Wednesday, May 01, 2019

First stringy steps: how a young fieldist expands her mind to become a string theorist

And yes, "she" is probably but not necessarily a young man

Three days ago, I mentioned that a "string theorist" is a description of expertise that includes most of "quantum field theory" but it goes beyond it, too. Seeing the world in the stringy way opens new perspectives, new ways to look at everything, and unleashes new powerful tools to theoretically wrestle with all the world's scholarly problems.

In practice, string theory isn't some philosophical superconstruction on top of quantum field theory (QFT) that is very different from the QFT foundations. Instead, string theory calculations are almost entirely identical to QFT calculations – but QFT calculations with new interpretations and new previously neglected effects. Most of the fundamental insights of string theory are irreversible, nearly mathematically rigorous insights about previously neglected properties and abilities of QFTs and especially previously overlooked properties of some special QFTs.

What are the limitations of a QFT student that prevent her from seeing physics through the new, stringy eyes? Let me look at these matters a little bit technically.

OK, let's first review the QFT. The Standard Model is the most "practical yet comprehensive" QFT relevant for the experiments that are actually being made. All the details are technical and only roughly 100,000+ people in the world understand them well enough. But the "verbal summary" is rather concise.

QFT is a special kind of quantum mechanics (QM). So we calculate probabilities of possible outcomes of observations. These probabilities are computed as the squared absolute values of the complex probability amplitudes – some matrix elements of linear operators on a complex Hilbert space.

In practice, in QFTs, those are computed as sums of the Feynman diagrams, such as one at the top. The internal lines are "propagators", linked to the two-point functions of quantum fields (and to the bilinear terms in the Lagrangian) and representing "virtual particles" that are seen neither in the initial state nor in the final one. The vertices come from higher-than-linear terms in the Lagrangian and they are needed for all interactions.

These Feynman rules – probability amplitudes are sums of Feynman diagrams – are derived either from some Dyson-like operator approach or from the Feynman sum over histories, the path integral. Each Feynman diagram translates to an integral – over locations of the vertices in the spacetime or over momenta of the propagators.

In the Standard Model or any particular QFT, there is a spectrum of possible propagators. They correspond to spin 0 or 1/2 or 1 particles in the Lagrangian. Some of them are gauge fields, you learn about the gauge symmetry, and if you're a bit advanced, you really master the renormalization, renormalization group, and non-perturbative effects such as instantons, among a few other things. I wanted to be really concise – so that's it. You must only understand that these several simple paragraphs translate to some 1,000 pages from textbooks if you really want to understand what my words mean – so that you can use the QFT apparatus! ;-)

Now, what are the new objects or treatments that string theory adds? How do you upgrade yourself from "one of the 100,000" QFT experts to "one of the 2,000" more or less string theorists?

Open a basic textbook on string theory such as Polchinski's book. I could only open Volume I of Polchinski because Nima Arkani-Hamed has borrowed my Volume II, I think, and he hasn't returned it yet. ;-) Already the initial chapters and sections of the textbook bombard the reader with great new insights that are spiritually "beyond" the mundane QFT apparatus sketched enough – apparatus optimized for the scattering amplitudes in the Standard Model. But I want to present the novelties independently.

The first novelty is that there are scale-invariant, conformal field theories (CFTs) and they have some special characteristics and allow new constructions and objects.

In the primitive QFT as sketched above, it doesn't matter much whether a particle is massive. A propagator may contain an extra \(-m^2\) or not. It's not a big deal. Gauge bosons and gravitons really have to be massless at the fundamental level – well, gauge bosons may get masses through the Higgs mechanism – but the calculational framework isn't affected much. At most, the massless particles are a pain in the buttocks because they may add long distance, infrared divergences and related problems.

In CFTs, massless particles aren't a liability. They are a virtue if not a necessity. Well, CFTs don't really allow massive elementary particles because those carry a special mass scale \(m\) and the corresponding special distance scale \(1/m\) which would destroy the scale invariance of the theory. Theories with massless particles are the beef of any CFT research. And CFTs bring new spacetime symmetries beyond the Poincaré group of translations, rotations, and boosts: the scaling and conformal symmetries.

