## Sunday, December 05, 2021 ... //

### Predictive power of the world sheet conformal symmetry

Physics is a natural science but it is one that is really capable of explaining a lot (and very precisely) by assuming a very little, by assuming some important principles that actually imply powerful predictions that are often verified to be correct. The precise laws of physics are often derivable from some propositions that are almost comprehensible to linguists. The search for the deepest such principles has been vaguely underway since the conception of physics (in the strict sense) by Galileo, Newton, and pals but it was elevated to a new, industrial level when Einstein and pals switched physics to the era of modern physics.

A natural metaphysical hypothesis is that a single deep principle is sufficient to derive all correct predictions in physics, and therefore in natural sciences. There is no guarantee that such an assumption about a "theory of everything" and its underlying principle is a correct expectation. To say the least, it motivates some people. Other people (and sometimes the same ones) are satisfied with making some progress which, in the case of fundamental physics, always includes some deepening and refining of the primary assumptions, and some extending of their domain of validity towards more universal explanations than what the previous theories and principles could cover.

There are lots of incredibly powerful conceptual physics ideas that imply a lot. For example, we have the conservation laws that allow us to make predictions about tons of seemingly diverse systems and exclude lots of possible scenarios that would otherwise be conceivable. Even more deeply, Noether's theorem shows the equivalence of such conservation laws and some symmetries of the laws of Nature. The existence of a conserved quantity that deserves to be called energy is equivalent to the symmetry of the laws of physics with respect to translations in time (i.e. the fact that the laws of physics don't change in time; and it's true with some possible clarifications but perhaps not others); the symmetry with respect to rotations is equivalent to the existence of a conserved pseudovector that we call the angular momentum, and so on.

The principle of relativity, basically the Lorentz symmetry that allows us to use all inertial frames equally to formulate the laws of physics, has lots of implications through the reasoning of the special theory of relativity. The speed of light is universal, it is the maximum speed at which the information may propagate, the existence of the conserved energy implies the existence of momentum (or vice versa), the existence of the electric fields implies the existence of the magnetic fields (and vice versa), and many more things like that. The previous sentence is a bit simplified and an extensive discussion would be needed to clarify what the postulates of relativity do actually imply and what they only do if some extra assumptions are added. But morally speaking, the simple claims about the "power of relativity" are almost precisely right.

Quantum mechanics has transformed the very Ansatz of the laws of physics more profoundly than relativity did, and in some sense, it may be said to be a physics framework that is also derived from principles (the universal postulates of quantum mechanics). But from a different perspective, quantum mechanics is not a theory where lots of quantitative predictions may be immediately made from "verbal principles". For quite some time, the quantum mechanical theories that people used to describe the world were just "quantizations" of some corresponding classical theories. So if they couldn't have derived some dynamical details about the right theories within the classical framework, they couldn't have done it in the quantum mechanical framework, either.

At the end, the view that a quantum theory is the same as a classical theory with "hats" (a decoration or clothes that turn observables to operators; read this sentence both literally mathematically as well as poetically) is incorrect, as was increasingly clear in the last decades of the 20th century. Quantum mechanics really imposes qualitatively different rules and the "space of possible theories" only seems to coincide (well, be in one-to-one correspondence) with the "space of possible classical theories" if we only focus on quantum mechanical theories that have a classical limit, and on the regime where the limit is relevant. However, deeply in the quantum realm where the classical approximation becomes O(100%) wrong, the map to classical theories becomes wrong, misleading, impossible, and the whole classical thinking is an incorrect guide. We know that there are extra conditions (like the anomaly cancellation) that ban many quantum mechanical theories although these non-existent or inconsistent theories' classical counterparts seem to be OK; there are classical unknown monodromies; there are quantum mechanical theories whose observables just aren't continuous or they don't admit a classical limit for other reasons; and the "classical guide" to do quantum mechanics is always a sign of someone's immature approach to modern physics.

However, it is still true that even today, a large majority of the quantum mechanical theories that we use are "constructed" from pieces, much like "their classical counterparts", and they do have some classical counterparts, after all. The search for the simplest or textbook-like quantum mechanical theories is an engineering-like construction which requires one to invent some objects or pieces or variables; and some terms in the equations (or the Hamiltonian) etc. However, when we get to more specific quantum mechanical theories, they respect a general form, an Ansatz. And it is surely the case of gauge theories that describe all non-gravitational forces.

