancel in the Standard Model and string theory will produce an identical structure of IR divergences because it agrees with a QFT at long distances – but that isn't an inconsistency. It isn't an inconsistency in QFT; and it isn't an inconsistency in string theory – for the same reason. It is a subtlety forcing you to ask questions and calculate answers more carefully. When you do everything carefully, you get perfectly consistent and finite answers to all questions that are actually experimentally testable.

Again, let me emphasize that while the interpretation of infrared divergences is the same in QFT and ST, because those agree at long distances, it isn't the case for UV divergences. At very short (stringy and substringy) distances, string theory is inequivalent to a quantum field theory – any quantum field theory – which is why it is capable of "eliminating the divergences altogether", even without any renormalization, which wouldn't be possible in any QFT.

Also, I want to point out that this ability of string theory to remove the ultraviolet divergences is special for the \(D=2\) world sheets. Higher-dimensional elementary objects could also unify "different particle species" and automatically "produce interactions from the free particles" because the world volume would be locally the same everywhere.

However, membranes and other higher-dimensional fundamental branes (beyond strings) would generate new fatal UV divergences in the world volume. The 2D world sheet is a theory of quantum gravity because the parameterization of the world sheet embedded in the spacetime mustn't matter. A funny thing is that the three components of the 2D metric tensor on the world sheet,\[


\] may be totally eliminated – set to some standard value such as \(h_{\alpha\beta}=\delta_{\alpha\beta}\) – by gauge transformations that are given by three parameters at each point,\[

\delta \sigma^1, \,\, \delta \sigma^2,\,\, \eta,

\] which parameterize the 2D diffeomorphisms and the Weyl rescaling of the metric. So the world sheet gravity may be locally eliminated away totally. That's why no sick "nonrenormalizable gravity" problems arise on the 2D world sheet. But they would arise on a 3D world volume of a membrane where the metric tensor has 6 components but you would have at most 3+1=4 parameters labeling the world volume diffeomorphisms plus the Weyl symmetry. So some polarizations of the graviton would survive, along with the nonrenormalizable UV divergences in the world volume.

In effect, if you tried to cure the quantized Einstein gravity's problems in the spacetime by using membranes, you would solve them indeed but equally serious problems and inconsistencies would re-emerge in the world volume of the membranes. The situation gets even more hopeless if you increase the dimension of the objects; \(h_{\alpha\beta}\) has about \(D^2/2\) components while the diffeomorphisms plus Weyl only depend on \(D+1\) parameters and the growth of the latter expression is slower.

Strings are the only simple fundamental finite-dimensional objects for which both the spacetime and world volume (world sheet) problems are eliminated. That doesn't mean that higher-dimensional objects never occur in physics – they do in string theory (D-branes and other branes) – but what it does mean is that you can't expect as simple and as consistent description of the higher-dimensional objects' internal dynamics as we know in the case of the strings. For example, the dynamics of D-branes may be described by fundamental open strings attached to these D-branes by both end points; you need some "new string theory" even for the world volume that naive old physicists would describe by an effective quantum field theory.

Gravity (dynamical spacetime geometry) is automatically implied by string theory

You may consider the first-quantized equations for a single particle propagating on a curved spacetime. However, the spacetime arena is fixed. The particle is affected by it but cannot dynamically curve it and play with the spacetime around it.

It's very different in string theory. String theory predicts gravity. Gravity was observed by the monkeys (and bacteria) well before they understood string theory which is a pity and a historical accident that psychologically prevents some people from realizing how incredible this prediction made by string theory has been. But logically, string theory is certainly making this prediction – or post-diction, if you wish – and it surely increases the probability that it's right in the eyes of an intelligent beholder.

Why does string theory automatically allow the spacetime to dynamically twist and oscillate and wiggle? Why is the spacetime gravity an unavoidable consequence of string theory?
It may look technical but it's not so bad. The reason is that any infinitesimal change of the spacetime geometry on which a string propagates is physically indistinguishable from the addition of coherent states of closed strings in certain particular vibration patterns – strings oscillating as gravitons – to all processes you may compute.
To sketch how it works in the case of the \(D=26\) bosonic string – the case of the superstring has many more indices and technical complications that don't change anything about the main message – try to realize that when you integrate over all the histories of oscillating, splitting, and joining strings, via Feynman's path integral, you are basically using some world sheet action that looks something like\[

S_{2D} = \int d^2 \sigma \, \sqrt{h}h^{\alpha\beta} \partial_\alpha X^\mu \partial_\beta X^\nu \cdot g_{\mu\nu}^{\rm spacetime}(X^\kappa).