In the \(d\)-dimensional spacetime where one of the dimensions is time, the Lorentz group is \(SO(d-1,1)\), isn't it? The conformal group is \(SO(d,2)\), I added one temporal and one spatial dimension. Relatively to the smaller Lorentz group, we have the extra \(J_{+-}\) that generates the scaling, \(J_{+i}\) that generates the usual translations, and \(J_{-i}\) that generates special conformal transformations. In two spacetime dimensions, there is an exception: the conformal group is infinite-dimensional, at least locally: any holomorphic function of the complex variable \(z\) preserves the angle at every point of the plane (we are talking about the Euclideanized spacetime or world sheet – the relationships to the Minkowski-signature ones is obtained by a Wick rotation or a similar analytical continuation). This is what you learn in the complex analysis – a mathematics course – as an undergraduate. Now, the action is invariant under these conformal transformations.

For example, an infinite cylinder is equivalent to the plane with the origin removed – the exponential map \(z=\exp(-iw)\) is how you do it, the situation is clarified by the picture. So the insertion of some operator at the point \(z=0\) is equivalent to the information about the state inserted to the evolution of the infinite cylinder at \(w\to -i\infty\). Quite generally, in CFT, you do want to study operators inserted at particular points, including very complicated operators, and the behavior of the theory when two or many such operators are inserted somewhere.

This is new relatively to the mundane QFT at the top. The mundane QFT really tells you that you should better not insert too many operators to several points, especially not nearby points, because that's a way to get ultraviolet (UV) divergences, i.e. short distance divergences, and those are a liability. But in CFTs which don't care about scales, there's nothing wrong about short distance and UV (just like there's nothing wrong about the IR) because all distances are physically equivalent by the scaling symmetry. So in fact, you do want to play with correlators of operators that are very close to each other. These correlators encode – in a new way – all the physical information about the interactions at the "finite" distances.

CFTs are generally important in QFT – they're the "fixed points" of the renormalization group, and therefore an essential starting point to understand the set of all QFTs according to the renormalization group paradigm. But CFTs are also vitally important in string theory. While the mundane QFT doesn't tell you anything about CFTs, as an upgraded QFTist or string theorist you must be ready to probe special properties of QFTs with massless particles and scale invariance and new constructions that are only possible when the conformal symmetry works.

In string theory, CFTs are important in the AdS/CFT realization of holography – CFTs on boundaries of the anti de Sitter space are equivalent to the full quantum gravitational (string/M) theory in the AdS bulk. But 2D CFTs are also the defining theories of any perturbative string theory – whose predictions are always calculated from the appropriate world sheet CFT.

You need to recall what you should have learned when you studied the holomorphic maps. How do you write down a complex holomorphic function that maps one region of the complex plane to another? You may need many of these things, especially the most elementary ones such as the exponential, logarithm, and the rational function \(z'=(az+b)/(cz+d)\).

State-operator correspondence

You should understand why the spectrum of "states on a closed string" is the same as the spectrum of "operators inserted at \(z=0\)". It has to be so because the states or the operators are needed to clarify what's happening at \(z=0\) i.e. \(w\to -i\infty\) and the rest of the 2D spacetime, the plane or the cylinder, is equivalent through the conformal transformation.

This correspondence, SOC, is the only good thing in the Universe that starts with "soc", the rest is some social, societal, and socialist junk.

New important spacetime symmetries

You need to learn the basic mathematics of the conformal symmetries. Why are the angle-preserving transformations isomorphic to a Lorentz group in a higher-dimensional spacetime? How do these maps work? What about the spherical inversion? Why is the CFT invariant under the spherical inversion?

Shocking new equivalences: bosonization and fermionization

Especially when the masses are zero, and you deal with CFTs, there are some new equivalences between theories that would sound impossible from the mundane QFT viewpoint. One of them is the equivalence of bosons and fermions. In QFT, you think that the Fock space built from a bosonic field is totally different from the fermionic Fock space. It's different from a pair of fermionic Fock spaces, too (OK, by the pair, I really meant the tensor product of two fermionic Fock spaces, sorry). If the occupation numbers are any non-negative integers, it must be a totally different spectrum than the spectrum of a theory where the occupation numbers are either zero or one, right?