Within the space of gauge theories, there are again some principles that imply a lot but they are somewhat more engineering-style principles because they tell you what kind of variables and interactions the "good" theories are ultimately going to have. Gauge symmetries are a great example of a principle to build "nice" quantum mechanical theories. The gauge symmetry under $$SU(3)\times SU(2)\times U(1)_Y$$ of the Standard Model implies "the behavior of all the non-gravitational forces". Electromagnetism, a $$U(1)$$ residual of the breaking of the $$SU(2)\times U(1)$$, leads to long-range Coulomb-like forces that drop like $$1/r^2$$; the breaking itself needs something like the spin-zero Higgs boson; the broken part of the forces generates the massive W-bosons and Z-bosons. And then there is the whole QCD with the colorful $$SU(3)$$ group that ends up being a short-range force as well because all finite-energy objects that are allowed to exist in isolation must be "color-neutral", and the forces between color-neutral objects are guaranteed to drop quickly with the distance (like the forces between protons and neutrons).

It is totally clear that gauge symmetries are very powerful and we need gauge theories for a reasonably effective description of the experiments in particle physics, especially (but not only) the high-energy ones such as the LHC collisions. On the other hand, the gauge theories are obviously not the final word (e.g. because they don't explain gravity or the spectrum of quarks and leptons) and there should be deeper principles that may be used as the starting point to derive an even wider and deeper set of correct statements about Nature, including the statement that "gauge symmetries in the spacetime are bound to arise and be useful to define the laws of physics". I am obviously talking about string/M-theory now which implies the existence of vacua whose dynamics may be approximated by "gauge theories coupled to Einstein's general relativity", the best classical theory of gravity, at long distances.

People may decide not to be interested in the deeper starting point which means to decide that they just want to remain shallow forever. But it is a fact that despite our ignorance about "which vacuum of string theory is really right", string theory is a deeper and more predictive system of laws that is not ruled out as of now and that seems to nontrivially imply lots of profound predictions about phenomena and patterns in fundamental physics. But is "string theory" a principle like the conservation laws of Noether's theorem? It sounds like an "object", a particular theory.

While string theory may be imagined as a system derived from "even deeper and more unifying principles" than e.g. gauge theories or Einstein's general relativity, string theory has been studied as a "totally constructive theory", almost by "engineers", for many decades. People who studied it between 1968 and the mid 1990s were facing some restrictions "what was possible and what dimension had to be picked" etc. but engineers are facing such restrictions, too. The derivation of all of string theory from some "deep principle" or "principles" (which would make it really analogous to the theories of relativity) remained a wishful thinking for quite some time. But in the 1990s, the situation changed and people started to have a clear idea how the "string theories" they had previously studied were connected or disconnected, and what the landscape of "all possible environments or 'theories' of this kind" looked like.

We might say that before the 1990s, people were playing on isolated "islands", like some ancient people with lame boats, but around the mid 1990s, the epoch of explorers arrived and people finally got credible plans to "map the whole world" of theories in this class (really "one theory" once they realized something about the "ocean floor" connecting the islands etc.). Finally, physicists could classify lots of things and see that the "accidentally constructed examples" are not just some random findings among googols of totally qualitatively different options. Instead, just like there are 5 or 6 continents, there are 5 or 6 maximally decompactified vacua of string theory in the maximal dimensions of the spacetime, ten or eleven.

There have been lots of progress that took us in unexpected directions etc. But at the end, I believe that the "massive industry of the mid 1980s", perturbative string theory, remains the proudest example of the "principles deeper than gauge symmetries that imply more". We know that what these deeper perturbative stringy principles imply only reliably applies to the case of the weak string coupling, $$g_s\ll 1$$, which are regimes analogous to boats that aren't far from the shores. Less constructive or less universal descriptions have to be exploited if we want to sail very far from the shores of the islands or continents.

But the vicinity of the shores is an important region and in some counting, we know that it contains a very big part of the truly unequivalent insights or qualitatively different types of behavior. This is a view that people didn't have when they were playing with lame boats near the beaches. They saw a seemingly infinite ocean, to use the analogy further, and it filled them with humility. There must be so many things over there. But some down-to-Earth people didn't ever want to sail too far and in some sense, they were proven right once the world was mapped away from the shores. OK, if you are a Spanish queen, you can send a Columbus somewhere but what will he find? Some Americas which don't differ from Spain much, a few potatoes, and savages. Columbus found the old natives and could leave some whites there who transformed into new nutty savages 250 years after they established their most famous country, anyway.

