# A Ed Witten on Symmetry and Emergence

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#### king vitamin

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Ed Witten posted an interesting article on arXiv a few days ago on the fate of global symmetries in physics beyond the Standard Model. You can read it here.

In particular, Witten argues that the global symmetries of the Standard Model are all approximate and emergent at low-energy, and they should be violated at the GUT and Planck scales. In particular, a quantum gravity theory should only contain conservation laws associated with gauged interactions. The arguments are likely familiar to experts, but I thought it was a nice and short self-contained lecture on the idea.

Related Beyond the Standard Model News on Phys.org

#### Demystifier

Science Advisor
2018 Award
A quote form the paper:
"... global symmetry is a property of a system, but gauge symmetry in general is a property of a description of a system. ... The meaning of global symmetries is clear: they act on physical observables. Gauge symmetries are more elusive as they typically do not act on physical observables. Gauge symmetries are redundancies in the mathematical description of a physical system rather than properties of the system itself."

#### DrDu

Science Advisor
The meaning of global symmetries is clear: they act on physical observables.
Is this so? E.g. a global U(1) symmetry is introduced to explain the superselection rule of electrical charge. The point is that it does not act on the observables, i.e. they are invariant under the transformation. On the other hand the field operators which are able to create or destroy a particle aren't invariant.

#### kith

Science Advisor
A basic question: What is the "operator dimension" he is talking about? What spaces are these operators acting on?

(My knowledge level: I'm deeply familiar with QM but only superficially with QFT and the Standard Model)

#### Fra

Just keeping in mind that my perspective is to reconstruct measurement theory, i see another way to conceptualize this, that can motivate wittens vision not just be the historical arguments and notes from string theory, but from the point of view of constructing principles for a truly relational measurement theory.

Then the logic of emergent symmetries can be interpreted as emerging in the internal state of the OBSERVER. If we insist that the process of observation and observers, is not anything different than physical interactions between the variety of systems. Then one can understand the "internal structure" of a system (say the hydrogen atom) from the point of view of the lab observer O5 as contain miniatures of Alice and Bob, or electron and quarks or however we label them. We can also name then just O1,O2,O3, ...

Here a gauge symmetry can be understood as emergent from the point of view of the O5, and this symmetry is inferrable by many experience where we for sample study the effect of "testobservers" that we inject into the system. Here we can see a particle accelerator as a way to "inject" test observers into the system of study. And in this view, I think it is deeply misleading to call gauge symmetris for MATHEMATICAL redundancy. I think of them as the freedom to consider ANY part of the systems as a "testobserver". This in fact reflects the INTERNAL structure of the system, and IMHO has nothing todo with mathematical redundancy. Beacase the specific form of mathematical redundancy has a physical origin.

Witten writes this though
witten said:
To put it differently, global symmetry is a property of a system, but gauge symmetry in general
is a property of a description of a system. What we really learn from the centrality of gauge
symmetry in modern physics is that physics is described by subtle laws that are “geometrical.”
This concept is hard to define, but what it means in practice is that the laws of Nature are subtle
in a way that defies efforts to make them explicit without making choices. The difficulty of making
these laws explicit in a natural and non-redundant way is the reason for “gauge symmetry.
I think he was not cleary clear on the meaning of this. This point is where i have antoher perspective that to me follows from constructing principles.

Sorry for the silly picture but a quick improvised illuststration :D
'

This is not a litteral picture or any interaction diagrams, its just a conceptual picture of how the hierarchy of observations might actuall work conceptually. As we see threr are LAYERS here of observers and sub-observers. And symmetries depend on the level of where the observer sits. And if you adopt this picture, the vision of Witten that ONLY gauge interactions have a place in the fundamental theory, can be easily understood because its the only type of mechanism that has an explanatory value in the inferential perspective (ie if you take the instrumentalist interpretation to extreems, like i suggested here.

The non-gauge theory the inferentially corresponds to a non-inferrable symmetry, which explains why its typically approximate.

Another conclusion is that we here have a hiearchy of observers, which corresponds to energy scale. And the symmetries are emergent ONLY when parts interact.
The symmetry emerges in the observing systems internal structyre. At least its how i see it.

But if this interpretation proves right, then the strings themselves should also be emergent from something even deeper. Because it would break the consistency of reasoning to have all this nice stuff, but STARTING from the non-inferrable concept of string in embedded space. I also see this realted to the landscape problem.

