- Beauty, Witten, and Fields medal
- Measuring the depth of ideas
- Values in physics
- Evolution of beauty in physics
- Gell-Mann on beauty and truth in physics

and dozens of others. However, I want to look at different types and manifestations of symmetries, with the special focus on the question which of them are "beautiful" in the sense that they increase the chances that the physicists are on the right path. I will also try to demystify the ability of "beauty" to increase the probability of beautiful theories.

The symmetry is certainly not the only aspect of beauty in physics but it continues to be the most well-known one, and I will concentrate on it.

In physics, symmetries are classified and described by the mathematical notion of a group. There exist various types of groups and ways how they can manifest themselves in the laws of physics, especially the following:

Every symmetry can be discrete or continuous: in the former case, the group is finite or countable; in the latter case, the group is uncountable as a set. The discreteness or continuity of the symmetries are separately compatible with all the adjectives below. For example, even discrete symmetries may be local i.e. gauge symmetries: if they're local, there must exist "cosmic strings" (or e.g. D7-branes, in 10 dimensions) so that the monodromy around them generates the desired element of the group.

There are good reasons to think that in quantum gravity, all symmetries must be local symmetries - or at least subgroups of bigger symmetry groups that are naturally local. That's why e.g. the Lorentz symmetry must be extended to the whole diffeomorphism group or the general covariance. It's morally the case because every measurement or transformation can be done locally in general relativity.

That doesn't mean that we should imagine that global symmetries play no role in physics. Quite on the contrary. They're important. Only global symmetries are genuine symmetries that act on physical objects. Gauge symmetries are redundancies of the description - and all acceptable configurations of matter must actually be invariant (i.e. singlets) under all gauge symmetries. Moreover, the nature of a gauge symmetry often depends on the description of the physical system that you choose.

The previous paragraph may confuse you: so which of the symmetries, local or global, are truly physical? Well, only global symmetries are physical in the sense that they don't depend on the description. On the other hand, there must always exist a description in which all these global symmetries are extended to a gauge group and this gauge group is not an invariant property of physics. ;-)

For example, in AdS/CFT, the AdS isometry group is extended to the whole diffeomorphism group in the bulk. However, there is an alternative description, in terms of a gauge theory on the boundary, that has no diffeomorphism group to start with: it is, in some sense, replaced by a completely different U(N) group.

Is it confusing? Do you think that there's a contradiction? There's none. The global symmetries - a special group of "large transformations" inside the extended group of local symmetries - still make a physical sense because they usually act on the asymptotic conditions in space or time non-trivially. Such "large transformations" are actually not required to annihilate the physical states (only the "small" transformations in the gauge groups have to do so); the physical states may transform nontrivially under them.

The global symmetries include translations in space and time, rotations, and Lorentz transformations (or the unbroken isometries of any background whose superselection sector you consider). In the context of gravity, these are "large transformations" within a larger and local diffeomorphism group. But because the physical states can transform nontrivially under these large transformations, these symmetries are useful to classify physical states, according to representations. That's why we can assign physical states with the energy, momentum, and angular momentum, among other things.

Other global symmetries are discrete. They probably include the baryon number - a U(1) symmetry that changes the phases of fields that carry a nonzero baryon charge.

By Emmy Noether's profound theorem, each global symmetry is linked to a conservation law and vice versa. The conservation of energy or momentum or angular momentum is linked to the translational symmetry in time or space or the rotational symmetry. The Lorentz or Galilean symmetry is associated with the conserved velocity of the center-of-mass.

Other discrete symmetries of Nature include parity, charge conjugation, and time reversal. All these discrete Z2 symmetries are actually broken by the weak nuclear interactions and the overall CPT transformation is the only one that has to be preserved as long as the Lorentz symmetry is preserved, too (as Wolfgang Pauli has proved). The C and P symmetries are broken massively by the neutrino sector: there doesn't even exist a right-handed neutrino field that could be mapped to the ordinary left-handed neutrino field (or its complex conjugate) by the C or P transformation. The CP symmetry is broken by even subtler effects: the required "symmetric fields" are available but a subtle dynamical phase in the CKM matrix breaks the symmetry.

For a fixed choice of degrees of freedom, every kind of symmetry - local or global - relates objects, interactions, and phenomena that were previously thought of as independent. That's why every symmetry that is compatible with the known physical phenomena reduces the number of independent concepts in physics. A priori, there is no reason to think that a symmetry should be compatible with the known phenomena. So whenever it is, such an insight makes our understanding of physics more compact, tightly, unified, and beautiful.

Internal symmetries are able to relate a large number of seemingly or previously unrelated fields and interactions. Spacetime symmetries play a special role because every physical phenomenon must respect the existence of spacetime, at least in the long-distance approximation. ;-) Because all the fields and particles share the same spacetime, the spacetime symmetries are universal for all of them. Translations, rotations, and Lorentz boosts are spacetime symmetries (much like the broken parity, i.e. the left-right mirror reflection). Supersymmetry is partially a spacetime symmetry, partially an internal symmetry: it has to act both on internal and spacetime properties of fields.

The commutator of two supersymmetry transformations is a translation, so you must clearly include supersymmetry among the symmetries that do act on spacetime. That's another point that the laymen (including most experimenters) usually misunderstand. Supersymmetry manifests itself by relating different particle species - bosons and fermions - and by doing so, it resembles the internal symmetries. However, the critics argue that no superpartners of known particles have been observed so far, so this whole structure is redundant and supersymmetry doesn't constrain anything because the "power to constrain" only emerges after the field content is doubled.