In CFTs, this "obvious" conclusion is wrong. In fact, a free boson is equivalent to two free fermions. Some generalizations of this statement exist for interacting bosons and fermions, too. A boson with a sine self-interaction is equivalent to fermions with a quartic interaction in \(d=2\) CFTs. How is it possible?

I believe that it's a good idea for a "mundane QFTist who is just upgrading herself to a string theorist" to verify this equivalent up to the extent that convinces her that something really works here – or perhaps more rigorously than that. One check is to count the degeneracies of excited states on an open or closed string. Two fermions may lead to the same degeneracies at each level as a free boson, assuming the corresponding matching choice of boundary conditions in both theories.

Another one – which is equivalent to the counting of the states above – is through operators. The fermions may be defined as exponentials:\[

\psi = \exp(i\phi), \quad \bar\psi = \exp(-i\phi)

\] Well, there should also be some "ordering" sign, \(:\exp(\dots):\), which you need to master once you study these things really seriously. The exponential mapping between operators may sound very strange from a mundane QFT viewpoint but it's natural in CFTs. The bosonic fields \(\phi\) may be viewed as "generators of some operations" so if you exponentiate them, you may get a finite operation which may be equivalent to the insertion or destruction of a fermion. In effect, the exponential of the bosonic field creates a "kink", a discontinuity that can't be combined with another copy of the same discontinuity, so it ends up having the Fermi statistics (Pauli's exclusion principle).

The inverse relationship is bilinear, of the form \(\phi\sim \bar \psi \cdot \psi\), because you need to cancel the charges of the fermionic fields. The current for this \(U(1)\) charge is \(\partial\phi=\bar\psi\partial \psi\). You need to study this equivalence – bosonization or fermionization – to be sure that the mundane QFT viewpoint prevented you from seeing some relationships that are clearly true and almost certainly important.

Operator product expansions (OPEs)

The mundane QFT apparatus allows you to think in terms of "states" most of the time, like the people who think that QM is about states and not operators. However, the advanced QFT or string theory really forces you to admit that actual physics is about operators. So for example, in a QFT textbook, you could have learned about the anomalous dimensions of operators. But you didn't care – you didn't need such stuff for the computation of scattering amplitudes which seemingly included "all the interesting physics".

In CFT, you need anomalous dimensions of operators. In mundane QFTs, the anomalous dimensions start with terms proportional to \(g^2\) etc., the squared coupling constant (that's also how the couplings "run" etc.). In CFTs, the anomalous dimensions may be "non-adjustable", fractional numbers such as \(1/16\). It's all very exciting. You may see interesting, both free or interacting, CFTs that can't be understood as deformations of a free QFT with an interaction that has a coupling constant. Instead, the coupling constant seems to be "fixed". Even for the free fermion, the spin field that creates a special point making the fermion antiperiodic around the location of the spin field insertion happens to have the dimension of \(\Delta=1/16\). You couldn't have constructed fields of dimensions \(1/16\) in the mundane QFT, could you? All dimensions were integer multiples of \(1/2\). You thought that only "de facto polynomial" functions of the fields and their derivatives were possible and more complex dimensions were impossible for that reason. But that conclusion was premature.

So you need to learn what happens when two operators are inserted next to each other. There is some singularity. You know that the commutator of two operators \(F(\vec x)\) and \(G(\vec y)\) in mundane QFTs may produce a delta-function. But the simple product is harder – and the leading term when \(|\vec x-\vec y| \to 0\) is encoded in some Green's functions. In CFT, you need to focus on these things from advanced chapters of QFT textbooks that looked as "useless complications".

The insertion of the two operators at points \(\vec z\) and \(0\) may be replaced by the insertion of one operator at \(\vec z =0\). You may expand this new operator in some power series in \(\vec z\). The leading terms are the singularities, usually \(c\)-numbers, that may be extracted from the Green's functions. These OPEs end up being important because they encode the transformation of operators under various symmetries generated by other operators, stringy scattering amplitudes in some limits, and more.

Monodromies: operators orbiting each other

I mentioned many new things about QFTs that emerge when you study CFTs in any dimension. But the stringy world sheet has \(d=2\) where many new things occur. In particular, in a plane, a point may orbit another point and this is a topologically non-trivial operation. One may generate a phase or something nontrivial when one operator completes a full orbit around another.