/Fredrik

#### king vitamin

Science Advisor
Gold Member
Is this so? E.g. a global U(1) symmetry is introduced to explain the superselection rule of electrical charge. The point is that it does not act on the observables, i.e. they are invariant under the transformation. On the other hand the field operators which are able to create or destroy a particle aren't invariant.
Electric charge is a symmetry which derives from gauge theory. I believe Witten's use of "global symmetry" does not include any charges associated with gauge groups. The superselection rule you cite is essentially his point - without such a huge constraint, one cannot have a conservation law.

A basic question: What is the "operator dimension" he is talking about? What spaces are these operators acting on?
I will try to explain this in a concise way, but as a result I will need to skip over some concepts.

The operators in question are composite operators of fields with act the same as every other field in the Standard Model. The "dimension" of an operator in this context (perturbation theory around a non-interacting QFT) is literally the dimensions of units it has. In high energy physics, one usually takes natural units $\hbar = c = 1$ which allows you to express all quantities in units of energy. When a high energy physicist says "a dimensions # operator," they mean "an operator with dimensions of energy^# in natural units."

(A quick warning. Some of the above is ONLY true when perturbing around a free theory. In an interacting theory, the operators carry two types of dimensions, their "engineering dimension" (the actual dimension of the operator) and their "scaling dimension" (which would require its own post to explain). These are the same for a free theory, but in interacting theories it is the scaling dimension which matters.)

Now, in the old school (Feynman/Schwinger/Tomonaga) viewpoint of QFT, one requires renormalizability for the theory to make sense. Furthermore, one can show that an operator with dimension>4 is non-renormalizable, so it is not allowed in an acceptable theory. Thus, the global symmetries are exact!

.....or not. It turns out that a non-renormalizable QFT makes sense if one is ok with a cutoff - and signs point to the cutoff being real. So we should include operators with dimension>4, and as Witten says, these operators result in violation of global symmetries.

#### Fra

Electric charge is a symmetry which derives from gauge theory. I believe Witten's use of "global symmetry" does not include any charges associated with gauge groups. The superselection rule you cite is essentially his point - without such a huge constraint, one cannot have a conservation law.
Yes i agree, thats what he must mean. For this reason there is a mixup of global vs local, physical vs gauge. The real message in wittens papers is not really global vs gauge, it is physical vs gauge! But choosing the right words isnt always easy. I might well also "overinterpret" his us of "mathematical redundancy" in my comment.

Its just due to habit that usually global = physical, and local = gauge, but there are exceptions to both. So physical vs gauge is better.

A paper that also mentions this and a bit more describes the different things with a better labelling of things

From physical symmetries to emergent gauge symmetries

"Gauge symmetries indicate redundancies in the description of the relevant degrees of freedom of a given field theory and restrict the nature of observable quantities. One of the problems faced by emergent theories of relativistic fields is to understand how gauge symmetries can show up in systems that contain no trace of these symmetries at a more fundamental level...
...of the Noether currents associated with physical symmetries leads to emergent gauge symmetries in specific situations. An example of a relativistic field theory of a vector field is worked out in detail in order to make explicit how this mechanism works and to clarify the physics behind it. The interplay of these ideas with well-known results of importance to the emergent gravity program, such as the Weinberg-Witten theorem, are discussed."
-- https://arxiv.org/abs/1608.07473

/Fredrik

#### Jimster41

Gold Member
One of the problems faced by emergent theories of relativistic fields is to understand how gauge symmetries can show up in systems that contain no trace of these symmetries at a more fundamental level...
Are multi-fractals believed to be capable of such a trick?
I'm not having any luck finding any papers on this specific question but I did find a paper on an application that seems to use some-kind of theory of it. Unfortunately it's not a free paper.
https://link.springer.com/chapter/10.1007/978-0-387-98154-3_8

#### Fra

Are multi-fractals believed to be capable of such a trick?
I'm not having any luck finding any papers on this specific question but I did find a paper on an application that seems to use some-kind of theory of it. Unfortunately it's not a free paper.
https://link.springer.com/chapter/10.1007/978-0-387-98154-3_8
I never thought about this in fractal terms, so i can not comment wether there is a connection somewhere. But I don't see it.

The trick i envision is that its only when a collective of similarly functioning units are allowed to interact, that NEW structures are formed, and its on these higher structures the emergence symmetries live. The "similarly functioning units" in my thinking are information processing agents. To be modelled as an evolving structure that codes information.

For example if you belive in strings, an isolated string in space is one thing. It seems easy to think that the embedding space could be almost anything. But if you imagine that the space its embedded in, is made up of relations to an environment populated but other strings, and if you have any idea of HOW strings interact with an unknown environment (not with other strings in a a priori given space) then you seem like holding a clue to understanding the selection principles for spacetime? The question is what kind of mathematical or algorithmic framework is needed to model this? At some level i think it has to be interacting independent processes, that are competing about controlling each other.