What these people neglect is the fact that supersymmetry belongs among the spacetime symmetries. The rotational symmetry doesn't reduce the number of particle species, either. But it reduces the spectrum of allowed interactions, and so does supersymmetry. The fact that the rotational symmetry, despite its strict rules, is compatible with all known experiments is highly nontrivial and increases the beauty and unity of the laws of physics - and by the Bayesian inference (because the theory has passed a difficult test), the probability that the whole framework is valid.

The same thing holds for supersymmetry which is also constraining: the main difference is that supersymmetry is spontaneously broken (if it exists). As we will argue at the very end, this difference doesn't and can't influence the degree of internal beauty of supersymmetry.

We haven't discussed the three other pairs of adjectives yet. Symmetries can be exact or they can fail to be exact. There are several qualitatively different reasons why the exactness may disappear. And depending on the reason, the conclusions about the beauty and our knowledge about the right theory - as constrained by the symmetries - are very different.

First, a symmetry can fail to be exact, even at the fundamental level. If that's the case, it obviously cannot help us to construct a top-down theory from the scratch - because the symmetry doesn't really hold if you look accurately enough. Nevertheless, such approximate symmetries are extremely important in model building and in our search for approximate or effective theories. For example, the "strangeness" (essentially the number of strange quarks minus antiquarks) was known to be partially conserved but only in some interactions. This insight was very useful to determine certain properties of the nuclear interactions - and it helped to discover the concept of quarks.

So I would like to summarize the concept of approximate symmetries by saying that the knowledge about them can strengthen our ability to quickly eliminate theories that disagree with certain observations - observations that are approximately compatible with certain symmetries. On the other hand, approximate symmetries are not real "qualitative features" that would lead us to consider qualitatively different types of theories. After all, approximate symmetries are deduced from a complete theory, not in the other way around. Only exact symmetries may be a good starting point to fully determine or guess a new theory.

An important subclass of approximate symmetries are accidental symmetries - such as the baryon number (a charge expressed via a U(1) group transforming the phases of fields, by a rate proportional to their baryon content). Accidental symmetries hold in effective theories because mathematics just happens to allow no interaction terms of a specified type that would violate the symmetry. For example, if you consider the field content of the Standard Model and try to add all renormalizable interactions that preserve the Lorentz symmetry and gauge symmetries, you won't be able to write down any interactions that violate the baryon charge.

That's why B is an accidental symmetry: it must hold at the level of the renormalizable effective theories controlling a pre-determined field content. Needless to say, nonrenormalizable interactions - such as those obtained by integrating out new high-energy stuff beyond the Standard Model - can violate and probably do violate B. Black holes are almost certainly able to evaporate and destroy the baryon charge of the initial star, too. The accidental symmetry therefore only holds at a certain level of approximation. By the way, B-L (the baryon minus lepton number) may also be an accidental symmetry but it may also be an exact gauge symmetry.

Another effect that can spoil the exactness of a symmetry are anomalies. Anomalies are quantum effects - usually represented by 1-loop Feynman diagrams - that imply that the full quantum theory cannot preserve all the symmetries that the classical limiting theory (i.e. the theory reduced to tree level diagrams) does. It is obvious that at the fundamental, quantum level, anomalous symmetries don't exist at all.

On the other hand, gauge symmetries are necessary to get rid of unphysical and/or negative-norm, time-like and/or longitudinal polarizations of spin 1 (or higher) particles. It means that any anomaly that you find in a gauge symmetry (typically a Yang-Mills symmetry or a local Lorentz group) kills your theory. For example, until 1984, it was generally believed that type I superstring theory had to be anomalous, because of some rough arguments. The Green-Schwarz calculation showed a remarkably exact and surprising cancellation of all these anomalies for the SO(32) type I string, restoring the consistency of superstring theory and sparking the first superstring revolution.

On the other hand, anomalies in global symmetries don't make a theory inconsistent but they profoundly modify its dynamics. Again, an anomalous symmetry could have existed at the classical level, i.e. on the classical paper, but as soon as the effects of quantum mechanics matter, the symmetry is simply not there.

Consider a realistic theory with three quark flavors, u,d,s. Each of the fermionic fields is composed of a left-handed and a right-handed 2-component spinor. They (u,d,s) can be rotated independently by U(3)_{left} and U(3)_{right} transformations. One can define diagonal U(3) transformations by making the two U(3)_{left/right} transformations simultaneously. The U(1) part of this U(3) is the accidentally conserved baryon charge discussed above while the SU(3) part is an approximate symmetry between the three flavors.

The remaining "chiral" generators, those that rotate the left-handed and right-handed spinors differently (or even oppositely), are broken. The U(1) is broken differently than the SU(3) part: it is anomalous and it can be ignored. The SU(3) part is just spontaneously broken. It remains an approximate symmetry and the existence of such a broken symmetry implies the existence of pseudo (because the symmetry is approximate!) Nambu-Goldstone bosons, i.e. spin-0 bosons that are much lighter than the QCD scale expectation exactly because the symmetry holds much more accurately than a randomly broken non-symmetry.

To summarize, anomalous symmetries are just illusions of a classical theory but they're not really there in the full quantum theory. Approximate and accidental symmetries are useful approximate concepts used to study effective and phenomenological theories. Anomalous gauge symmetries make a theory inconsistent.

