In many previous texts, most recently in the essay posted two blog posts ago, I expressed the idea that string theory may be interpreted as the wisdom of quantum mechanics that is taken really seriously – and that is applied to everything, including the most basic aspects of the spacetime, matter, and information.
People like me are impressed by the power of string theory because it really builds on quantum mechanics in a critical way to deduce things that would have been impossible before. On the contrary, morons typically dislike string theory because their mezzoscopic peabrains are already stretched to the limit when they think about quantum mechanics – while string theory requires the stretching to go beyond these limits. Peabrains unavoidably crack and morons, writing things that are not even wrong about their trouble with physics, end up lost in math.
Other physicists have also made the statement – usually in less colorful ways – that string theory is quantum mechanics on steroids. It may be a good idea to explain what all of us mean – why string theory depends on quantum mechanics so much and why the power of quantum mechanics is given the opportunity to achieve some new amazing things within string theory.
At the beginning, I must say that the non-experts (including many pompous fools who call themselves "experts") usually overlook the whole "beef" of string theory just like they overlook the "beef" of quantum mechanics.
They imagine that quantum mechanics "is" a new equation, Schr̦dinger's equation, that plays the same role as Newton's, Maxwell's, Einstein's, and other equations. But quantum mechanics is much more Рand much more universal and revolutionary Рthan another addition to classical physics. The actual heart of quantum mechanics is that the objects in its equations are connected to the observations very differently than the classical counterparts have been.
In the same way, they imagine that string theory is a theory of a new random dynamical object, a rubber band, and they imagine either downright classical vibrating strings or quantum mechanical strings that just don't differ from other quantum mechanical objects. But this understanding doesn't go beyond the (unavoidably oversimplified) name of string theory. If you analyze the composition of the term "string theory" as a linguist, you may think it's just a "theory of some strings". But that's not really the lesson one should draw. The real lesson is that if certain operations are done well with particular things, one ends with some amazing set of equations that may explain lots of things about the Universe.
Strings are exceptionally powerful – and only exceptionally powerful – at the quantum level. And the point of string theory isn't that it's a theory of another object. The point is that string theory is special among theories that would initially look "analogous".
Why is it special? And why is the magic of string theory so intertwined with quantum mechanics?
Discrete types of Nature's building blocks
For centuries, people knew something about chemistry. Matter around us is made of compounds which are mixtures of elements – such as hydrogen, helium, lithium, and I am sure you have memorized the rest. The number of types of atoms around us is finite. If arbitrarily large nuclei were allowed or stable, it would be countably infinite. But the number would still be discrete – not continuous.
For some century, people realized that the elements are probably made out of identical atoms. Each element has its own kind of atoms. The concept of atoms was first promoted by Democritus in ancient Greece. But in chemistry, atoms became more specific.
Sometime in the late 19th and early 20th century, people began to understand that the atom isn't as indivisible as the Greek name suggested. It is composed of a dense nucleus and electrons that live somewhere around the nucleus. Nucleus was later found to be composed of protons and neutrons. Quantum mechanics of 1925 allowed the physicists to study the quantized motion of electrons around the nuclei – and the motion of the electrons is the crucial thing that decides about the energy levels of all atoms and, consequently, their chemical properties.
In the 1960s, protons and neutrons were found to be composite as well. First, matter was composed of atoms – different kinds of building blocks for every element. Later, matter was reduced to bound states of electrons, protons, and neutrons. Later, protons and neutrons were replaced with quarks while electrons remained and became an important example of leptons, a group of fermions that is considered "on par" with quarks. The Standard Model deals with fermions, namely quarks and leptons, and bosons, namely the gauge boson and the Higgs boson. The bosons are particularly capable of mediating forces between all the fermions (and bosons).
But even in this "nearly final" picture, there are still finitely many but relatively many species of elementary particles. Their number is slightly lower than the number of atoms that were considered indivisible a century earlier. But the difference isn't too big – neither qualitatively nor quantitatively. We have dozens of types of basic "atoms" or "elementary particles" and each of them must be equipped with some properties (yes, the properties of elementary particles in the Standard Model look more precise and fundamental than the properties of atoms of the elements used to). The different particle species amount to many independent assumptions about Nature that have to be added to the mix to build a viable theory.
Can we do better? Can we derive the species from a smaller number of assumptions – and from one kind of matter?
String theory – let's assume that Nature is described by a weakly-coupled heterotic string theory (closed strings only), to make it simpler – describes all elementary particles, bosons and fermions, as discrete energy eigenstates of a vibrating closed string. All interactions boil down to splitting and merging of these oscillating strings. Quantum mechanics is needed for the energy levels to be discrete – just like in the case of the energy levels of atoms. But for the first time, there is only one underlying building block in Nature, a vibrating closed string.
Like in atomic and molecular physics, quantum mechanics is needed for the discrete – finite or countable – number of species of small bound objects that exist.
Also, the number of spacetime dimensions was always arbitrary in classical physics. When constructing a theory, you had to assume a particular number – in other words, you had to add the coordinates \(t,x,y,z\) to your theory manually, one by one – and because the choice of the spacetime dimension was one of the first steps in the construction of any theory, there was no way to treat the theories in different spacetime dimensions simultaneously, and there were consequently no conceptual ways how to derive the right spacetime dimension.