If I were into strings, thats the place where i would dig, as from the outside i see its the key to solve the landscape problem. but i think the only way todo so is to consider the strings themselvs as emergent as well. After all a string is a very non-trivial starting point, that IF it has anything at all todo with reality begs a deeper explanation.

I think to understand this, one has to step outside the QFT formalism. It cant be the right formalism.

/Fredrik

#### Urs Schreiber

Science Advisor
Gold Member
This popular statement that gauge symmetries are "just a redundancy" in the description of a system needs qualification. If boundaries are taken into account, the gauge symmetries leave their physically detectable imprint there.

A famous example is Chern-Simons theory with gauge group $G$. On its boundary sits the WZW model, for which $G$ is no longer a gauge group, but the target space and thus manifestly not a "redundancy". The target space of the WZW model is the boundary values of the gauge transformations of the corresponding CS-model.

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#### Crass_Oscillator

A famous example is Chern-Simons theory with gauge group $G$. On its boundary sits the WZW model, for which $G$ is no longer a gauge group, but the target space and thus manifestly not a "redundancy". The target space of the WZW model is the boundary values of the gauge transformations of the corresponding CS-model.
Does this model have any experimental manifestations?

#### Fra

This popular statement that gauge symmetries are "just a redundancy" in the description of a system needs qualification. If boundaries are taken into account, the gauge symmetries leave their physically detectable imprint there.
Yes, this is a good point.

Technical details tend to be very model specific and its hard to see what is model specific and what is generic properties. But I think the conceptual understanding of this is that the "boundary" is where the observer interfaces with the system. And this indeed MUST break the gauge invariance. If there was no boundary at all, then also there no measurements or inferences would be possible. Also if the boundary at some point, somehow did not break the gauge invariance during the measurement then the gauge symmetry would be a trivial.

This is similar to how you make a measurement on an isolated system. Obivoulsy if it was truly isolated, no interactions (and thus no measuments) are possible.

It was also the idea behind the below lines
Another conclusion is that we here have a hiearchy of observers, which corresponds to energy scale. And the symmetries are emergent ONLY when parts interact.
And there "interaction" of internal parts are typically different at the boundary. So symmetry can be "broken" either by a boundary of some kind, or by an elevated temperature that separates the parts. In a sense they two scenarious seems to me to have the same effect on symmetries

/Fredrik

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#### Urs Schreiber

Science Advisor
Gold Member
Yes, this is a good point.
Technical details tend to be very model specific and its hard to see what is model specific and what is generic properties.
It's not so bad in the present case. By definition (e.g. here), a gauge symmetry is a parameterizable transformation of the fields that leaves the Lagrangian invariant up to a total spacetime derivative, and in the present of boundaries that total spacetime derivative is itself, by the Stokes theorem, a contribution on the boundary.

Therefore the phenomenon that gauge symmetry imprints itself on boundaries happens whenever a Lagrangian is gauge invariant only up to total spacetime derivatives. This famously happens for Chern-Simons theory but not, for instance, for Yang-Mills theory.

Nevertheless, a related effect also affects Yang-Mills theory: Here the "instanton sectors" are controlled by the Chern-Simons Lagrangian (really: by the Chern-Simons 2-gerbe, see here, but anyway). Now the classification of instantons does depend on the topology of the gauge group $G$: for $\Sigma^\ast$ the one-point compactification of spacetime, then $G$ Yang-Mills instantons are classified by $H^{1}(\Sigma^\ast, B G)$.

So in the example of "S-dual" super Yang-Mills theories with gauge groups $SO$ or $Sp$, I would think that in general they are distinuishable after all if one looks at their instanton sectors.

Does this model have any experimental manifestations?
While these instanton sectors are not directly being observed, they are argued to control the non-perturbative vacuum of Yang-Mills theory ("instanton sea model"), also baryogenesis.

#### Urs Schreiber

Science Advisor
Gold Member
You could have just said "No."
That would have been wrong, though.

The very subject of Witten's article, and hence of this thread here, requires to be willing to think just a tad beyond. If your attitude is to not be interested in theoretical development that aims to go beyond what is presently strictly verified, then there is nothing to be seen here.

#### Crass_Oscillator

I'm just pointing out that it is odd that you would talk about a model which is not grounded in reality (i.e. experiments) rather than other examples of the non-redundancy of the gauge degree of freedom which are, such as the AB effect.

