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Big advances in our understanding of the character of symmetries in Nature

Exact symmetries can't be global and all similar qualitative predictions of string theory seem to be getting experimental confirmations

Luke and Don tried to read

Symmetry and Emergence,
a written version of Edward Witten's talk for the American Physical Society in Utah, April 2016. Luke has decided that the stuff is hard and the text is not too comprehensible. I read it and it's a nice, comprehensible summary of the progress in physicists' understanding of symmetries in Nature. Well, I know this stuff so everything is comprehensible to me. Can I do better in explaining these things? Am I more patient than Witten? I am not sure. My texts about similar topics e.g. in 2009 and 2011 were not significantly more popular than Witten's.



But let me try.

What is a symmetry? Symmetry is an important idea in mathematics and physics. In the mathematical and physical understanding of the word, the symmetry isn't just "any kind of beauty" or "aesthetically pleasing virtue" of an object – which could be imagined by someone who is really detached from the exact content of the phrase. Symmetries are transformations you can do with an object – or the history of the Universe – so that the object looks the same afterwards (in the case of the object) or the history still obeys the same laws of physics (in the case of the symmetries of the laws).




For example, I embedded a picture which has some symmetry. In mathematical language which is also used in physics, the 2D drawing has the \(\ZZ_4\) symmetry. Symmetries are "sets of transformations" and these transformations may be applied one after another, successively. Mathematically, a set of transformations with a rule how to compose them – rules saying which transformation you get if you compose transformations \(a\) and \(b\), and you get \(a\circ b\) – which is associative and has a neutral "do nothing", identity element and the inverse transformation for every transformation, is called a group.

The symmetry \(\ZZ_4\) of the picture above is referred to as the cyclic group. The picture may be rotated by 90 times \(k\) degrees where \(k=0,1,2,3\) and it remains the same. So there are four symmetry transformations, four elements in the group \(\ZZ_4\). Note that the picture was drawn in such a way that it is not the same if you look into a mirror. However, the three-dimensional object depicted by the picture can be flipped and stay the same. Rotate it around an axis in the plane of the picture – so that the top becomes the bottom and vice versa. For this reason, the 3D object has a larger symmetry group, \(D_4\), the dihedral group, which has 8 elements: 4 rotations and 4 mirroring operations along 4 different axes (or, equivalently, the mirroring with respect to a chosen axis combined with one of the 4 rotations).




In mathematics and physics, possible symmetries like that and all their properties and implications for the objects and laws are analyzed by the methods of group theory. There are lots of groups one may talk about. The cyclic group \(\ZZ_n\) for \(n\in \ZZ\), the dihedral group \(D_n\), the permutation group \(S_n\) of all transformations of \(n\) elements, the alternating group \(A_n\) of "even" permutations only, and then Lie groups such as \(O(n)\) or \(SO(n)\) or \(U(n)\) or \(SU(n)\) or \(Sp(n)\) or \(USp(n)\) or \(G_2,F_4,E_6,E_7,E_8\), and others.

The Lie groups are groups of continuously infinitely many transformations. For example, a sphere has an \(O(3)\) symmetry. You can rotate it by any angle \(\gamma\) around an axis that intersects the sphere at the point with latitude \(\theta\) and longitude \(\phi\). It means that there are three continuous parameters, \((\gamma,\theta,\phi)\), that determine which transformation you want.

Now, it should be self-evident to everybody with any talent for physics that groups and symmetries are important in physics.

Those who are willing to consider whole books of tirades against symmetry and beauty in physics are urged to stop reading now because they're not the "material ready for physics".

Groups have been important but they may have become more important and less important in physics at various points and the status of many fundamental questions – independent of the precise choice of the group – has been transformed many times. In recent years or 2-4 decades, some of the transformations were really deep.

But let's return deeper to the history, to the 1950s. At that time, people found something shocking.