In string theory, it's different because even the spacetime dimensions – scalar fields on the world sheet – are "things" that contribute to various quantities (such as the conformal anomaly) and string theory is therefore capable of picking the preferred (critical) dimension of the spacetime. Even the individual spacetime dimensions are sort of made of the "same convertible stuff" within string theory. This would be unthinkable in classical physics.
Prediction of gravity and other special forces: state-operator correspondence
String theory is not only the world's only known theory that allows Einsteinian gravity in \(D\geq 4\) to co-exist with quantum mechanics. String theory makes the Einsteinian gravity unavoidable. It predicts gravitons, spin-two particles that interact in agreement with the equivalence principle (all objects accelerate at the same acceleration in a gravitational field).
Why is it so? I gave an explanation e.g. in 2007. It is because a particular energy level of the vibrating closed string looks like a spin-two massless particle and it may be shown that the addition of a coherent state of such "graviton strings" into a spacetime is equivalent to the change of the classical geometry on which all other objects – all other vibrating strings – propagate. In this way, the dynamical curved geometry (or at least any finite change of it) may be literally built out of these gravitons.
(Similarly, the addition of strings in another mode, the photon mode, may have the effect that is indistinguishable from the modification of the background electromagnetic field and it is true for all other low-energy fields, too.)
Why is it so? What is the most important "miracle" or a property of string theory that allows this to work? I have picked the state-operator correspondence. And the state-operator correspondence is an entirely quantum mechanical relationship – something that wouldn't be possible in a classical world.
What is the state-operator correspondence? Consider a closed string. It has some Hilbert space. In terms of energy eigenstates, the Hilbert space has a zero mode described by the usual \(x_0,p_0\) degrees of freedom that make the string behave as a quantum mechanical particle. And then the strings may be stretched and the amount of vibrations may be increased by adding oscillators – excitations by creation operators of many quantum harmonic oscillators. So a basis vector in this energy basis of the closed string's Hilbert space is e.g.\[
\alpha^\kappa_{-2}\alpha^\lambda_{-3} \tilde \alpha^\mu_{-4} \tilde\alpha_{-1}^\nu \ket{0; p^\rho}.
\] What is this state? It looks like a momentum eigenstate of a particle whose spacetime momentum is \(p^\rho\). However, for a string, the "lightest" state with this momentum is just a ground state of an infinite-dimensional harmonic oscillator. We may excite that ground state with the oscillators \(\alpha\). These excitations are vaguely analogous to the kicking of the electrons in the atoms from the ground state to higher states, e.g. from \(1s\) to \(2p\). Those oscillators without a tilde are left-moving, those with a tilde are right-moving waves on the string. The (negative) subscript labels the number of periods along the closed string (which Fourier mode we pick). The superscript \(\kappa\) etc. labels in which transverse spacetime direction the string's oscillation is increased.
The total squared mass is given by \(2+3=4+1\) in some string units. The sum of the tilded and untilded subscripts must be equal (five, in this case) for the "beginning" of the closed string to be immaterial, technically because \(L_0-\tilde L_0 = 0\). Great. This was a basis of the closed string's Hilbert space.
But we may also discuss the linear operators on that Hilbert space. They're constructed as functionals of \(X^\kappa(\sigma)\) and \(P^\kappa(\sigma)\) – I am omitting some extra fields (ghosts) that are needed in some descriptions, plus I am omitting a discussion about the difference between transverse and longitudinal directions of the excitations etc. – there are numerous technicalities you have to master when you study string theory at the expert level but they don't really affect the main message I want to convey.
OK, the Hilbert space is infinite-dimensional but its dimension \(d\) must be squared, to get \(d^2\), if you want to quantify the dimension of the space of matrices on that space, OK? A matrix is "larger" than a column vector. The number \(d^2\) looks much higher than \(d\) but nevertheless, for \(d=\infty\), as long as it is the right "stringy infinity", there exists a very natural one-to-one map between the states and the local operators. Let me immediately tell you what is the operator corresponding to the state above:\[
(\partial_z)^2 X^\kappa
(\partial_z)^3 X^\lambda
(\partial_{\bar z})^4 X^\mu
(\partial_{\bar z})^1 X^\nu
\exp(ip\cdot X(\sigma))
\] There should be some normal ordering here. All the four operators \(X^{\kappa,\lambda,\mu,\nu}\) are evaluated at the point of the string \(\sigma\), too. You see that the superscripts \(\kappa,\lambda,\mu,\nu\) were copied to natural places, the subscripts \(2,3,4,1\) were translated to powers of the world sheet derivative with respect to \(z\) or \(\bar z\), the holomorphic or antiholomorphic complex coordinates on the Euclideanized worldsheet. Tilded and untilded oscillators were translated to the holomorphic and antiholomorphic derivatives. An exponential of \(X^\rho\) operator was inserted to encode the ordinary "zero mode", particle-like total momentum of the string. And the total operator looks like some very general product of a function of \(X^\rho\) – the imaginary exponentials are a good basis, ask Mr Fourier why it is so – and its derivatives (of arbitrarily high orders). By the combination of the "Fourier basis wisdom" and a simple decomposition to monomials, every function of \(X^\rho\) and its worldsheet derivatives may be expanded to a sum of such terms.