But my main motivation to write this essay are the spontaneously broken symmetries. These are symmetries that are exactly satisfied by the laws of physics, including all of their quantum and nonperturbative corrections, but they're not directly seen in the low-energy physical phenomena, for reasons that will be explained. Because, as you will see, the asymmetry in the observations is a derivable fact that has nothing to do with the beauty of the laws of physics, it doesn't diminish the beauty of the underlying theory at all.

In other words, a theory with a spontaneously broken symmetry is as constrained and as beautiful, in the refined sense, as a theory with an unbroken symmetry. Let me explain why.

The classical popular example of a broken symmetry is a pencil that you want to place symmetrically on its tip. If you do so, its position will show the rotationally symmetry around the vertical axis - i.e. the democracy between all the directions - very clearly. However, in the real world, something else happens. A tiny fluctuation (even one whose existence is guaranteed by the uncertainty principle) will be enough for the pencil to wobble. It will randomly choose a direction and collapse. Once it does so, the democracy between the directions is broken.

It is broken spontaneously because both the laws of physics as well as the pencil are rotationally symmetric. But the pencil just spontaneously decided to ignore this symmetry because an asymmetric position of the pencil actually lowers the energy. As far as the potential energy goes, it doesn't matter which direction the pencil chooses: again, all the directions have the same energy. But the pencil is damn sure that it should fall down. ;-)

A similar mechanism appears in the electroweak theory, among others. The SU(2) x U(1) symmetry is an exact symmetry of the laws of Nature. However, there exists something like the Higgs field "h(x,y,z,t)" at every point of space and time. And much like the pencil, this Higgs field just doesn't want to sit at the symmetric point "h=0". In fact, the minimum energy is obtained if the length of the complex vector "h" is equal to "v", the ultimate vacuum expectation value. So the Higgs field randomly chooses a direction, much like the pencil, and the original SU(2) x U(1) symmetry is spontaneously broken.

It is important to realize that the symmetry is still there and it is exact. All physical objects or configurations that are related by the symmetry transformation have the same energy. However, this is just the "equality of opportunities" which doesn't imply the "equality of outcomes" in the real world. In fact, the Higgs field surely wants to choose an asymmetric value, instead of "h=0". Other fields' masses (and other properties) depend on whether or not they're aligned in the same "direction" as the Higgs field.

The potential energy that is sufficient to convince the Higgs to pick an asymmetric vacuum expectation value may be as simple as "x^4 / 4 - x^2 / 2", the classical starting point for the renormalizable theory ("x" is the same thing as "h"). This function is even (and, more generally, symmetric with respect to the SU(2) x U(1) rotations). However, its global minimum is not the symmetric point "x=0". Instead, the minima of the function are located at "x=+1" and "x=-1" (or any other vector whose length is one, if you promote "x" to a complex vector).

If the potential energy were "x^4 / 4 + x^2 / 2", the minimum would be at "x=0" only and the symmetry would remain unbroken. So you should ask, is the theory with "x^4 / 4 - x^2 / 2" uglier than the theory with "x^4 / 4 + x^2 / 2"? Obviously, it can't be uglier. It only differs by a minus sign. All the fundamental symmetries are equally large and equally pretty: you surely don't want to discriminate against theories with a minus sign. ;-) However, the minus sign leads to a different physical behavior.

With the minus sign, the Higgs boson wants to choose a nonzero value. This nonzero value will influence all low-energy phenomena. On the other hand, the vacuum expectation value "v" of the Higgs has the dimension of a mass, and at energy scales "m" much higher than this "v", this "v" may be neglected. That's why the symmetry is restored in very high-energy phenomena. This conclusion is quite general: symmetries tend to be restored at high energies, high temperatures, or short distances and broken at low energies, low temperatures, or long distances.

The spontaneously broken symmetries are as accurate as the unbroken ones: they just dynamically lead to a vacuum (analogous to the fallen pencil) that is not a singlet under the symmetry. This necessary choice of the vacuum (and its superselection sector) makes all the objects and phenomena built upon the vacuum look asymmetric, even though the symmetry exactly holds and is as constraining as an unbroken symmetry: it just relates events in a given vacuum with events in a physically equivalent but different vacuum, which is much less "physical" an action than a symmetry that relates objects that can exist in the same vacuum.

As in many other cases, I think that the ultimate reason why people like Tommaso Dorigo can't understand why spontaneously broken symmetries are equally pretty is politics. Dorigo is a hardcore Marxist who makes Friedrich Engels look like Friedrich Hayek in comparison. He just doesn't distinguish the "equality of opportunities" and the "equality of outcomes". If he doesn't see a world where everything is equal and symmetric, it is "ugly" for him.

But that's a very deep misunderstanding. Much like the justice in a decent society only requires the "equality of opportunities" and says nothing about the "equality of outcomes", the beauty of the laws of physics only depends on the symmetry of the underlying equations, not on the symmetries of the actual configurations that solve the physical equations and that will evolve in the real world. It is not a purpose of the laws of physics to produce symmetric objects. In fact, the spontaneous breaking is quite a generic fate of many underlying symmetries and it is often as necessary for the existence of life as income/wealth inequality is critical for a decent GDP growth in any society.

The same comments hold not only for the electroweak symmetry but also for other symmetries that may exist in the real world but that are spontaneously broken, such as grand unified symmetries and supersymmetry. In the latter case, the mathematics needed to describe their spontaneous breaking is more sophisticated than a simple quartic function - and the "minimal" gadget to break the symmetry is not as unique as it is in the case of a single Higgs doublet (also known as the Weinberg toilet). But you should still understand that the situation is qualitatively isomorphic to the case of the electroweak symmetry.