\] Here, \(X^\mu\) and \(h_{\alpha\beta}\) are world sheet fields i.e. functions of two coordinates \(\sigma^\alpha\). I suppressed the dependence (and the overall coefficient) to make the equation more readable. At any rate, \(h\) is the world sheet metric and \(g\) is the spacetime metric which is a predetermined function of \(X^\mu\), the spacetime coordinates. But to calculate the world sheet action in string theory, you substitute the value of the world sheet field \(X^\kappa(\sigma^\alpha)\) as arguments into the function \(g_{\mu\nu}(X^\kappa)\).

What happens if you infinitesimally change the spacetime metric \(g\)? Differentiate with respect to this \(g\). You will effectively produce a term in the scattering amplitude that contains the extra prefactor of the type \(h^{\alpha\beta}\partial_\alpha X^\mu \partial_\beta X^\nu\) with some coefficients \(\delta g_{\mu\nu}\) to contract the indices \(\mu,\nu\).

But the addition of similar prefactors inside the path integral is exactly the string-theoretical rule to add external particles to a process. If you allow me some jargon, it's because external particles attached as "cylinders" to a world sheet may be conformally mapped to local operators (while the infinitely long thin cylinder is amputated) and there's a one-to-one map between the states of an oscillating closed string and the operators you may insert in the bulk of the world sheet. This map is the so-called "state-operator correspondence", a technical insight in any conformal field theory that you probably need to grasp before you fully comprehend why string theory predicts gravity.

And the structure of this prefactor, the so-called "vertex operator", in this case \(\partial X\cdot \partial X\) (the product of two world sheet derivatives of the scalar fields representing the spacetime coordinates), is exactly the vertex operator for a "graviton", a particular massless excitation of a closed string. It's a marginal operator – one whose addition keeps the world sheet action scale-invariant – and this "marginality" is demonstrably the right condition on both sides (consistent deformation of the spacetime background; or the vertex operator of an allowed string excitation in the spectrum).

We proved it for the graviton but it holds in complete generality:
Any consistent/allowed infinitesimal deformation of "the string theory" – the background and properties of the theory governing the propagation of a string – is in one-to-one correspondence with the addition of a string in an oscillation state that is predicted by the unperturbed background.
So the spectrum unavoidably includes the closed string states (graviton states) that exactly correspond to the infinitesimal deformation of the spacetime geometry, and so on. (Only the deformations of the spacetime geometry that obey the relevant equations – basically Einstein's equations – lead to a precisely conformal and therefore consistent string theory so only the deformations and gravitons obeying the Einstein's equations are allowed.) Similarly, if you want to change the string coupling or the dilaton, as we discussed previously, you will find a string state that is predicted in the spectrum whose effect is exactly like that. Gauge fields, Wilson lines, other scalar fields etc. work in the same way. All of them may be viewed either as "allowed deformations of the pre-existing background" or "excitations predicted by the original background".

That's why the undeformed and (infinitesimally but consistently) deformed string theory are always exactly physically equivalent. That is why there don't exist any inequivalent deformations of string theory. String theory is completely unique and because you may define consistent dynamics of a string on an arbitrary Ricci-flat etc. background, such string theory always predicts dynamical gravity, too.

Witten has tried to explain the same point so if I failed to convey this important observation, you should try to get the same message from his review.


To summarize, the most obvious first pedagogic approach to learn how to define string theory and do calculations in string theory deals with the first-quantized formalism, a generalization of the "one-dimensional world lines of particles" to the case of "two dimensions of the stringy world sheet". It's easy to see that analogous rules produce physical predictions that not only share the same qualitative virtues with those in the point-like-particle case and are equally consistent. Instead, the stringy version of this formalism is more consistent and has other advantages.

The stringy version of this computational framework is demonstrably superior from all known viewpoints. And the evidence is overwhelming that there exist particular non-perturbatively exact answers to all the physically meaningful questions and at least in many backgrounds (e.g. those envisioned in matrix models as well as the AdS/CFT correspondence, any version of them), we actually know how to compute them in principle and in the case of a large number of interesting observables, in practice.

Everyone who tries to dispute these claims is either incompetent or a liar or – and it's the case of the system manager – both.

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