The map between operators and states isn't quite one-to-one. We only considered "local operators at point \(\sigma\) of the string" where the value of \(\sigma\) remains unspecified. But the "number of possible values of \(\sigma\)" looks like a smaller factor than the factor \(d\) that distinguishes \(d,d^2\), the dimension of the Hilbert space and the space of operators, so the state-operator correspondence is "almost" a one-to-one map.
Such a map would be unthinkable in classical physics. In classical physics, a pure state would be a point in the phase space. On the other hand, the observable of classical physics is any coordinate on the phase space – such as \(x\) or \(p\) or \(ax^2+bp^2\). Is there a canonical way to assign a coordinate on the phase space – a scalar function on the phase space – to a particular point \((x,p)\) on that space? There's clearly none. These mathematical objects carry completely different information – and the choice of the coordinate depends on much more information. You would have a chance to map a probability distribution (another scalar function) on the phase space to a general coordinate on the phase space – except that the former is non-negative. But that map wouldn't be shocking in quantum mechanics, either, because the probability distribution is upgraded to a density matrix which is a similar matrix as the observables. The magic of string theory is that there is a dictionary between pure states and operators.
This state-operator correspondence is important – it is a part of the most conceptual proof of the string theory's prediction of the Einsteinian gravity. Why does the state-operator correspondence exist? What is the recipe underlying this magic?
Well, you can prove the state-operator correspondence by considering a path integral on an infinite cylinder. By conformal transformations – symmetries of the world sheet theory – the infinite cylinder may be mapped to the plane with the origin removed. The boundary conditions on the tiny removed circle at the origin (boundary conditions rephrased as a linear insertion in the path integral) correspond to a pure state; but the specification of these boundary conditions must also be equivalent to a linear action at the origin, i.e. a local operator.
Another "magic player" that appeared in the previous paragraph – a chain of my explanations – is the conformal symmetry. A solution to the world sheet theory works even if you conformally transform it (a conformal transformation is a diffeomorphism that doesn't change the angles even if you keep the old metric tensor field). Conformal symmetries exist even in purely classical field theories. Lots of the self-similar or scale-invariant "critical" behavior exhibits the conformal symmetry in one way or another. But what's cool about the combination of conformal symmetry and quantum mechanics is that a particular, fully specified pure state (and the ground state of a string or another object, e.g. the spacetime vacuum) may be equivalent to a particular state of the self-similar fog.
The combination of quantum mechanics and conformal symmetry is therefore responsible for many nontrivial abilities of string theory such as the state-operator correspondence (see above) or holography in the AdS/CFT correspondence. At the classical level, the conformal symmetry of the boundary theory is already isomorphic to the isometry of the AdS bulk. But that wouldn't be enough for the equivalence between "field theory" in spacetimes of different dimensions. Holography i.e. the ability to remove the holographic dimension in quantum gravity may only exist when the conformal symmetry exists within a quantum mechanical framework.
Dualities, unexpected enhanced symmetries, unexpected numerous descriptions
The first quantum mechanical X-factor of quantum mechanics is the state-operator correspondence and its consequences – either on the world sheet (including the prediction of forces mediated by string modes) or on in the boundary CFT in the holographic AdS/CFT correspondence.
To make the basic skeleton of this blog post simple, I will only discuss the second class of stringy quantum muscles as one package – the unexpected symmetries, enhanced symmetries, and numerous descriptions. For some discussion of the enhanced symmetries, try e.g. this 2012 blog post.
In theoretical physicists' jargon, dualities are relationships between seemingly different descriptions that shouldn't represent the same physics but for some deep, nontrivial, and surprising reasons, the physical behavior is completely equivalent, including the quantitative properties such as the mass spectrum of some bound states etc.
The enhanced symmetries such as the \(SU(2)\) gauge group of the compactification on a self-dual circle (under T-duality) are a special example of dualities, too. The action of this \(SU(2)\), except for the simple \(U(1)\) subgroup, looks like some weird mixing of states with different winding numbers etc. Nothing like that could be a symmetry in classical physics. In particular, we need quantum mechanics to make the momenta quantized – just like the winding numbers (the integer saying how many times a string is wound around a non-contractible circle in the spacetime) are quantized – if we want to exchange momenta and windings as in T-duality. But within string theory, those symmetries become possible.
Many stringy vacua have larger symmetry groups than expected classically. You may identify 16+16 fermions on the heterotic string's world sheet and figure out that the theory will have an \(SO(16)\times SO(16)\) symmetry. But if you look carefully, the group is actually enhanced to an \(E_8\times E_8\). Similarly, a string theory on the Leech lattice could be expected to have a Conway group of symmetries – the isometry of such a lattice – but instead, you get a much cooler, larger, and sexier monster group of symmetries, the largest sporadic finite group.