Finally, Tommaso Dorigo has offered us some popular laymen's misconceptions about supersymmetry breaking, namely that it depends on 130 parameters. Well, 130 parameters only count the number of parameters in an effective theory - a theory describing how supersymmetry breaking affects low-energy physics, one that is not trying to understand the fundamental origin of the supersymmetry breaking. But if one actually constructs, knows, or finds a well-defined top-down theory with the right fields that break the supersymmetry and other fields that communicate it to the Standard Model fields, most or all of these 130 parameters become calculable.

So these soft-SUSY-breaking 130 parameters certainly don't make the underlying theory uglier - they just inform us that there are many details about its implementation that are not yet known to us. But this "ignorance" is something completely different than "ugliness". More obviously, as soon as the right supersymmetry breaking scenario is found and/or the parameters are measured, the ignorance disappears. If the ignorance were a real ugliness, it could never disappear.

In the most complete and almost certainly correct top-down framework we have, namely string theory, the supersymmetry breaking scenario is fully determined by the choice of the compactification. It's fashionable for the stupid people (who have problems to imagine numbers greater than 4 or so) to say that there are 10^{500} semirealistic vacuum solutions to the equations of string theory - but they completely misunderstand what this fact means and doesn't mean, even after the long years when this topic was explained. Instead, they prefer to swallow the same shit by idiotic crackpots like Lee Smolin and Peter Woit who were popular in 2006, smacking their lips, and praising the food's precious taste.

String theory, like any theory sufficiently complex to account for wide portions of the real world, has many solutions. It is a good, and probably paramount feature because this multiplicity is almost certainly needed to account for the seemingly complex character of low-energy particle physics. Nevertheless, the number of the stabilized solutions is countable, and they're solutions to the same equations, not different "versions" of a theory, as the duality revolution has demonstrated. String theory has no "different versions" and no continuously adjustable dimensionless parameters.

In fact, even if 10^{500} were the set to choose from and the right vacuum would be "random", the choice can be described by 208 bytes or 3 lines of text. The validity of string theory together with 3 lines of text that determine the compactification is enough to predict anything in the world with an arbitrary accuracy. The correct compactification - i.e. the vacuum solution that is directly relevant for the Universe around us - may be a special one or a generic one and the anthropic vs misanthropic camps differ in this question.

However, it's not really important when it comes to the beauty in physics as measured by symmetries and related rational concepts. The number of discrete solutions doesn't change the symmetry of the theory at all. If a theory has more solutions than what you can directly check, it may mean that you are more ignorant - and face a harder job - than what you used to think. But it surely cannot mean that the theory is less beautiful or likely to be true, in the very technical and rationally justifiable sense that I tried to sketch.

The symmetry is certainly not the only aspect of beauty in physics but it continues to be the most well-known one, and I will concentrate on it.

**Yellow pages: types of symmetries**In physics, symmetries are classified and described by the mathematical notion of a group. There exist various types of groups and ways how they can manifest themselves in the laws of physics, especially the following:

- discrete vs. continuous symmetries
- global vs. local symmetries
- internal vs. spacetime symmetries
- exact vs. approximate symmetries
- broken vs. unbroken symmetries
- anomalous vs. anomaly-free symmetries

Not all 64 combinations of these "six bits" are allowed so it may be a good idea to spend some time with the relationships between these adjectives.

**Discrete, continuous, global, and local symmetries**

Every symmetry can be discrete or continuous: in the former case, the group is finite or countable; in the latter case, the group is uncountable as a set. The discreteness or continuity of the symmetries are separately compatible with all the adjectives below. For example, even discrete symmetries may be local i.e. gauge symmetries: if they're local, there must exist "cosmic strings" (or e.g. D7-branes, in 10 dimensions) so that the monodromy around them generates the desired element of the group.

There are good reasons to think that in quantum gravity, all symmetries must be local symmetries - or at least subgroups of bigger symmetry groups that are naturally local. That's why e.g. the Lorentz symmetry must be extended to the whole diffeomorphism group or the general covariance. It's morally the case because every measurement or transformation can be done locally in general relativity.

That doesn't mean that we should imagine that global symmetries play no role in physics. Quite on the contrary. They're important. Only global symmetries are genuine symmetries that act on physical objects. Gauge symmetries are redundancies of the description - and all acceptable configurations of matter must actually be invariant (i.e. singlets) under all gauge symmetries. Moreover, the nature of a gauge symmetry often depends on the description of the physical system that you choose.

The previous paragraph may confuse you: so which of the symmetries, local or global, are truly physical? Well, only global symmetries are physical in the sense that they don't depend on the description. On the other hand, there must always exist a description in which all these global symmetries are extended to a gauge group and this gauge group is not an invariant property of physics. ;-)

For example, in AdS/CFT, the AdS isometry group is extended to the whole diffeomorphism group in the bulk. However, there is an alternative description, in terms of a gauge theory on the boundary, that has no diffeomorphism group to start with: it is, in some sense, replaced by a completely different U(N) group.

Is it confusing? Do you think that there's a contradiction? There's none. The global symmetries - a special group of "large transformations" inside the extended group of local symmetries - still make a physical sense because they usually act on the asymptotic conditions in space or time non-trivially. Such "large transformations" are actually not required to annihilate the physical states (only the "small" transformations in the gauge groups have to do so); the physical states may transform nontrivially under them.