This butterly is pretty and left-right, \(\ZZ_2\), symmetric. You flip the left side and the right side and it looks the same. Our faces are approximately symmetric, too (the face of Cyborg-Witten is an exception). But we're not quite symmetric. Our hearts are typically on the left side. Why is there no heart on the right side? It's probably some accident in biology. But it's not just big organs. Here you have the DNA



The proper DNA, B-DNA, winds in a right-handed direction. Z-DNA, the mirror image, exists but is rare. All the molecules seem to be spinning in the same direction. That may be some accident in molecular biology, too. Up to the 1950s, people expected the laws of physics to be left-right-symmetric. This symmetry is called the parity \(P\) and generates a group isomorphic to \(\ZZ_2\). Well, every group with two elements is isomorphic to \(\ZZ_2\): they're groups which have the identity element \(1\) which does nothing, and another element \(g\) which "flips" something, and if you "flip" it twice, you get the identity element i.e. \(g^2=1\).

It was assumed that if you look at some processes allowed by the laws of physics through the mirror, what you see is allowed by the laws of physics, too.

But an experiment in the 1950s showed it was wrong. Even elementary particles, like antineutrinos, are spinning in a way that look like the right-handed screw or the DNA molecule above. And there's no left-handed antineutrino. By "there is none", I don't just mean that we have run out of them, or we would have to travel very far to find one. I mean that those objects are prohibited by the laws of Nature in the whole Universe. So even elementary particles seem to have left-right preferences analogous to our having the heart on the left side.

OK, people actually had a perfect mathematical description of such particles – neutrinos – in terms of Weyl spinors already since the mid 1920s. They accepted that the laws are left-right-asymmetric if you look carefully. Fine. The neutrinos are left-handed and antineutrinos are right-handed. Up to the 1960s, they believed a weaker principle. While parity \(P\) as well as \(C\), the charge conjugation – replacement of particles with antiparticles and vice versa – aren't symmetries of nature, their combination, the \(CP\) conjugation, should be.
Anniversary: Steven Weinberg published A model of leptons (PDF) exactly 50 years ago today. The electroweak part of the Standard Model was completed – including the Higgs fields which Weinberg called just some "physical" fields including the quotes LOL, however. Google Scholars shows those 2.5 pages have 14,693 citations now – a pretty good efficiency. ;-)
But even that idea was experimentally falsified, in the 1960s. Even if you use a mirror and exchange neutrinos and antineutrinos etc., you get phenomena that don't follow the laws of physics if the original ones did. But this violation is even weaker – through the small complex phase in the CKM matrix that encodes the mixing of quarks' flavors (and maybe a similar one in the neutrino sector – which isn't settled yet). The combined \(CPT\) symmetry is still believed to be a perfect symmetry and there's a good reason for that: \(CPT\) is basically continued to the rotation of the "Euclidean time" and "one spatial coordinate" by 180 degrees, and that should be a symmetry – because the Lorentz symmetry is a symmetry for any value of the "hyperbolic angle", including complex ones (which are real angles).

You may deduce a more general lesson: the discrete symmetry, like the left-right symmetry or the exchange of particles with antiparticles, tend to be broken in Nature. It seems that the "gauge symmetries" are the only ones that can be exact at the end: all other symmetries seem to be broken. Recall that a gauge (or local) symmetry is one that allows you to pick the transformation separately and differently at each point of the spacetime – with the assumption that if the transformations are a continuous group, the chosen transformation should continuously depend on the spacetime coordinates.

What are gauge symmetries in the real world? We describe particle physics by the Standard Model – which still seems to be enough for all directly observed non-gravitational particles and forces. The quarks come in three "colors" (just a funny name or metaphor picked by physicists) which are sometimes called "red, green, blue" (that's an even funnier metaphor – there is no relationship of the quarks' colors with the colors as we see them with our eyes or their wavelengths). And these three basic colors may be not only permuted by the \(S_3\) permutations but even continuously transformed to mixtures of each other, by the \(SU(3)\) symmetry. On top of that, up-type and down-type quarks; as well as charged leptons (such as the electron) and their neutrino are related by a similar \(SU(2)\) symmetry. At some level, Nature behaves the same when we exchange electrons with neutrinos but this symmetry is broken. And there's also the hypercharge \(U(1)\) group which rotates an abstract circle.