Two fermions on the world sheet may be bosonized – they are equivalent to one boson. This is also a simple example of a "stringy duality" between two seemingly very different theories. The conformal symmetry and/or the relative scarcity of the number of possible conformal field theories may be used in a proof of this equivalence. Wess-Zumino-Witten models involving strings propagating on group manifolds are equivalent to other "simple" theories, too.
I don't want to elaborate on all the examples – their number is really huge and I have discussed many of them in the past. They may often be found in different chapters of string theory textbooks. Here, I want to emphasize their general spirit and where this spirit comes from. Quantum mechanics is absolutely essential for this phenomenon.
Why is it so? Why don't we see almost any of these enhanced symmetries, dualities, and equivalences between descriptions in classical physics? An easy answer is unlikely to be a rigorous proof but it may be rather apt, anyway. My simplest explanation would be: You don't see dualities and other things in classical physics because classical physics allows you the "infinite sharpness and resolution" which means that if two things look different, they almost certainly are different.
(Well, some symmetries do exist classically. For example, Maxwell's equations – with added magnetic monopoles or subtracted electric charges – have the symmetry of exchanging the electric fields with the magnetic fields, \(\vec E\to \vec B\), \(\vec B\to -\vec E\). This is a classical seed of the stringy S-dualities – and of stringy T-dualities if the electromagnetic duality is performed on a world sheet. But quantum mechanics is needed for the electromagnetic duality to work in the presence of particles with well-defined non-zero charges in the S-duality case; and in the presence of quantized stringy winding charges in the T-duality example because the T-dual momenta have to be quantized as well.)
On the other hand, quantum mechanics brings you the uncertainty principle which introduces some fog and fuzziness. The objects don't have sharp boundaries and shapes given by ordinary classical functions. Instead, the boundaries are fuzzy and may be interpreted in various ways. It doesn't mean that the whole theory is ill-defined. Quantum mechanics is completely quantitative and allows an arbitrarily high precision.
Instead, the quantum mechanical description often leads to a discrete spectrum and allows you to describe all the "invariant" properties of an energy-like operator by its discrete spectrum – by several or countably many eigenvalues. And there are many classical models whose quantization may yield the same spectrum. The spectrum – perhaps with an extra information package that is still relatively small – may capture all the physically measurable, invariant properties of the physical theory.
We may see the seed of this multiplicity of descriptions in basic quantum mechanics. The multiplicity exists because there are many – and many clever – unitary transformations on the Hilbert space and many bases and clever bases we may pick. The Fourier-like transformation from one basis to another makes the theory look very different than before. Such integral transformations would be very unnatural in classical physics because they would map a local theory to a non-local one. But in quantum mechanics, both descriptions may often be equally local.
OK, so string theory, due to its being a special theory that maximizes the number of clever ways in which the novel features of quantum mechanics are exploited, is the world champion in predicting things that were believed to be "irreducible assumptions whose 'why' questions could never be answered by science" and allowing new perspectives to look at the same physical phenomena. String theory allows to derive the spacetime dimension, the spectrum of elementary particles (given some discrete information about the choice of the compactification, a vacuum solution of the stringy equations), and it allows you to describe the same physics by bosonized or fermionized descriptions, descriptions related by S-dualities, T-dualities (including mirror symmetries), U-dualities, string-string-dualities which exhibit enhanced gauge symmetries, holography as in the AdS/CFT correspondence, the matrix model description representing any system as a state of bound D-branes with off-diagonal matrix entries for each coordinate, the ER-EPR correspondence for black holes, and many other things.
If you feel why quantum mechanics smells like progress relatively to classical physics, string theory should smell like progress relatively to the previous quantum mechanical theories because the "quantum mechanical thinking" is applied even to things that were envisioned as independent classical assumptions. That's why string theory is quantum mechanics squared, quantum mechanics with an X-factor, or quantum mechanics on steroids. Deep thinkers who have loved the quantum revolution and who have looked into string theory carefully are likely to end up loving string theory, and those who have had psychological problems with quantum mechanics must have even worse problems with string theory.
Throughout the text above, I have repeatedly said that "quantum mechanics is applied to new properties and objects" within string theory. When I was proofreading my remarks, I felt uneasy about these formulations because the comment about the "application" indicates that we just wanted to use quantum mechanics more universally and seriously, and it was guaranteed that we could have done so. But this isn't the case. The existence of string theory (where the deeper derivations of seemingly irreducible classical assumptions about the world may arise) is a sort of a miracle, much like the existence of quantum mechanics itself. (Well, a miracle squared.) Before 1925, people didn't know quantum mechanics. They didn't know it was possible. But it was possible. Quantum mechanics was discovered as a highly constrained, qualitatively different replacement for classical physics that nevertheless agrees with the empirical data – and allows us to derive many more things correctly. In the same way, string theory is a replacement for local quantum field theories that works in almost the same way but not quite. Just like quantum mechanics allows us to derive the spectrum and states of atoms from a deeper point, string theory allows us to derive the properties of elementary particles and even the spacetime dimension and other things from a deeper, more starting point. Like quantum mechanics itself, string theory feels like something important that wasn't invented or constructed by humans. It pre-existed and it was discovered.