I am not saying that the vast ocean or the Mariana Trench or the exotic life forms that live in the very deep ocean are uninteresting. But most of the stuff that is found there may be described using the concepts that are already useful near the shores, and in this explorers' sense, with down-to-Earth concepts. The ocean is vast but it is also a mostly boring place that (almost) only exists in order to consistently connect the landmasses and to make some intercontinental flights annoyingly long.

With this new perspective that looks at string/M-theory as a single connected theory that is partially known in its entirety (although we still don't have the completely precise definition that would cover all the places of the landscape, not even in principle), two seemingly contradictory changes took place. On one hand, our focus was redirected away from the shores to the ocean and the routes that exist there (which includes duality, the equivalences between previously "different" string theories or landmasses; India=America is an example of a wrong duality LOL but it is exactly the kind of dualities that the new tools allowed to establish). On the other hand, we could see that "the world is nice but it's the best thing to live at home". The European shores (and even landlocked countries like Czechia) basically contain everything that you need to know. The remaining places in the world are just combinations of three coordinates. Aside from the potatoes, some nutcases in the Silicon Valley, and McDonald's hamburgers that will only exist after a Czech American puts a small company on steroids, there is nothing interesting about the U.S. ;-)

Even the routes in the ocean (e.g. dualities) and other properties away from the shores have been derived from the concepts and laws that only claim to work very well near the shores, from perturbative string theory. Finally, what is really perturbative string based upon? What is the principle at least in this near-shore approximation?

I think that the "conformal symmetry on the world sheet" may be said to be the main technical principle that replaces the Yang-Mills theory and other "principles" of the quantum field theory in the spacetime. The conformal symmetry is a residual symmetry after some gauge-fixing, something that is left in a theory with a larger symmetry, the world sheet diffeomorphism (general coordinate transformations) times the Weyl symmetry (scaling of the world sheet metric tensor by a place-dependent scalar coefficient) after you make sure that the metric tensor looks locally Minkowskian (which is always possible in 2 dimensions of the world sheet).

Finally, some simple equations will start to appear. That is why you read it up to here.

Perturbative string theory applies to all environments in string/M-theory which contains light (low-tension) strings that are weakly interacting. In these regimes, the lightest massive objects may be described as slightly excited (near-ground-states of) vibrating strings in the spacetime. The qualitative behavior of such strings is pretty much universal, there is some quasi-exponential growth of the number of excited string states with the mass, and some general calculus how to compute the spectrum and interactions of all these strings, despite the fact that the strings are allowed in "many environments" that differ by some technical traits (the shape of the Calabi-Yau manifolds etc.).

A string is a 1-dimensional curve in the spacetime and its shape may be captured by the vector-valued function $$X^\mu(\sigma)$$ where $$\sigma$$ (read "sigma", Mr Biden and Fauci) is both a future variant of Covid-19 as well as a real coordinate parameterizing the one-dimensional string. Because the strings continue to exist in time, we need another time coordinate $$\tau$$ (tau, also a virus) which may be identified with the spacetime time $$X^0$$ in a certain choice of coordinates, but those coordinates are not really convenient at all. So we deal with two-dimensional world sheets (history of the shape of strings how they exist and evolve in the spacetime), $$X^\mu(\sigma,\tau)$$.

These $$X$$'s are classical fields, soon to be turned into operators, and the whole relevant theory that we want to study requires operators in a spacetime-like space (the world sheet) labeled by the coordinates $$\sigma,\tau$$. It is a two-dimensional quantum field theory. Even classically, the two-dimensional world sheet inherits some proper distances and therefore a metric tensor $$h_{\alpha\beta}(\sigma,\tau)$$ from the embedding of the world sheet into the spacetime. It is useful to define this $$h_{\alpha\beta}$$ on the world sheet but not directly calculate it from the spacetime metric. Instead, it is better to allow $$h_{\alpha\beta}$$ to be a scalar-rescaled induced metric extracted from the spacetime. Once we allow this Weyl rescaling, the 3 components of the symmetric tensor, $$h_{\sigma\sigma},h_{\sigma\tau},h_{\tau\tau}$$ may be locally set to the normal Minkowski form by 3-parameter (picked at each point) transformations, 2 for the general change of coordinates in 2D and one for the scaling of the metric tensor $$h_{\alpha\beta}$$.