#### no-ir

I'm just pointing out that it is odd that you would talk about a model which is not grounded in reality (i.e. experiments) rather than other examples of the non-redundancy of the gauge degree of freedom which are, such as the AB effect.
Not my specific area of expertise (I study quantum spin liquids), but from what I understand the model of a Chern-Simons theory with a boundary that Urs Schreiber is referring to (if I understand correctly) is directly applicable at least to the quantum Hall effect. In the integer case an abelian CS theory describes the 2+1 dimensional bulk and it's precisely its gauge noninvariance due to the presence of a boundary (the edge of the material) that gives rise to topologically-protected gapless edge states described by a 1+1 dimensional chiral Weyl fermion theory (see e.g. https://arxiv.org/abs/hep-th/9902115v1). The fractional quantum Hall effect can be likewise modeled by a nonabelian CS theory with a boundary. These things are eminently experimentally accessible so I really don't see what the problem here is with Urs Schreiber giving this example.

In any case, the boundary-induced states in the quantum Hall effect are just one specific example of a general bulk-edge (or bulk-boundary) correspondence that happens in a whole host of different materials and also goes under the name of topologically-protected edge states or topological phases of matter - quite a hot topic in condensed matter physics these past few years. You might also recall that last year's physics Nobel prize was given precisely "for theoretical discoveries of topological phase transitions and topological phases of matter".

But even if CS theory did not have this huge experimental support and if Nobel prizes were not given out for these ideas I still don't see why a mathematically consistent model, even if it didn't have exprimental support, could not serve as a valid counterexample to the mathematically precise, but incorrect, statement that gauge symmetry is always "just a redundancy". This forum is called "Beyond the Standard Model" for a reason.

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#### Demystifier

Science Advisor
2018 Award
A quote form the paper:
"... global symmetry is a property of a system, but gauge symmetry in general is a property of a description of a system. ... The meaning of global symmetries is clear: they act on physical observables. Gauge symmetries are more elusive as they typically do not act on physical observables. Gauge symmetries are redundancies in the mathematical description of a physical system rather than properties of the system itself."
I have just found a similar statement in Schwartz's "Quantum Field Theory and the Standard Model", Sec. 8.6:
"Gauge invariance is not physical. It is not observable and is not a symmetry of nature.
Global symmetries are physical, since they have physical consequences, namely conserva-
tion of charge. That is, we measure the total charge in a region, and if nothing leaves that
region, whenever we measure it again the total charge will be exactly the same. There is no
such thing that you can actually measure associated with gauge invariance. We introduce
gauge invariance to have a local description of massless spin-1 particles. The existence of
these particles, with only two polarizations, is physical, but the gauge invariance is merely
a redundancy of description we introduce to be able to describe the theory with a local
Lagrangian."

#### star apple

I have just found a similar statement in Schwartz's "Quantum Field Theory and the Standard Model", Sec. 8.6:
"Gauge invariance is not physical. It is not observable and is not a symmetry of nature.
Global symmetries are physical, since they have physical consequences, namely conserva-
tion of charge. That is, we measure the total charge in a region, and if nothing leaves that
region, whenever we measure it again the total charge will be exactly the same. There is no
such thing that you can actually measure associated with gauge invariance. We introduce
gauge invariance to have a local description of massless spin-1 particles. The existence of
these particles, with only two polarizations, is physical, but the gauge invariance is merely
a redundancy of description we introduce to be able to describe the theory with a local
Lagrangian."
I don't get it. We knew since before that gauge symmetry were redundancies in the mathematical description of a physical system rather than properties of the system itself. Why are people emphasizing it now.. is it because some people believe they were physical or because there is still possibility it can be physical and some want to persuade themselves the point they were not?

#### Demystifier

Science Advisor
2018 Award
Why are people emphasizing it now.. is it because some people believe they were physical
I guess that's the reason.

#### star apple

I guess that's the reason.
When I first read "Deep Down Things" by Bruce Schumm. The author didn't say flat out it was not physical. Quoting him:
"Without the extension provided by sophisticated scientific apparatus, our senses, our biochemical senses, are just too anemic to invoke or perceive motion in the spaces we have introduced and discussed in this chapter. Might it be possible that the internal spaces of this chapter are in fact no less real than the three-dimensional space of our everyday perception? Could it be that, absent any way to influence or perceive motion in these realms, we simply lack the tools and motivation to evolve the ability to perceive them?
I'll leave you to mull this over while moving on to the second and final discussion of this section: that of an important qualification of the nature of the symmetry transformations we've been considering..."