People like me are impressed by the power of string theory because it really builds on quantum mechanics in a critical way to deduce things that would have been impossible before. On the contrary, morons typically dislike string theory because their mezzoscopic peabrains are already stretched to the limit when they think about quantum mechanics – while string theory requires the stretching to go beyond these limits. Peabrains unavoidably crack and morons, writing things that are not even wrong about their trouble with physics, end up lost in math.
Other physicists have also made the statement – usually in less colorful ways – that string theory is quantum mechanics on steroids. It may be a good idea to explain what all of us mean – why string theory depends on quantum mechanics so much and why the power of quantum mechanics is given the opportunity to achieve some new amazing things within string theory.
At the beginning, I must say that the non-experts (including many pompous fools who call themselves "experts") usually overlook the whole "beef" of string theory just like they overlook the "beef" of quantum mechanics.
They imagine that quantum mechanics "is" a new equation, Schr̦dinger's equation, that plays the same role as Newton's, Maxwell's, Einstein's, and other equations. But quantum mechanics is much more Рand much more universal and revolutionary Рthan another addition to classical physics. The actual heart of quantum mechanics is that the objects in its equations are connected to the observations very differently than the classical counterparts have been.
In the same way, they imagine that string theory is a theory of a new random dynamical object, a rubber band, and they imagine either downright classical vibrating strings or quantum mechanical strings that just don't differ from other quantum mechanical objects. But this understanding doesn't go beyond the (unavoidably oversimplified) name of string theory. If you analyze the composition of the term "string theory" as a linguist, you may think it's just a "theory of some strings". But that's not really the lesson one should draw. The real lesson is that if certain operations are done well with particular things, one ends with some amazing set of equations that may explain lots of things about the Universe.
Strings are exceptionally powerful – and only exceptionally powerful – at the quantum level. And the point of string theory isn't that it's a theory of another object. The point is that string theory is special among theories that would initially look "analogous".
Why is it special? And why is the magic of string theory so intertwined with quantum mechanics?
Discrete types of Nature's building blocks
For centuries, people knew something about chemistry. Matter around us is made of compounds which are mixtures of elements – such as hydrogen, helium, lithium, and I am sure you have memorized the rest. The number of types of atoms around us is finite. If arbitrarily large nuclei were allowed or stable, it would be countably infinite. But the number would still be discrete – not continuous.
For some century, people realized that the elements are probably made out of identical atoms. Each element has its own kind of atoms. The concept of atoms was first promoted by Democritus in ancient Greece. But in chemistry, atoms became more specific.
Sometime in the late 19th and early 20th century, people began to understand that the atom isn't as indivisible as the Greek name suggested. It is composed of a dense nucleus and electrons that live somewhere around the nucleus. Nucleus was later found to be composed of protons and neutrons. Quantum mechanics of 1925 allowed the physicists to study the quantized motion of electrons around the nuclei – and the motion of the electrons is the crucial thing that decides about the energy levels of all atoms and, consequently, their chemical properties.
In the 1960s, protons and neutrons were found to be composite as well. First, matter was composed of atoms – different kinds of building blocks for every element. Later, matter was reduced to bound states of electrons, protons, and neutrons. Later, protons and neutrons were replaced with quarks while electrons remained and became an important example of leptons, a group of fermions that is considered "on par" with quarks. The Standard Model deals with fermions, namely quarks and leptons, and bosons, namely the gauge boson and the Higgs boson. The bosons are particularly capable of mediating forces between all the fermions (and bosons).
But even in this "nearly final" picture, there are still finitely many but relatively many species of elementary particles. Their number is slightly lower than the number of atoms that were considered indivisible a century earlier. But the difference isn't too big – neither qualitatively nor quantitatively. We have dozens of types of basic "atoms" or "elementary particles" and each of them must be equipped with some properties (yes, the properties of elementary particles in the Standard Model look more precise and fundamental than the properties of atoms of the elements used to). The different particle species amount to many independent assumptions about Nature that have to be added to the mix to build a viable theory.
Can we do better? Can we derive the species from a smaller number of assumptions – and from one kind of matter?
String theory – let's assume that Nature is described by a weakly-coupled heterotic string theory (closed strings only), to make it simpler – describes all elementary particles, bosons and fermions, as discrete energy eigenstates of a vibrating closed string. All interactions boil down to splitting and merging of these oscillating strings. Quantum mechanics is needed for the energy levels to be discrete – just like in the case of the energy levels of atoms. But for the first time, there is only one underlying building block in Nature, a vibrating closed string.
Like in atomic and molecular physics, quantum mechanics is needed for the discrete – finite or countable – number of species of small bound objects that exist.
Also, the number of spacetime dimensions was always arbitrary in classical physics. When constructing a theory, you had to assume a particular number – in other words, you had to add the coordinates \(t,x,y,z\) to your theory manually, one by one – and because the choice of the spacetime dimension was one of the first steps in the construction of any theory, there was no way to treat the theories in different spacetime dimensions simultaneously, and there were consequently no conceptual ways how to derive the right spacetime dimension.