You need to understand how these operations may be linked to boundary conditions on a closed string. You need to understand that the situation in which the orbiting does "nothing" is special, we say that the operators are mutually local. And you need to learn how to calculate such things not only for the "basic" operators such as \(\phi,\psi,\bar\psi\) I mentioned above; but also for operators such as \(\exp(a\phi)\).

Quite generally, the calculations involving the operator \(\exp(a\phi)\) where \(a\) is a number and \(\phi\) is a bosonic field are very important in CFTs. That's another fact that would look shocking from a mundane QFT viewpoint – that viewpoint only "encouraged" you to consider polynomial operators made of the basic fields. But I mentioned that these exponential operators with a particular value of \(a\) – well, there should have been \(1/2\) in my bosonization exponents, I can tell you now, at least in the normal conventions – are important for bosonization and fermionization.

But these operators are needed to define string states with a generic momentum, too. You should learn how to compute their anomalous dimensions which scales like \(a^2\) and is related to the mass of the string. You should learn how to orbit these operators around each other, and more. There was nothing special about "exponential of fields" in the mundane QFTs but these objects are important and omnipresent in CFTs and string theory that uses world sheet CFT.

Virasoro algebra

It is an infinite-dimensional Lie algebra generating all the reparameterizations of a circle, a periodic \(\sigma\) variable. It's generated by \(L_m\) and the commutator is \[

[L_m,L_n] = (m-n) L_{m+n}

\] in the simplest case. You should understand how it works, perhaps learn the central charge extension of the algebra as well, and basics of how to look for its representations. It is important because this algebra is a residual symmetry on the world sheet. It plays a similar role as the Yang-Mills symmetry or diffeomorphism symmetry (of GR) in the spacetime. On the other hand, the unphysical states of the Virasoro symmetry on the world sheet may be matched to the unphysical states in the spacetime – due to the Yang-Mills and diff symmetries. The world sheet gauge symmetry principles "produce" all the spacetime gauge symmetries that you need.

There are less and more rigorous ways to deal with the Virasoro algebra, the BRST treatment is a modern advanced one.

Topologies of world sheets, cohomology etc.

The higher-order string scattering amplitudes may be written as path integrals over world sheets of harder topologies – pants-like diagrams where strings merge and split, a sort of thickened versions of Feynman diagrams. Up to conformal transformations, the moduli spaces of possible shapes of such higher-genus Riemann surfaces are finite-dimensional. You should understand what the dimensions are, why they're finite at all, how the moduli spaces roughly look, and understand something about why the unitary S-matrix in string theory requires you to integrate over the moduli spaces in the most natural way, and what the most natural way is.

The genus \(h\) topologies have some non-contractible loops. This is a kind of "topology 101" – and algebraic geometry – that you may need to analyze spacetime (compactification spaces), too. Homology, cohomology, their relationships with forms and cycles matter.

CFT on sphere, torus, and other important low-genus topologies

The world sheet is normally considered compact – because all the infinite cylinders corresponding to the external particles may be "shrunk" and conformally mapped to disks. You should know the moduli space of such low-genus diagrams, with and without extra operator insertions. For the sphere, which is conformally equivalent to a plane by a stereographic projection, you need to see the Mobius \((az+b)/(cz+d)\) transformations.

A half-plane is a \(\ZZ_2\) quotient of the plane, the \(\ZZ_2\) is generated by the complex conjugation of \(z\).

But the torus is a "one-loop" diagram and has some special mathematics. A torus is a plane modulo a 2D lattice. The lattices that produce the same tori are equivalent via the \(SL(2,\ZZ)\), the modular group. The 2D torus may be read as a spacetime diagram in two different ways: the Euclideanized time is either the vertical or the horizontal direction. This gives you an equality between two different partition sums for different, basically inverse, temperatures! You should roughly know why it works – and then how it works precisely.

At the mathematical level, you have a great opportunity to learn the modular forms, eta and theta functions, and similar stuff to express these partition sums and their symmetry properties (under the modular group in particular).