When it's done, we may see that even the most natural action describing the embedding of the world sheet, namely the proper area (which involves square roots and looks contrived), may be transformed to a nice, bilinear Klein-Gordon action on the world sheet, basically $\int \partial_\alpha X^\mu \partial^\alpha X_\mu.$ The scaling of the $$h$$ tensor was a local symmetry on the world sheet, much like the Yang-Mills symmetry is a local symmetry in the spacetime (and the coordinate transformations of GR may be local symmetries both in the spacetime and the world sheet). That is why the residual symmetry coming out of this Weyl-diff symmetry must be required to annihilate physical states etc. It's really just the overall scaling of the world sheet (or the holomorphic functions of a complex variable which have just 1D-worth of parameters to be described, not 2D) and the invariance of the world sheet theories and of the physical states under the simple scaling of the world sheet metric is actually the single most powerful principle that implies all the big predictions, including

• the existence of spin-2 gravitons in the spacetime that interact just like Einstein's GR dictates
• the existence of spin-1 gauge bosons in the spacetime that interacts just like Yang-Mills theories based on the spacetime gauge symmetries demand
• the existence of Dirac/Weyl/Majorana fermions in the spacetime that obey the expected kind of spacetime equations, plus the same for new scalars, Higgses/axions/inflatons, with their equations etc.
• lots of new insights that string theory discovered for the first time but that may indeed be verified to be important in the "effective quantum field theories in the spacetime" as well, including holography and other dualities
How does the existence of a spin-two graviton follow from the "new powerful principle", the scale invariance of the world sheet coordinates? Well, the physical states of a vibrating string (energy or mass eigenstates) may be mapped to composite local operators on the world sheet. The map is really natural (and the only natural one) and it geometrically boils down to the fact that an infinite cyllinder (a history of a single closed string, one of a circular shape) is conformally equivalent to the whole complex plane.

In this state-operator correspondence which is a nice property of the 2D world sheet theories (depending on the conformal transformations of the complex plane and regions in it) and which is a major explanation "why the conformal symmetry ends up being so powerful", the overall momentum of the string $$p$$ gets mapped to a prefactor in a seemingly artificial non-polynomial operator $$\exp(ip\cdot X(\sigma,\tau))$$ in the world sheet-based quantum field theory. The Hilbert space of a string ends up being a Fock space (infinite-dimensional harmonic oscillator) and every excitation by $$\alpha^\mu_{-n}$$ is mapped to an extra prefactor of $$\partial_z^n$$ that you insert in front of the local operator in the 2D quantum field theory. (The corresponding $$\partial_{\bar z}^n$$ is similarly representing the right-moving excitations $$\tilde \alpha_{-n}^\mu$$, I don't want to go to similar technicalities such as the existence of the left-movers and right-movers in the Fock space.)

Now, on the world sheet, the fields $$X^\mu(\sigma,\tau)$$ are dimensionless, like Klein-Gordon fields in 2D are (the Lagrangian density must be squared-mass which is just enough to cover the two derivative operators in the Klein-Gordon term; an alternative explanation is that $$X^\mu$$ is an actual location in the spacetime so it mustn't scale in any way if you scale the auxiliary world sheet coordinates, so it must be dimensionless on the world sheet). Now, massless particles admit zero-momentum states where $$\exp(ip\cdot X)$$ reduces to the identity operator. But the overall operator must still be a scale-invariant density (because its world sheet integral is the only natural operator representing the string state) so it must have the mass dimension two, too. It follows that the simplest local operators representing the simplest zero-momentum states of a string are$\partial_z X^\mu \partial_{\bar z} X^\nu.$ I simply needed two world sheet derivatives to get the mass dimension of two, so that the operator is a world sheet density. There was one $$z$$ and one $$\bar z$$-derivative in order for the operator to be invariant under the rotations as well (those are also residual conformal symmetries, aside from the scaling, I forgot to mention that the multiplication by a complex number may both scale and rotate). We needed to insert two derivatives to obtain the right mass dimension on the world sheet, but because we needed to use the fields encoding the embedding of the world sheet into the spacetime, we produced two spacetime indices $$\mu,\nu$$, and that is why we get a spin-two tensor-worth of spacetime states. Now, it is absolutely cool, and I have sketched it in an older blog post, that these massless string states integrate exactly as if they change the background metric in the spacetime. It is really possible to roughly see this fact from the very property that the operator above resembles the Klein-Gordon action in the world sheet. The Klein-Gordon action is actually also proportional to the spacetime metric and this "vertex operator for the graviton" clearly has the same form as the "variation of the world sheet action with respect to the change of the spacetime metric". So the presence of a coherent state of gravitons with this mode is exactly equivalent to an infinitesimal change of the theory resulting from the infinitesimal change of the spacetime metric.