The global symmetries include translations in space and time, rotations, and Lorentz transformations (or the unbroken isometries of any background whose superselection sector you consider). In the context of gravity, these are "large transformations" within a larger and local diffeomorphism group. But because the physical states can transform nontrivially under these large transformations, these symmetries are useful to classify physical states, according to representations. That's why we can assign physical states with the energy, momentum, and angular momentum, among other things.

Other global symmetries are discrete. They probably include the baryon number - a U(1) symmetry that changes the phases of fields that carry a nonzero baryon charge.

By Emmy Noether's profound theorem, each global symmetry is linked to a conservation law and vice versa. The conservation of energy or momentum or angular momentum is linked to the translational symmetry in time or space or the rotational symmetry. The Lorentz or Galilean symmetry is associated with the conserved velocity of the center-of-mass.

Other discrete symmetries of Nature include parity, charge conjugation, and time reversal. All these discrete Z2 symmetries are actually broken by the weak nuclear interactions and the overall CPT transformation is the only one that has to be preserved as long as the Lorentz symmetry is preserved, too (as Wolfgang Pauli has proved). The C and P symmetries are broken massively by the neutrino sector: there doesn't even exist a right-handed neutrino field that could be mapped to the ordinary left-handed neutrino field (or its complex conjugate) by the C or P transformation. The CP symmetry is broken by even subtler effects: the required "symmetric fields" are available but a subtle dynamical phase in the CKM matrix breaks the symmetry.

For a fixed choice of degrees of freedom, every kind of symmetry - local or global - relates objects, interactions, and phenomena that were previously thought of as independent. That's why every symmetry that is compatible with the known physical phenomena reduces the number of independent concepts in physics. A priori, there is no reason to think that a symmetry should be compatible with the known phenomena. So whenever it is, such an insight makes our understanding of physics more compact, tightly, unified, and beautiful.

Internal symmetries are able to relate a large number of seemingly or previously unrelated fields and interactions. Spacetime symmetries play a special role because every physical phenomenon must respect the existence of spacetime, at least in the long-distance approximation. ;-) Because all the fields and particles share the same spacetime, the spacetime symmetries are universal for all of them. Translations, rotations, and Lorentz boosts are spacetime symmetries (much like the broken parity, i.e. the left-right mirror reflection). Supersymmetry is partially a spacetime symmetry, partially an internal symmetry: it has to act both on internal and spacetime properties of fields.

The commutator of two supersymmetry transformations is a translation, so you must clearly include supersymmetry among the symmetries that do act on spacetime. That's another point that the laymen (including most experimenters) usually misunderstand. Supersymmetry manifests itself by relating different particle species - bosons and fermions - and by doing so, it resembles the internal symmetries. However, the critics argue that no superpartners of known particles have been observed so far, so this whole structure is redundant and supersymmetry doesn't constrain anything because the "power to constrain" only emerges after the field content is doubled.

What these people neglect is the fact that supersymmetry belongs among the spacetime symmetries. The rotational symmetry doesn't reduce the number of particle species, either. But it reduces the spectrum of allowed interactions, and so does supersymmetry. The fact that the rotational symmetry, despite its strict rules, is compatible with all known experiments is highly nontrivial and increases the beauty and unity of the laws of physics - and by the Bayesian inference (because the theory has passed a difficult test), the probability that the whole framework is valid.

The same thing holds for supersymmetry which is also constraining: the main difference is that supersymmetry is spontaneously broken (if it exists). As we will argue at the very end, this difference doesn't and can't influence the degree of internal beauty of supersymmetry.

Broken, anomalous, approximate, and accidental symmetries

Broken, anomalous, approximate, and accidental symmetries

We haven't discussed the three other pairs of adjectives yet. Symmetries can be exact or they can fail to be exact. There are several qualitatively different reasons why the exactness may disappear. And depending on the reason, the conclusions about the beauty and our knowledge about the right theory - as constrained by the symmetries - are very different.

First, a symmetry can fail to be exact, even at the fundamental level. If that's the case, it obviously cannot help us to construct a top-down theory from the scratch - because the symmetry doesn't really hold if you look accurately enough. Nevertheless, such approximate symmetries are extremely important in model building and in our search for approximate or effective theories. For example, the "strangeness" (essentially the number of strange quarks minus antiquarks) was known to be partially conserved but only in some interactions. This insight was very useful to determine certain properties of the nuclear interactions - and it helped to discover the concept of quarks.

So I would like to summarize the concept of approximate symmetries by saying that the knowledge about them can strengthen our ability to quickly eliminate theories that disagree with certain observations - observations that are approximately compatible with certain symmetries. On the other hand, approximate symmetries are not real "qualitative features" that would lead us to consider qualitatively different types of theories. After all, approximate symmetries are deduced from a complete theory, not in the other way around. Only exact symmetries may be a good starting point to fully determine or guess a new theory.

An important subclass of approximate symmetries are accidental symmetries - such as the baryon number (a charge expressed via a U(1) group transforming the phases of fields, by a rate proportional to their baryon content). Accidental symmetries hold in effective theories because mathematics just happens to allow no interaction terms of a specified type that would violate the symmetry. For example, if you consider the field content of the Standard Model and try to add all renormalizable interactions that preserve the Lorentz symmetry and gauge symmetries, you won't be able to write down any interactions that violate the baryon charge.