The Standard Model has the \(SU(3)\times SU(2)\times U(1)\) gauge group. The \(SU(3)\) for the colors is exact and remains unbroken (although the color is "confined" – the colorful objects are only allowed to exist in isolation if they're combined to "grey" bound states such as protons and neutrons where the red-green-blue colors are balanced). The remaining, "electroweak" group is broken to another \(U(1)\), the electromagnetic \(U(1)\), which is a combination of the hypercharge \(U(1)\) and one generator in the \(SU(2)\). OK, some people know these things, some don't. I can't explain everything here.

Emmy Noether has taught us that for every symmetry, there is a conservation law and vice versa. Why is it so? It's easy to see in quantum mechanics. If the quantity \(L\) is conserved, \(dL/dt=0\). But the Heisenberg equations of motion say that this derivative is \((i/\hbar)[H,L]\). It is proportional to the commutator. The Hamiltonian generates the evolution in time and if it commutes with the operator \(L\), the operator \(L\) doesn't change. On the other hand, the operator \(L\) also generates something, a potential symmetry, and if \([L,H]=0\), it means that the Hamiltonian doesn't change under these transformations associated with \(L\). So \([L,H]=0\) is the same as \([H,L]=0\) and it can be read in two ways: either the Hamiltonian, i.e. the laws of physics, are symmetric under \(L\). Or the quantity \(L\) is symmetric under the translation in time given by \(H\). Conservation laws for \(L\) and symmetries generated by \(L\) are inseparable!

Key examples: the energy is conserved, \([H,H]=0\), because the laws of physics don't change with time. The momentum is conserved, \([\vec p,H]=0\), because the laws of physics don't change if we translate objects linearly in space. \([\vec J,H]=0\) says that the angular momentum is conserved and/or that the laws of physics don't change under rotations in space. The electromagnetic \(U(1)\) is really a symmetry because the corresponding electric charge is conserved, and so on. This has been known for a century. In fact, even the parity symmetry \(P\) has a corresponding conservation law – also named parity. If the phenomena are left-right-symmetric in space, you may associate a sign, \(P=\pm 1\), which is conserved in interactions and which is multiplicative (the combination of two objects with \(P=-1\) has \(P=+1\) etc.).

As Witten argues, the global i.e. non-gauge symmetries tend to be broken and this lesson has been learned both from particular examples as well as from the general rules and examples of descriptions in string theory. Those two ways of learning – experiments and formal research in string theory – agree in these general lessons about the symmetries. One should also interpret this agreement as successfully experimentally or at least phenomenologically verified predictions made by string theory.

Baryon and lepton number.

OK, half a century ago, people tended to think that the lepton number \(L\) – the number of electrons plus muons plus tau leptons plus their neutrinos minus the number of all their antiparticles – is exactly conserved. The same was believed for the baryon number \(B\) – the number of protons plus neutrons minus the number of their antiparticles. But even though we haven't experimentally detected a violation of the lepton number and the baryon number, we're almost certain that they're not exactly conserved. Why?

Well, the Standard Model allows interactions like\[

{\mathcal L } = \frac{1}{M} HH LL.

\] OK, in the field theory Lagrangian, one writes a product of the Higgs field (two copies) and some lepton fields (two copies). This is not a renormalizable interaction but more generally, even such interactions are expected "to be there" with some tiny coefficient \(1/M\) where \(M\) is basically the mass scale – probably well above the LHC scale – where the renormalizability of the Standard Model really breaks down because this "naughty" interaction becomes comparably strong as the well-behaved ones.

The interaction above may be shown to be gauge-invariant (and symmetric under all other required symmetries) with a proper contraction of indices. It violates the lepton number. On the other hand, there is no renormalizable interaction that would violate the lepton number. That's why at low enough energies, the lepton number seems conserved: there is just no possible, gauge-invariant way to violate the lepton number. Quite accidentally, the requirement of Lorentz and gauge symmetries (which we really need) plus renormalizability of the interactions is sufficient to preserve the lepton number. That's why we call the lepton and baryon number accidental symmetries. They seem to be approximately there and there's some reason which didn't have to exist but it does exist – there's just no allowed way to violate the law. The accidental symmetries aren't important principles and they're not considered "fundamental" but Nature ends up obeying them, anyway, for reasons that are rather "derived".