T-dualities and other equivalences

The T-duality is a reparametrization of the fields on the world sheet that is somewhat analogous to the fermionization and bosonization but the basic form only requires bosonic fields. There's a way to switch from a bosonic field \(X\) to the T-dual field \(\tilde X\) on the world sheet. What actually happens is that \(X\) may be split into the left-moving and right-moving part (or the holomorphic and antiholomorphic modes, if you use the Euclideanized world sheet).

And the T-duality is the reflection \(X_L \to -X_L\) that "mirror reflects" the spacetime coordinate \(X\), a field describing the embedding of the world sheet into the spacetime, but the T-duality only reflects the left-moving part of \(X\) while the right-moving one is conservatively kept fixed! (Or vice versa, but physicists' conventions admit that it's more natural for the right-movers and right-wingers to be conservative.)

If you already know how string theory amplitudes are extracted from the 2D world sheet CFT, you will realize that this implies the equivalence of string theory on two totally different spacetimes.

Derivation of Einstein's equations and other spacetime effective equations

2D CFTs are rather rare. They include the free bosons, free fermions – with lots of equivalences between the two – then things like the Ising models and minimal models. The latter are basically "countable", there is a spectrum of "exceptions" that still manage to be CFTs.

But there are also CFTs with lots of parameters. The non-linear sigma model is the most important master example. The kinetic term \(\partial_\alpha X^\mu \partial^\alpha X_\mu\) in the world sheet Lagrangian is generalized by its being multiplied and contracted with a general function of \(X\),\[

g_{\mu\nu}(X^\gamma)\cdot \partial_\alpha X^\mu \partial^\alpha X^\nu

\] So all the values of the function \(g_{\mu\nu}\), for every value (point in spacetime) \(X^\gamma\) of the argument and for every choice of spacetime vector indices \(\mu,\nu\), is adjustable. It's exactly the information that defines a metric tensor field in the spacetime. Great. For every spacetime geometry, you may write down a theory for strings propagating on that spacetime.

This theory looks conformal for every choice of the tensor. However, there are quantum effects that also violate the scale invariance in general. In particular, for each point \(X^\gamma\) and each choice of \(\mu,\nu\), the coupling constant \(g_{\mu\nu}\) has its \(\beta\)-function encoding its "running with scale", and that \(\beta\)-function has to vanish for the world sheet theory to be actually scale-invariant at the quantum level.

And the cancellation of these "anomalies" actually tells you that the spacetime metric tensor must obey Einstein's equations! The \(\beta\)-function for the coupling \(g_{\mu\nu}(X^\gamma)\) ends up being basically the Ricci tensor at the same point, \(R_{\mu\nu}(X^\gamma)\). Its vanishing requires the Ricci flatness i.e. Einstein's equations in the vacuum. You may derive the defining equations of general relativity just from the requirement that the "conformal" strings may propagate on that spacetime!

This is true for all other effective field equations in the spacetime. If open or closed string modes produce gluons or electrons, their Yang-Mills or Dirac equations may be deduced from the conformal invariance of the world sheet theory at the quantum level! The right hand side of Einstein's equations (and all other spacetime equations) also correctly emerges if you calculate other contributions to the \(\beta\)-function.

Lots of extra technicalities

Weyl and diffeomorphism symmetry of the world sheet dynamics, fixed into the conformal symmetry, \(bc\) ghosts needed for that. Closed and open strings, various boundary conditions, how it affects both the states and the operators (open string vertex operators live on the boundary of open world sheets). Orbifolds and how their consistency requires something to work for the toroidal world sheets (modular invariance I mentioned). D-branes and how T-duality changes the dimension of the locus where open strings end. How the D-branes carry new fields. Why their dynamics is often Yang-Mills like. Addition of fermions to the world sheet, superstrings. Unorientable strings, orientifolds, and world sheet diagrams that are the projective sphere, Möbius strip, and Klein bottle – all those may be obtained from the sphere and the torus. And infinitely many harder topologies with boundaries and crosscaps.

And of course the critical dimension. Why the scale invariance of the world sheet theory at the quantum level implies \(D=26\) for bosonic string theory and \(D=10\) for the superstring. Polchinski calculates \(D=26\) in seven different ways, to assure a sensible reader that there's some "deep truth" about that result.