That's why B is an accidental symmetry: it must hold at the level of the renormalizable effective theories controlling a pre-determined field content. Needless to say, nonrenormalizable interactions - such as those obtained by integrating out new high-energy stuff beyond the Standard Model - can violate and probably do violate B. Black holes are almost certainly able to evaporate and destroy the baryon charge of the initial star, too. The accidental symmetry therefore only holds at a certain level of approximation. By the way, B-L (the baryon minus lepton number) may also be an accidental symmetry but it may also be an exact gauge symmetry.

Another effect that can spoil the exactness of a symmetry are anomalies. Anomalies are quantum effects - usually represented by 1-loop Feynman diagrams - that imply that the full quantum theory cannot preserve all the symmetries that the classical limiting theory (i.e. the theory reduced to tree level diagrams) does. It is obvious that at the fundamental, quantum level, anomalous symmetries don't exist at all.

On the other hand, gauge symmetries are necessary to get rid of unphysical and/or negative-norm, time-like and/or longitudinal polarizations of spin 1 (or higher) particles. It means that any anomaly that you find in a gauge symmetry (typically a Yang-Mills symmetry or a local Lorentz group) kills your theory. For example, until 1984, it was generally believed that type I superstring theory had to be anomalous, because of some rough arguments. The Green-Schwarz calculation showed a remarkably exact and surprising cancellation of all these anomalies for the SO(32) type I string, restoring the consistency of superstring theory and sparking the first superstring revolution.

On the other hand, anomalies in global symmetries don't make a theory inconsistent but they profoundly modify its dynamics. Again, an anomalous symmetry could have existed at the classical level, i.e. on the classical paper, but as soon as the effects of quantum mechanics matter, the symmetry is simply not there.

Consider a realistic theory with three quark flavors, u,d,s. Each of the fermionic fields is composed of a left-handed and a right-handed 2-component spinor. They (u,d,s) can be rotated independently by U(3)_{left} and U(3)_{right} transformations. One can define diagonal U(3) transformations by making the two U(3)_{left/right} transformations simultaneously. The U(1) part of this U(3) is the accidentally conserved baryon charge discussed above while the SU(3) part is an approximate symmetry between the three flavors.

The remaining "chiral" generators, those that rotate the left-handed and right-handed spinors differently (or even oppositely), are broken. The U(1) is broken differently than the SU(3) part: it is anomalous and it can be ignored. The SU(3) part is just spontaneously broken. It remains an approximate symmetry and the existence of such a broken symmetry implies the existence of pseudo (because the symmetry is approximate!) Nambu-Goldstone bosons, i.e. spin-0 bosons that are much lighter than the QCD scale expectation exactly because the symmetry holds much more accurately than a randomly broken non-symmetry.

To summarize, anomalous symmetries are just illusions of a classical theory but they're not really there in the full quantum theory. Approximate and accidental symmetries are useful approximate concepts used to study effective and phenomenological theories. Anomalous gauge symmetries make a theory inconsistent.

Spontaneously broken symmetries

Spontaneously broken symmetries

But my main motivation to write this essay are the spontaneously broken symmetries. These are symmetries that are exactly satisfied by the laws of physics, including all of their quantum and nonperturbative corrections, but they're not directly seen in the low-energy physical phenomena, for reasons that will be explained. Because, as you will see, the asymmetry in the observations is a derivable fact that has nothing to do with the beauty of the laws of physics, it doesn't diminish the beauty of the underlying theory at all.

In other words, a theory with a spontaneously broken symmetry is as constrained and as beautiful, in the refined sense, as a theory with an unbroken symmetry. Let me explain why.

The classical popular example of a broken symmetry is a pencil that you want to place symmetrically on its tip. If you do so, its position will show the rotationally symmetry around the vertical axis - i.e. the democracy between all the directions - very clearly. However, in the real world, something else happens. A tiny fluctuation (even one whose existence is guaranteed by the uncertainty principle) will be enough for the pencil to wobble. It will randomly choose a direction and collapse. Once it does so, the democracy between the directions is broken.

It is broken spontaneously because both the laws of physics as well as the pencil are rotationally symmetric. But the pencil just spontaneously decided to ignore this symmetry because an asymmetric position of the pencil actually lowers the energy. As far as the potential energy goes, it doesn't matter which direction the pencil chooses: again, all the directions have the same energy. But the pencil is damn sure that it should fall down. ;-)

A similar mechanism appears in the electroweak theory, among others. The SU(2) x U(1) symmetry is an exact symmetry of the laws of Nature. However, there exists something like the Higgs field "h(x,y,z,t)" at every point of space and time. And much like the pencil, this Higgs field just doesn't want to sit at the symmetric point "h=0". In fact, the minimum energy is obtained if the length of the complex vector "h" is equal to "v", the ultimate vacuum expectation value. So the Higgs field randomly chooses a direction, much like the pencil, and the original SU(2) x U(1) symmetry is spontaneously broken.

It is important to realize that the symmetry is still there and it is exact. All physical objects or configurations that are related by the symmetry transformation have the same energy. However, this is just the "equality of opportunities" which doesn't imply the "equality of outcomes" in the real world. In fact, the Higgs field surely wants to choose an asymmetric value, instead of "h=0". Other fields' masses (and other properties) depend on whether or not they're aligned in the same "direction" as the Higgs field.

The potential energy that is sufficient to convince the Higgs to pick an asymmetric vacuum expectation value may be as simple as "x^4 / 4 - x^2 / 2", the classical starting point for the renormalizable theory ("x" is the same thing as "h"). This function is even (and, more generally, symmetric with respect to the SU(2) x U(1) rotations). However, its global minimum is not the symmetric point "x=0". Instead, the minima of the function are located at "x=+1" and "x=-1" (or any other vector whose length is one, if you promote "x" to a complex vector).