But beyond the renormalizable interactions, it's possible to violate it and we are pretty sure that it is violated. Why? Well, the interaction above, \(HHLL\), actually gives neutrinos their Majorana masses. We know that the effectively observed neutrinos have some masses because those are needed for the neutrino oscillations that we observe. Strictly speaking, we only know that the electron lepton number may be converted to the muon or tau lepton number, so the differences \(L_e-L_\mu\) etc. are not conserved.

However, it's very likely that \(L_e\) is non-conserved even separately. That's true if the masses are of the Majorana and not Dirac type. And it's true because a star which has a nonzero \(B\) and \(L\), and even a nonzero \(B-L\), often collapses to a black hole which ultimately evaporates to photons and gravitons only whose \(B=L=0\). So the black hole creation-plus-evaporation violates the lepton number and baryon number, too. The nonzero initial baryon and lepton numbers of the stars are just eradicated.

The same conclusion – all non-gauge, global symmetries seem to be violated – also followed from string theory. For example, some situations in string theory may be described by the AdS/CFT correspondence. One may see that any global symmetry on the boundary CFT gives rise to a local, gauge symmetry in the AdS bulk space where gravity exists. And there can be no purely global symmetries in the AdS bulk. That's not shocking: gravity makes everything "local" or "gauged". In particular, the acceleration and gravity exists or doesn't exist depending on your reference frame: an accelerated reference frame creates a "fictitious" gravity or it may cancel it. So whether something moves, and even whether it accelerates, depends on the coordinate system and the coordinate system may be chosen non-linearly and differently stretched or curved in every region.

There seem to be more general arguments beyond AdS/CFT implying that symmetries in string theory are either violated or gauge.

A special discussion has to be made for local but discrete symmetries. A discrete symmetry doesn't produce any "gauge field", like the electromagnetic field. But one may decide whether it's local or global e.g. according to the existence of "cosmic strings" with a "monodromy". What is it?

Let me begin with a cool science-fiction model – which cannot really be true in our Universe, but for "professional" reasons. Imagine that there is a huge pole in the street – a vertical string. If you run around it, your path becomes a non-contractible loop which may be distinguished from the trivial paths where you didn't encircle anything, OK? And that's why special things may happen. You may turn into antimatter.

Just imagine that. You go around a pole and every proton in your body becomes an antiproton, every electron becomes a positron, and vice versa, and so on. That's cool. And if \(C\) were an exact symmetry of Nature (it's not, as shown in the 1950s), it would even be possible. You might annihilate against the matter objects that didn't make the round trip around the magic pole, e.g. with your relatives.

The interchange of matter and antimatter, expressed as the operator of the charge conjugation \(C\), is a symmetry transformation. But the funny thing may be in principle defined for any symmetry transformation \(g\). There may exist cosmic strings – poles like above – that have the property that if you go around, all things \(\ket\psi\) get transformed to \(g\ket\psi\). In fact, even the Möbius strip may be considered an example of that for the parity operator \(P\). If a chicken with the "R" Rutgers university symbol on its T-shirt makes a round trip around the strip, it becomes a left-right-reflected chicken living on the Möbius strip with the Russian "Я" on its T-shirt.



The Möbius steak is delicious.

Now, if the corresponding magic pole – the cosmic string – exists for the transformation \(g\), we say that \(g\) is a local or gauge symmetry, and if it doesn't exist, it's a global symmetry. Why is it equivalent? Well, if the transformation is local, it may be different in different regions – and there's no global constraint saying that the transformation has to return to itself once you encircle the cosmic string. Again, the diverse evidence in string theory suggests that all the symmetry transformations \(g\) are either broken – just approximate, broken symmetries – or they are gauge symmetries i.e. the correspoding cosmic string should exist!