Well. After asking physicists later if the gauge symmetry could be physical. They said no. But then is it possible there are physicists who think they are physical? Any paper that describe they could be physical?

We don't know the "dna" of physical law as Fra put it. But what if the gauge symmetry was real in the "dna" dynamics of physical law?

#### Demystifier

Science Advisor
2018 Award
a gauge symmetry is a parameterizable transformation of the fields that leaves the Lagrangian invariant up to a total spacetime derivative, and in the present of boundaries that total spacetime derivative is itself, by the Stokes theorem, a contribution on the boundary.
But if there is a boundary contribution then action does not have a gauge symmetry, so we cannot say that the theory has gauge symmetry.

#### Urs Schreiber

Science Advisor
Gold Member
I don't get it. We knew since before that gauge symmetry were redundancies in the mathematical description of a physical system rather than properties of the system itself. Why are people emphasizing it now..
I share the puzzlement about the sociological processes, but maybe it's an occasion to emphasize that it was never true that gauge symmetry is just a reduncancy.

What you are all thinking of is the gauge equivalence relation, which checks whether two field histories or states are related by some gauge transformation or not.

Indeed, whenever you have an equivalence relation on some set, then to any operation on that set which respects the equivalence relation this relation embodies a mere redundancy, and you may without restriction simply pass to the set of equivalence classes and ignore the original set on which the equivalence relation was defined.

But the point about gauge symmetry in physics is that there is more information than just the equivalence relation saying whether two field histories are gauge equivalent. Namely there is also the information how they are being gauge transformed into each other in a given situation. Because in general there is more than one gauge transformation that relates two gauge equivalent field histories or two states.

In particular, generally every field history or state is gauge equivalent to itself in more than one way. For instance in the archetypical situation of an abelian 1-form gauge field, every spacetime-constant gauge transformation takes all field histories to themselves.

This refinement of a mere equivalence relation to a situation where one has information about how things are equivalent to each other is called a "groupoid" or "stack".

A famous example that you may have heard of is the stack of complex tori. Naively, a complex torus is a equivalence class of a point in the upper half plane by the equivalence relation given by the action of $SL(2,\mathbb{Z})$. But for classical reasons (here) it is a bad idea to think that the action of the "gauge group" $SL(2,\mathbb{Z})$ is just a redundancy. This is because there are some points in the upper half plane which are fixed by this action, hence which are gauge equivalent to themselves in non-trivial ways. Acordingly, if one pays attention then the $SL(2,\mathbb{Z})$ action on the upper half plane is not just a redundancy. It is a redundancy plus extra information. It is an ancient insight in mathematics that it is important to remember that extra information. There is no reason why physicists should be ignorant of such old insights.

Further exposition of the relevance in physics of keeping track of how field histories are gauge equivalent, on top of knowing that they are, is here.

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#### Fra

but maybe it's an occasion to emphasize that it was never true that gauge symmetry is just a reduncancy.
I share this view, and its easy to get confused with mathematical redundancy, physical redundancy etc.

One reason to discuss this is that it may seem paradoxal:if this is just a mathematical redundancy, then where is the explanatory power?
A mathematical redundancy is a triviality, a matter or labelling, thats clearly not quite what we have here.

I also think you can see this in different view. If you are on formalising and axiomatising physical theories, then you will see this in one way.
But I see this from a different angle. For me this has to do what you consider to be observable, or measurable or inferrable. As with the other words, you can mean slightly different things with these. In formal QM or QFT, there are precise mathematical meanings, but again i see this from a BTSM view, and the context of reconstructing a measurement theory that ssolves some of the observer vs observed problem.

The question of what is "physical" IMO clearly depends on which observer you ask! Herein lies also the mystery of gauge theory. Thereof my silly picture in the beginning. I myself am sufficiently turned on at this, that it have a hard time to understand what there seems to be so little research in this direction.

This has bearing to many things, observables in GR for example. What are the "right way" to quantize? how do you "view" gauge theory conceptually, WHY is gauge theory so useful?
/Fredrik

#### Fra

Why are people emphasizing it now.. is it because some people believe they were physical or because there is still possibility it can be physical and some want to persuade themselves the point they were not?
The question is what we mean by physical? CLASSICAL gauge theory is one thing. Here the whole notion of "observer" is kind of nonexistant anyway. And gauge ina measurement theory is something else.

IMO, herein there is something fishy. What is physical anyway? What are the ontologies? I think some people think about what's out there, in a realist ense, while people like me think ontologies are inferrable states, and these are fundamentally observer dependent.

There are different thinking here. And this partly relates to inmperfection in QM if you ask me.

/Fredrik