If the potential energy were "x^4 / 4 + x^2 / 2", the minimum would be at "x=0" only and the symmetry would remain unbroken. So you should ask, is the theory with "x^4 / 4 - x^2 / 2" uglier than the theory with "x^4 / 4 + x^2 / 2"? Obviously, it can't be uglier. It only differs by a minus sign. All the fundamental symmetries are equally large and equally pretty: you surely don't want to discriminate against theories with a minus sign. ;-) However, the minus sign leads to a different physical behavior.

With the minus sign, the Higgs boson wants to choose a nonzero value. This nonzero value will influence all low-energy phenomena. On the other hand, the vacuum expectation value "v" of the Higgs has the dimension of a mass, and at energy scales "m" much higher than this "v", this "v" may be neglected. That's why the symmetry is restored in very high-energy phenomena. This conclusion is quite general: symmetries tend to be restored at high energies, high temperatures, or short distances and broken at low energies, low temperatures, or long distances.

The spontaneously broken symmetries are as accurate as the unbroken ones: they just dynamically lead to a vacuum (analogous to the fallen pencil) that is not a singlet under the symmetry. This necessary choice of the vacuum (and its superselection sector) makes all the objects and phenomena built upon the vacuum look asymmetric, even though the symmetry exactly holds and is as constraining as an unbroken symmetry: it just relates events in a given vacuum with events in a physically equivalent but different vacuum, which is much less "physical" an action than a symmetry that relates objects that can exist in the same vacuum.

As in many other cases, I think that the ultimate reason why people like Tommaso Dorigo can't understand why spontaneously broken symmetries are equally pretty is politics. Dorigo is a hardcore Marxist who makes Friedrich Engels look like Friedrich Hayek in comparison. He just doesn't distinguish the "equality of opportunities" and the "equality of outcomes". If he doesn't see a world where everything is equal and symmetric, it is "ugly" for him.

But that's a very deep misunderstanding. Much like the justice in a decent society only requires the "equality of opportunities" and says nothing about the "equality of outcomes", the beauty of the laws of physics only depends on the symmetry of the underlying equations, not on the symmetries of the actual configurations that solve the physical equations and that will evolve in the real world. It is not a purpose of the laws of physics to produce symmetric objects. In fact, the spontaneous breaking is quite a generic fate of many underlying symmetries and it is often as necessary for the existence of life as income/wealth inequality is critical for a decent GDP growth in any society.

The same comments hold not only for the electroweak symmetry but also for other symmetries that may exist in the real world but that are spontaneously broken, such as grand unified symmetries and supersymmetry. In the latter case, the mathematics needed to describe their spontaneous breaking is more sophisticated than a simple quartic function - and the "minimal" gadget to break the symmetry is not as unique as it is in the case of a single Higgs doublet (also known as the Weinberg toilet). But you should still understand that the situation is qualitatively isomorphic to the case of the electroweak symmetry.

Finally, Tommaso Dorigo has offered us some popular laymen's misconceptions about supersymmetry breaking, namely that it depends on 130 parameters. Well, 130 parameters only count the number of parameters in an effective theory - a theory describing how supersymmetry breaking affects low-energy physics, one that is not trying to understand the fundamental origin of the supersymmetry breaking. But if one actually constructs, knows, or finds a well-defined top-down theory with the right fields that break the supersymmetry and other fields that communicate it to the Standard Model fields, most or all of these 130 parameters become calculable.

So these soft-SUSY-breaking 130 parameters certainly don't make the underlying theory uglier - they just inform us that there are many details about its implementation that are not yet known to us. But this "ignorance" is something completely different than "ugliness". More obviously, as soon as the right supersymmetry breaking scenario is found and/or the parameters are measured, the ignorance disappears. If the ignorance were a real ugliness, it could never disappear.

Counting of vacua

Counting of vacua

In the most complete and almost certainly correct top-down framework we have, namely string theory, the supersymmetry breaking scenario is fully determined by the choice of the compactification. It's fashionable for the stupid people (who have problems to imagine numbers greater than 4 or so) to say that there are 10^{500} semirealistic vacuum solutions to the equations of string theory - but they completely misunderstand what this fact means and doesn't mean, even after the long years when this topic was explained. Instead, they prefer to swallow the same shit by idiotic crackpots like Lee Smolin and Peter Woit who were popular in 2006, smacking their lips, and praising the food's precious taste.

String theory, like any theory sufficiently complex to account for wide portions of the real world, has many solutions. It is a good, and probably paramount feature because this multiplicity is almost certainly needed to account for the seemingly complex character of low-energy particle physics. Nevertheless, the number of the stabilized solutions is countable, and they're solutions to the same equations, not different "versions" of a theory, as the duality revolution has demonstrated. String theory has no "different versions" and no continuously adjustable dimensionless parameters.

In fact, even if 10^{500} were the set to choose from and the right vacuum would be "random", the choice can be described by 208 bytes or 3 lines of text. The validity of string theory together with 3 lines of text that determine the compactification is enough to predict anything in the world with an arbitrary accuracy. The correct compactification - i.e. the vacuum solution that is directly relevant for the Universe around us - may be a special one or a generic one and the anthropic vs misanthropic camps differ in this question.