Witten also discusses that gauge symmetries may be emergent. Condensed matter physics allows e.g. a \(U(1)\) to emerge in superconductors – although I would say it's really the same \(U(1)\) as the electromagnetic one, with a doubled (Sheldon Cooper pair) elementary electric charge. But in string theory, we know lots of examples of "emergent" gauge symmetries and e.g. the enhanced gauge symmetries. An \(SO(32)\) group may get broken to \(U(1)^{16}\), things may change a bit in the environment, and \(U(1)^{16}\) gets extended to \(E_8 \times E_8\), a completely different group than the original \(SO(32)\) even though both groups have the rank \(16\) and the dimension equal to \(496\). Also, at a self-dual radius in the circular compactification, \(U(1)\) may get extended to an \(SU(2)\). Gauge symmetries are exact and have to be exact but what the actual group is depends on the "environment" or the "compactification". This is a relatively new lesson – in the past, people would probably automatically expect the negation of this proposition (a naive theoretical physicist used to assume – and the semi-laymen still currently assume – that one chooses the "right" gauge symmetry for Nature and the same group has to be relevant for all situations and environments allowed by these laws of physics: that's wrong). But both developments in string theory and in condensed matter physics, a highly experimentally rooted discipline of physics, have taught us that our prejudices were stupid and Nature is way more elegant than theoretical physicists used to think. Nature loves to bend the identity of the gauge symmetry group. As the conditions change, so can the group. It doesn't mean that there are no fixed laws of Nature. But fixed laws of Nature do not imply a fixed choice of the gauge group! The choice of the gauge group depends on the environment and the mathematical description optimized for that given environment.

You know, the emergent and flexible choice of the gauge symmetry group is one of the deep modern insights that allow me to identify incompetent "fake physicists" and crackpots within minutes. You encounter a would-be author of a theory of everything, let's call him Mr Weinstein, and he talks about a theory with a fixed gauge group for all solutions, situations, and environments. Well, just like when I hear a "singer" who is completely out of tune, I immediately know that he has no clue about the physics discoveries of the recent decades because we have learned that the gauge groups in any quantum theory of gravity are generally emergent and their choice must be unavoidably capable of changing when you change the conditions or the compactification. This trap of a "fixed fundamental gauge group" is extremely tempting and virtually all fake physicists get actually caught into this trap. And there are many similar traps – some assumptions that would-be physics revolutionaries without the proper intelligence or background are extremely likely to assume in one way, but the physicists who have listened to Nature, even in recent 50 years, know that the correct answer is qualitatively different.



Ms Bára Basiková, "The Symmetrical [Female One]". Cutely enough, the Czech synonym for a symmetry, "souměrnost" (="symetrie"), is a literal piecewise translation of the symmetry – from Greek sun- (with-) and metron (to measure).

I won't discuss the axions, strong CP-problem, and a few more extra lessons sketched in Witten's paper because there should be something left for those who want to return to Witten's paper after they tried this blog post.

Even if you didn't understand all the stuff above, you should understand – assuming that your IQ is above 70 – that progress has been made in the general understanding of the "role of symmetries in the Universe", their exactness, their global or local character, and relationships between these conditions. Several lines of evidence including experiments and string theory point to the same conclusion which is pretty and strengthens the philosophical lessons of the general theory of relativity: everything in physics that makes sense and is pure enough is "local" in character. No transformations should be demanded to be done just globally, i.e. one "centrally" chosen transformation per the whole spacetime. All regions of the spacetime should enjoy their autonomy and self-government, allow their local choice of the symmetry transformations, and so on.

I mentioned that string theory gave the right predictions for all these answers. They're qualitative questions and answers but they're still confirmed predictions of string theory. All the proposed "alternatives" to string theory fail in these criteria – just like they fail to produce consistent precise predictions of observable quantities. No other claimed "alternative" to string theory can correctly predict things like "the lepton number should ultimately be broken if there's no corresponding force". Quantum field theory may be said to be "agnostic" about this question – because the answer depends on whether we allow the non-renormalizable interactions or not.

Crackpots love to claim that string theory is "opposite" to the branches of physics that are rooted in experiments but that's really not the case. Many of the lessons we have learned from string theory are morally equivalent to the lessons we have learned from some technical observations in particle physics and even condensed matter physics. The lessons linked to the groups are morally equivalent; it's the non-stringy theorists who are strange and increasingly incompatible with the empirical science.

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