There are lots of wonderful insights about QFTs that happen to be CFTs – and especially CFTs in \(d=2\) which is appropriate for a string world sheet. These things can't ever disappear from physics again because they're really established mathematical facts about some classes of QFTs. If and when you study these things, and if you're intelligent, you will realize that it has been silly for you to be ignorant about them. You will know that they cannot be ignored. To "ban them" would be about as weird as banning molecular or nuclear physics or condensed matter physics (e.g. crystal lattices) for someone who has just mastered atomic physics.

Lots of special identities hold in CFTs or \(d=2\) CFTs and lots of new consistent objects may be defined and many consistent operations may be performed. There's a way to define a unitary S-matrix for states in the spacetime that looks just like one from an "advanced QFT" but also includes consistent quantum gravity. All these things look at least as natural as those in spacetime QFTs – but the gravity is added on top of that.

You will encounter some old objects – anomalous dimensions etc. – more often than in mundane QFT. You will learn some new functions, gamma functions for the tree-level amplitudes; eta and theta functions and modular forms for the toroidal partition sums and correlators. You will deal with some previously "unnatural" operators such as exponentials of bosonic fields. You will often treat the left-moving and right-moving (or holomorphic and antiholomorphic) parts of the fields separately, something that is impossible in \(d\gt 2\). Mundane QFT was telling you that "you shouldn't do certain things" but many of these things are extremely important, useful, and lead to new deep insights.

Already at the level of perturbative string theory, basically Volume I of Polchinski, you will see that too many things seem to work. The amount of great surprises and unbelievable consistency gets even more formidable once you study non-perturbative string theory, S-dualities, string-string duality, maps between D-branes and black \(p\)-branes, once you can microscopically calculate black hole entropy, geometerize the gauge symmetries in many new ways, find many more dualities (unexpected equivalences between vacua of string theory or QFTs), and more. The existence of string/M-theory "explains" all these particular coincidences and equivalences as well as other unexpectedly constrained yet consistent constructions – and it also "happens" to be a theory that is capable of producing all the predictions as the QFT class (plus consistent quantum gravity amplitudes).

At some psychological level, the transition from "one in 100,000 QFTists" to "one in 2,000 string theorists" in the world starts by realizing that the mundane QFT picture is not the whole story. It hides many wonderful, mathematically natural things that may be done with quantum fields and many of their properties. It hides many special QFTs, like CFTs or supersymmetric QFTs or CFTs, and even more special kinds of those, that have even more striking properties. You will only make the transition from a "quantum field theorist" to "string theorist" if you have the sufficient curiosity and desire to understand how "things really work"; and sufficient intelligence – so that you know that you haven't run out of your mental capacity once you got to the mundane QFT level.

Academically speaking, you don't need to be "certain" that string theory correctly describes our real Universe at a much better accuracy than any spacetime QFT. But if you actually master this material, so that you could get an A or B from most of the exercises e.g. in Polchinski's book, you will surely agree that it's utterly idiotic to ignore the existence of string theory or pretend that theoretical high-energy physics may continue or should continue while carefully avoiding all these stringy and similar (or similarly advanced) insights, constructions, and coincidences (that aren't quite "coincidental" because they're really "explained" by the existence of a unifying, deeper theory that unifies them all, string theory).

String theory is more than the mundane QFT but they are tightly connected and inseparable. They form one continuum of insights – one may be more or less familiar with that continuum but there exists no meaningful framework that could present "less familiar" as an advantage. You clearly become a better expert in the properties of QFTs once you master at least basics of string theory.

The left-wing establishment has restored propaganda, censorship, politically motivated dismissals etc. but they haven't revived the tradition of huge May Day parades yet. Check what the May 1st 1986 rally in Prague, five days after Chernobyl, looked like. Included are kids who are there for the first time, excited black students of biochemistry, history's criminals such as Marx, Engels, Lenin, and Gottwald, as well as the glorious Czechoslovak leaders of the mid 1980s. At the beginning of the march, you might have met the workers from the technological Tesla factory – some things aren't changing at all. I remember that such parades looked rather high-tech to me, at least in Pilsen, but when I watch this video, it is embarrassingly low-tech.

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