In string theory, it's different because even the spacetime dimensions – scalar fields on the world sheet – are "things" that contribute to various quantities (such as the conformal anomaly) and string theory is therefore capable of picking the preferred (critical) dimension of the spacetime. Even the individual spacetime dimensions are sort of made of the "same convertible stuff" within string theory. This would be unthinkable in classical physics.
Prediction of gravity and other special forces: state-operator correspondence
String theory is not only the world's only known theory that allows Einsteinian gravity in \(D\geq 4\) to co-exist with quantum mechanics. String theory makes the Einsteinian gravity unavoidable. It predicts gravitons, spin-two particles that interact in agreement with the equivalence principle (all objects accelerate at the same acceleration in a gravitational field).
Why is it so? I gave an explanation e.g. in 2007. It is because a particular energy level of the vibrating closed string looks like a spin-two massless particle and it may be shown that the addition of a coherent state of such "graviton strings" into a spacetime is equivalent to the change of the classical geometry on which all other objects – all other vibrating strings – propagate. In this way, the dynamical curved geometry (or at least any finite change of it) may be literally built out of these gravitons.
(Similarly, the addition of strings in another mode, the photon mode, may have the effect that is indistinguishable from the modification of the background electromagnetic field and it is true for all other low-energy fields, too.)
Why is it so? What is the most important "miracle" or a property of string theory that allows this to work? I have picked the state-operator correspondence. And the state-operator correspondence is an entirely quantum mechanical relationship – something that wouldn't be possible in a classical world.
What is the state-operator correspondence? Consider a closed string. It has some Hilbert space. In terms of energy eigenstates, the Hilbert space has a zero mode described by the usual \(x_0,p_0\) degrees of freedom that make the string behave as a quantum mechanical particle. And then the strings may be stretched and the amount of vibrations may be increased by adding oscillators – excitations by creation operators of many quantum harmonic oscillators. So a basis vector in this energy basis of the closed string's Hilbert space is e.g.\[
\alpha^\kappa_{-2}\alpha^\lambda_{-3} \tilde \alpha^\mu_{-4} \tilde\alpha_{-1}^\nu \ket{0; p^\rho}.
\] What is this state? It looks like a momentum eigenstate of a particle whose spacetime momentum is \(p^\rho\). However, for a string, the "lightest" state with this momentum is just a ground state of an infinite-dimensional harmonic oscillator. We may excite that ground state with the oscillators \(\alpha\). These excitations are vaguely analogous to the kicking of the electrons in the atoms from the ground state to higher states, e.g. from \(1s\) to \(2p\). Those oscillators without a tilde are left-moving, those with a tilde are right-moving waves on the string. The (negative) subscript labels the number of periods along the closed string (which Fourier mode we pick). The superscript \(\kappa\) etc. labels in which transverse spacetime direction the string's oscillation is increased.
The total squared mass is given by \(2+3=4+1\) in some string units. The sum of the tilded and untilded subscripts must be equal (five, in this case) for the "beginning" of the closed string to be immaterial, technically because \(L_0-\tilde L_0 = 0\). Great. This was a basis of the closed string's Hilbert space.
But we may also discuss the linear operators on that Hilbert space. They're constructed as functionals of \(X^\kappa(\sigma)\) and \(P^\kappa(\sigma)\) – I am omitting some extra fields (ghosts) that are needed in some descriptions, plus I am omitting a discussion about the difference between transverse and longitudinal directions of the excitations etc. – there are numerous technicalities you have to master when you study string theory at the expert level but they don't really affect the main message I want to convey.
OK, the Hilbert space is infinite-dimensional but its dimension \(d\) must be squared, to get \(d^2\), if you want to quantify the dimension of the space of matrices on that space, OK? A matrix is "larger" than a column vector. The number \(d^2\) looks much higher than \(d\) but nevertheless, for \(d=\infty\), as long as it is the right "stringy infinity", there exists a very natural one-to-one map between the states and the local operators. Let me immediately tell you what is the operator corresponding to the state above:\[
(\partial_z)^2 X^\kappa
(\partial_z)^3 X^\lambda
(\partial_{\bar z})^4 X^\mu
(\partial_{\bar z})^1 X^\nu
\exp(ip\cdot X(\sigma))
\] There should be some normal ordering here. All the four operators \(X^{\kappa,\lambda,\mu,\nu}\) are evaluated at the point of the string \(\sigma\), too. You see that the superscripts \(\kappa,\lambda,\mu,\nu\) were copied to natural places, the subscripts \(2,3,4,1\) were translated to powers of the world sheet derivative with respect to \(z\) or \(\bar z\), the holomorphic or antiholomorphic complex coordinates on the Euclideanized worldsheet. Tilded and untilded oscillators were translated to the holomorphic and antiholomorphic derivatives. An exponential of \(X^\rho\) operator was inserted to encode the ordinary "zero mode", particle-like total momentum of the string. And the total operator looks like some very general product of a function of \(X^\rho\) – the imaginary exponentials are a good basis, ask Mr Fourier why it is so – and its derivatives (of arbitrarily high orders). By the combination of the "Fourier basis wisdom" and a simple decomposition to monomials, every function of \(X^\rho\) and its worldsheet derivatives may be expanded to a sum of such terms.