However, it's not really important when it comes to the beauty in physics as measured by symmetries and related rational concepts. The number of discrete solutions doesn't change the symmetry of the theory at all. If a theory has more solutions than what you can directly check, it may mean that you are more ignorant - and face a harder job - than what you used to think. But it surely cannot mean that the theory is less beautiful or likely to be true, in the very technical and rationally justifiable sense that I tried to sketch.

And that's the memo.

Dear Professor:

ReplyDeleteThanks for your wonderful essay first.I have several questions:

1) Are there any conserved currents or charges to discrete symmetries by Emmy Noether's profound theorem?

2) Isometry is a global symmetry, but its Killing generators are usually spacetime dependent functions. It seems strange to me.

Global or local does not depend on their generators's properties. Is this right?

3) Would you like to give some easy-approached reasons to “There is no global continuous symmetries in quantum gravity.”?

Thanks for your answes.

By the way if the pencil point is fixed about a rotation point then this is a vertical pendulum; if the pencil is given a periodic motion vertically up and down at the point with a small displacement compared with the pencil length and with frequency (omega), then a small angular displacement of the pencil from the vertical satisfies a Mathieu equation in the angular displacement, involving the periodic vertical motion of the pendulum.

ReplyDeleteIt can be shown by a number of methods that there are regions of stabilty of the frequency of vertical oscillation (omega) for which the solutions of the Mathieu equation are stable - thus the vertical pendulum remains stable for these frequencies of vertical oscillation.

I don't know if the analogy carries over

Dear 玉书,

ReplyDeletethanks for your nice words. I don't really mind being called a "professor", but just to be sure, it is not a correct title for me now. ;-)

1) That's actually a great question I should have covered, too. Only continuous symmetries lead to conserved quantities that can go from minus infinity to plus infinity.

By the same Noether theorem, discrete (finite) symmetries are associated with conserved quantities that have a finite number of values. For example, if the theory is left-right symmetric (under parity), then a quantum number called "parity" is conserved.

It's clear in quantum mechanics - the parity is expressed by an operator "P" that commutes with the Hamiltonian which means that it is conserved (because the Hamiltonian generates time evolution), and it also means that the Hamiltonian is left-right symmetric (because the conjugation by P makes the mirror reflection).

The eigenvalues of P are +1 or -1 and are added multiplicatively. Similarly, one can have other finite symmetries - the eigenvalues of their operators are conserved and can be roots of unity etc.

Discrete infinite symmetries, like "Z", lead to a conserved quantity that is continuous but periodic. Think about the crystal and the electron's momenta that are only defined modulo something, i.e. in the Bril. zone. The group is always a kind of "Fourier transform" of the domain in which the conserved quantity takes values.

2) Yes, isometries are global symmetries, and they're conveniently determined by vector fields. What is global about those fields is that the configuration of these Killing fields is determined in the whole spacetime globally - you can't redefine the Killing vector field in different regions independently. That's why the symmetry is "global".

But indeed, it may be expressed using fields, i.e. as a special case of a local symmetry. Indeed, I have argued that global symmetries are typically subgroups of "extended" (including "large transformations") groups of local symmetries.

Global vs local really depends on the number of independent generators only. If there is a generator for every region, it is a local symmetry. Global symmetries come in "one copy per spacetime" only. There may be a way to get rid of the "small transformations" in a gauge group, which is really a redundancy, but the global transformations (which may nontrivially transform physical states) survive in all alternative, physically equivalent descriptions.

[to be continued]

...

ReplyDelete3) For spacetime translations, the principle that "all symmetries must be local" is nothing else than the equivalence principle. One is allowed to move or change coordinates arbitrarily at every point of space and time, so the translations and rotations must be extended to diffeomorphisms.

This must really be the case also for other symmetries, although it is less direct to see and depends on some lore (that is confirmed both by vague consistency arguments in quantum gravity as well as by robust arguments in string theory). For example, if there is a U(N) global symmetry, it must be possible to choose different U(N) transformation parameters in two (imagine!) disconnected portions of spacetime because these portions have no physical way to interact and agree about the same U(N) parameter.

By an extension, regions must be allowed to have independent U(N) parameters even if they are connected. This becomes very clear in Kaluza-Klein theory where Yang-Mills symmetries arise from isometries acting on extra duimensions: these higher-dimensional isometries must be extended to the full diffeomorphism group just like the ordinary 4D spacetime symmetries. So Yang-Mills symmetries automatically follow from the global ones, as long as you try to unify them with gravity in this natural way. If you add symmetries in non-Kaluza-Klein ways, this conclusion must still be valid although the proof above is no longer usable.

In perturbative string theory, one can show that (continuous) symmetries must be local by the (almost) same argument that proves that string theory has no adjustable non-dynamical dimensionless parameters.

If there were such a parameter, it would be given by a marginal deformation of the CFT, a (1,1) operator, but this operator could be multiplied by exp(ik.X) to get (the vertex operator for) a massless scalar field with any momentum.

In a similar way, if there is a global symmetry, it must carry a current "j" on the worldsheet, a (1,0) or (0,1) tensor, and one can multiply it by partial(X_mu).exp(ik.X) (the partial may also have a bar, to get a (1,1) tensor) to obtain the vertex operator for a spin-1 boson in spacetime, proving that the symmetry is automatically gauged.

That's a perturbative argument but experience suggests that it holds nonperturbatively, too. In fact, even discrete symmetries in string theory seem to be automatically "gauged", which means that a corresponding cosmic string with the monodromy "g" always exists (although its tension is not necessarily low).

Best wishes

Lubos

Dear Lubos,

ReplyDeleteThanks for your explaination.

Best regards

yushu