Now, you can derive that quantum mechanically, the number of these bosons $$X$$ must be right, the critical dimension. It's $$D=26$$ spacetime dimensions for bosonic string theory; and $$D=10$$ for the only (and truly) realistic improvement of that theory where you add fermions and the minimal supersymmetry relating the world sheet bosons and fermions. Because there are extra dimensions i.e. extra values for the indices $$\mu,\nu$$ aside from the values $$0,1,2,3$$ spanning the spacetime that we thought we knew, it means that aside from the spin-2 operators above, some components may also be $$\mu\nu=\mu 5$$ or $$55$$ so you may also produce massless states of the string that behave as spin-1 or spin-0 particles in the spacetime. With fermions, you may also get the spin 1/2 or 3/2 in a similar way. Elementary particles predicted from massless vibrating strings have the spins $$0,1/2,1,3/2,2$$. The gravitino with $$j=3/2$$ (the main new fermion predicted by supergravity theories) remains the so far only undetected particle that is being predicted by superstring theory.

Analogous arguments to those that imply that the spin-2 particles interact just like the spacetime gravitons are OK to prove that the spin-1 particles obey some Yang-Mills symmetries in the spacetime. You may derive the same "qualitative list of allowed fields, interactions, and symmetries" in the spacetime from a very different starting point than some "consistency conditions in the spacetime", from the scaling symmetry on the world sheet. It is surprising that a seemingly technical symmetry in a space that seems totally auxiliary from the viewpoint of a spacetime experimenter – a world sheet local symmetry – is at least equally powerful as the direct constraints "we need this and that to work in the spacetime, and that's why we need these fields in the spacetime" but it is so.

Aside from the right dimension of the vertex operators for the physical states (mass dimension of two for the densities), we may also derive lots of the characteristically new stringy phenomena like T-duality once we consider nontrivial topologies of the world sheet, starting with the torus. On the torus, another discrete residual symmetry of the diff-Weyl symmetry, the modular symmetry, is possible, and the theory must still be invariant under it which kills many wrong candidate theories with incorrectly allowed quasi-periodic boundary conditions on the closed string; and which is the first step to derive the T-dualities and other new important phenomena.

One usually needs to study things for some years "after she masters quantum field theory" to understand why the world sheet scaling-or-conformal symmetry is enough to predict and classify all these things and to indicate that the spacetime particle spectrum and interactions are indeed of the kind that we may expect for totally different reasons; i.e. why string theory implies the qualitatively right behavior near the shores. But the true power of string/M-theory ultimately seems to be the ability to connect all the islands, continents, and to allow us to sail far away from the shores. A principle that is equally potent as the "world sheet scaling symmetry" could underlie all the known truths about the behavior of the theory near the shores as well as in the deep ocean. Such an extension of the 2D world sheet and its scaling symmetry could be as straightforward as a space of values of infinite matrices and some unitary transformations of them; or something completely different, something that also appreciates that the topology change is equivalent to a quantum entanglement (ER=EPR) etc. To turn the voyages over the distant ocean completely controllable also means to understand maximum about the possible shapes of the islands and our "history of journeys" in between them. We may derive that we actually did come from Botswana and South Afica, perhaps just like our friend Omicron, and our ancestors simply had to first sail here or there, and that is why one compactification or another is the right one (or much more likely than others, to say the least). Again, one may ignore the deeper, more universal, and potentially more precise levels of explanations in physics but that means that he chooses shallowness and primitivism over wisdom, hard work, and curiosity. It is insane to represent this attitude as "good science".