The map between operators and states isn't quite one-to-one. We only considered "local operators at point \(\sigma\) of the string" where the value of \(\sigma\) remains unspecified. But the "number of possible values of \(\sigma\)" looks like a smaller factor than the factor \(d\) that distinguishes \(d,d^2\), the dimension of the Hilbert space and the space of operators, so the state-operator correspondence is "almost" a one-to-one map.
Such a map would be unthinkable in classical physics. In classical physics, a pure state would be a point in the phase space. On the other hand, the observable of classical physics is any coordinate on the phase space – such as \(x\) or \(p\) or \(ax^2+bp^2\). Is there a canonical way to assign a coordinate on the phase space – a scalar function on the phase space – to a particular point \((x,p)\) on that space? There's clearly none. These mathematical objects carry completely different information – and the choice of the coordinate depends on much more information. You would have a chance to map a probability distribution (another scalar function) on the phase space to a general coordinate on the phase space – except that the former is non-negative. But that map wouldn't be shocking in quantum mechanics, either, because the probability distribution is upgraded to a density matrix which is a similar matrix as the observables. The magic of string theory is that there is a dictionary between pure states and operators.
This state-operator correspondence is important – it is a part of the most conceptual proof of the string theory's prediction of the Einsteinian gravity. Why does the state-operator correspondence exist? What is the recipe underlying this magic?
Well, you can prove the state-operator correspondence by considering a path integral on an infinite cylinder. By conformal transformations – symmetries of the world sheet theory – the infinite cylinder may be mapped to the plane with the origin removed. The boundary conditions on the tiny removed circle at the origin (boundary conditions rephrased as a linear insertion in the path integral) correspond to a pure state; but the specification of these boundary conditions must also be equivalent to a linear action at the origin, i.e. a local operator.
Another "magic player" that appeared in the previous paragraph – a chain of my explanations – is the conformal symmetry. A solution to the world sheet theory works even if you conformally transform it (a conformal transformation is a diffeomorphism that doesn't change the angles even if you keep the old metric tensor field). Conformal symmetries exist even in purely classical field theories. Lots of the self-similar or scale-invariant "critical" behavior exhibits the conformal symmetry in one way or another. But what's cool about the combination of conformal symmetry and quantum mechanics is that a particular, fully specified pure state (and the ground state of a string or another object, e.g. the spacetime vacuum) may be equivalent to a particular state of the self-similar fog.
The combination of quantum mechanics and conformal symmetry is therefore responsible for many nontrivial abilities of string theory such as the state-operator correspondence (see above) or holography in the AdS/CFT correspondence. At the classical level, the conformal symmetry of the boundary theory is already isomorphic to the isometry of the AdS bulk. But that wouldn't be enough for the equivalence between "field theory" in spacetimes of different dimensions. Holography i.e. the ability to remove the holographic dimension in quantum gravity may only exist when the conformal symmetry exists within a quantum mechanical framework.
Dualities, unexpected enhanced symmetries, unexpected numerous descriptions
The first quantum mechanical X-factor of quantum mechanics is the state-operator correspondence and its consequences – either on the world sheet (including the prediction of forces mediated by string modes) or on in the boundary CFT in the holographic AdS/CFT correspondence.
To make the basic skeleton of this blog post simple, I will only discuss the second class of stringy quantum muscles as one package – the unexpected symmetries, enhanced symmetries, and numerous descriptions. For some discussion of the enhanced symmetries, try e.g. this 2012 blog post.
In theoretical physicists' jargon, dualities are relationships between seemingly different descriptions that shouldn't represent the same physics but for some deep, nontrivial, and surprising reasons, the physical behavior is completely equivalent, including the quantitative properties such as the mass spectrum of some bound states etc.
The enhanced symmetries such as the \(SU(2)\) gauge group of the compactification on a self-dual circle (under T-duality) are a special example of dualities, too. The action of this \(SU(2)\), except for the simple \(U(1)\) subgroup, looks like some weird mixing of states with different winding numbers etc. Nothing like that could be a symmetry in classical physics. In particular, we need quantum mechanics to make the momenta quantized – just like the winding numbers (the integer saying how many times a string is wound around a non-contractible circle in the spacetime) are quantized – if we want to exchange momenta and windings as in T-duality. But within string theory, those symmetries become possible.
Many stringy vacua have larger symmetry groups than expected classically. You may identify 16+16 fermions on the heterotic string's world sheet and figure out that the theory will have an \(SO(16)\times SO(16)\) symmetry. But if you look carefully, the group is actually enhanced to an \(E_8\times E_8\). Similarly, a string theory on the Leech lattice could be expected to have a Conway group of symmetries – the isometry of such a lattice – but instead, you get a much cooler, larger, and sexier monster group of symmetries, the largest sporadic finite group.
Two fermions on the world sheet may be bosonized – they are equivalent to one boson. This is also a simple example of a "stringy duality" between two seemingly very different theories. The conformal symmetry and/or the relative scarcity of the number of possible conformal field theories may be used in a proof of this equivalence. Wess-Zumino-Witten models involving strings propagating on group manifolds are equivalent to other "simple" theories, too.
I don't want to elaborate on all the examples – their number is really huge and I have discussed many of them in the past. They may often be found in different chapters of string theory textbooks. Here, I want to emphasize their general spirit and where this spirit comes from. Quantum mechanics is absolutely essential for this phenomenon.
Why is it so? Why don't we see almost any of these enhanced symmetries, dualities, and equivalences between descriptions in classical physics? An easy answer is unlikely to be a rigorous proof but it may be rather apt, anyway. My simplest explanation would be: You don't see dualities and other things in classical physics because classical physics allows you the "infinite sharpness and resolution" which means that if two things look different, they almost certainly are different.
(Well, some symmetries do exist classically. For example, Maxwell's equations – with added magnetic monopoles or subtracted electric charges – have the symmetry of exchanging the electric fields with the magnetic fields, \(\vec E\to \vec B\), \(\vec B\to -\vec E\). This is a classical seed of the stringy S-dualities – and of stringy T-dualities if the electromagnetic duality is performed on a world sheet. But quantum mechanics is needed for the electromagnetic duality to work in the presence of particles with well-defined non-zero charges in the S-duality case; and in the presence of quantized stringy winding charges in the T-duality example because the T-dual momenta have to be quantized as well.)
On the other hand, quantum mechanics brings you the uncertainty principle which introduces some fog and fuzziness. The objects don't have sharp boundaries and shapes given by ordinary classical functions. Instead, the boundaries are fuzzy and may be interpreted in various ways. It doesn't mean that the whole theory is ill-defined. Quantum mechanics is completely quantitative and allows an arbitrarily high precision.
Instead, the quantum mechanical description often leads to a discrete spectrum and allows you to describe all the "invariant" properties of an energy-like operator by its discrete spectrum – by several or countably many eigenvalues. And there are many classical models whose quantization may yield the same spectrum. The spectrum – perhaps with an extra information package that is still relatively small – may capture all the physically measurable, invariant properties of the physical theory.
We may see the seed of this multiplicity of descriptions in basic quantum mechanics. The multiplicity exists because there are many – and many clever – unitary transformations on the Hilbert space and many bases and clever bases we may pick. The Fourier-like transformation from one basis to another makes the theory look very different than before. Such integral transformations would be very unnatural in classical physics because they would map a local theory to a non-local one. But in quantum mechanics, both descriptions may often be equally local.
OK, so string theory, due to its being a special theory that maximizes the number of clever ways in which the novel features of quantum mechanics are exploited, is the world champion in predicting things that were believed to be "irreducible assumptions whose 'why' questions could never be answered by science" and allowing new perspectives to look at the same physical phenomena. String theory allows to derive the spacetime dimension, the spectrum of elementary particles (given some discrete information about the choice of the compactification, a vacuum solution of the stringy equations), and it allows you to describe the same physics by bosonized or fermionized descriptions, descriptions related by S-dualities, T-dualities (including mirror symmetries), U-dualities, string-string-dualities which exhibit enhanced gauge symmetries, holography as in the AdS/CFT correspondence, the matrix model description representing any system as a state of bound D-branes with off-diagonal matrix entries for each coordinate, the ER-EPR correspondence for black holes, and many other things.
If you feel why quantum mechanics smells like progress relatively to classical physics, string theory should smell like progress relatively to the previous quantum mechanical theories because the "quantum mechanical thinking" is applied even to things that were envisioned as independent classical assumptions. That's why string theory is quantum mechanics squared, quantum mechanics with an X-factor, or quantum mechanics on steroids. Deep thinkers who have loved the quantum revolution and who have looked into string theory carefully are likely to end up loving string theory, and those who have had psychological problems with quantum mechanics must have even worse problems with string theory.
Throughout the text above, I have repeatedly said that "quantum mechanics is applied to new properties and objects" within string theory. When I was proofreading my remarks, I felt uneasy about these formulations because the comment about the "application" indicates that we just wanted to use quantum mechanics more universally and seriously, and it was guaranteed that we could have done so. But this isn't the case. The existence of string theory (where the deeper derivations of seemingly irreducible classical assumptions about the world may arise) is a sort of a miracle, much like the existence of quantum mechanics itself. (Well, a miracle squared.) Before 1925, people didn't know quantum mechanics. They didn't know it was possible. But it was possible. Quantum mechanics was discovered as a highly constrained, qualitatively different replacement for classical physics that nevertheless agrees with the empirical data – and allows us to derive many more things correctly. In the same way, string theory is a replacement for local quantum field theories that works in almost the same way but not quite. Just like quantum mechanics allows us to derive the spectrum and states of atoms from a deeper point, string theory allows us to derive the properties of elementary particles and even the spacetime dimension and other things from a deeper, more starting point. Like quantum mechanics itself, string theory feels like something important that wasn't invented or constructed by humans. It pre-existed and it was discovered.
Why string theory is quantum mechanics on steroids
![Why string theory is quantum mechanics on steroids]()
Reviewed by MCH
on
September 10, 2018
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