# FeaturedA Ed Witten on Symmetry and Emergence

1. Nov 6, 2017

### star apple

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?

2. Nov 6, 2017

### Demystifier

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.

3. Nov 6, 2017

### Urs Schreiber

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.

Last edited: Nov 6, 2017
4. Nov 6, 2017

### Fra

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

5. Nov 6, 2017

### Fra

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

6. Nov 6, 2017

### star apple

I read this interesting passage in Deep Down Things:

"For the case of regular spin, we had to take spin-space seriously because
it was associated with a concrete, measurable, physical quantity—angular
momentum. This was only mildly uncomfortable because, although spinspace
has the somewhat hard-to-stomach property that you have to turn all
the way around twice to get back to your original condition, it’s otherwise
pretty much like regular space. Isospin space, however, is completely abstract;
it bears no relation whatsoever (other than through analogy) to anything
we can grasp with our faculties of perception. How could rotations in
such a space possibly have anything to do with the physical world? And yet
the physical manifestation of the invariance of the strong force with respect
to rotations in this space, the conservation of isospin, is a solidly established
fact in the world of experimental science.
So, what then is isospin-space from a physical point of view? Physicists
usually describe it as an internal symmetry space, but what’s that, really? It’s
your old buddy again, telling you that your car’s carburetion system “works
on a vacuum principle.” How’s that going to help you to understand and fix
the thing? It isn’t.
Regarding the physical interpretation of the notion of isospin space,
again your guess is as good as mine. Perhaps its experimental manifestations
are hinting at some new and deeper truth about the universe that lies just
beyond the current limits of our comprehension. Perhaps not. But one
thing, however, is true: The introduction of the idea of internal symmetry
spaces, of which isospin space was the first example, was an essential step
forward in our understanding of the universe and the nature of the laws that
govern it."

Note The invariance of physical laws with respect to rotations in ordinary space is associated by Noether's theorem with the conservation of angular momentum. The conserved quantity associated with invariance with respect to rotation in the abstract space of isospin is isospin itself. Any connection between the two?

For those so tired of pondering quantum interpretations. It is refreshing to instead ponder on interpretations of gauge symmetry.. lol..

7. Nov 7, 2017

### no-ir

True. We should be more precise: you can make the total action gauge invariant by adding to it a boundary term that precisely cancels the gauge noninvariant contribution of the bulk theory at the boundary. Then your total action is composed of the bulk action, which would be gauge invariant if there was no boundary, and the boundary action, which ensures gauge invariance of the total action (by canceling the gauge noninvariant part of the bulk action on the boundary). The boundary action is thus completely determined by the properties of the bulk lagrangian density under gauge transformations.

Explicitly, if the bulk lagrangian density $\mathcal{L}_{\rm bulk}$ transforms as $\mathcal{L}_{\rm bulk} \rightarrow \mathcal{L}'_{\rm bulk} = \mathcal{L}_{\rm bulk} + \partial_\mu \lambda^\mu$ under gauge transformations, then define the boundary lagrangian density $\mathcal{L}_{\rm boundary} = \partial_\mu l^\mu$ such that it transforms as $\mathcal{L}_{\rm boundary} \rightarrow \mathcal{L}'_{\rm boundary} = \mathcal{L}_{\rm boundary} - \partial_\mu \lambda^\mu = \partial_\mu (l^\mu - \lambda^\mu)$. Now the total lagrangian density $\mathcal{L} = \mathcal{L}_{\rm bulk} + \mathcal{L}_{\rm boundary}$ transforms trivially under gauge transformations $\mathcal{L} \rightarrow \mathcal{L}' = \mathcal{L}$ and so the total action is gauge invariant, and as $\mathcal{L}_{\rm boundary}$ is always a total spacetime derivative it, by Gauss's theorem, really does contribute only on the boundary

Thus we have spontaneously obtained a boundary theory (boundary action) from the non-trivial gauge transformation properties of the, otherwise gauge invariant (if there was no boundary), bulk lagrangian (a nonzero $l^\mu$), when we demanded that the total action be gauge invariant. This would thus be one instance of a bulk-boundary correspondence.

P.S. Of course, in general, if the boundary is not smooth, i.e. if it has corners, we might also consider the "boundary of the (individual pieces of the) boundary", which might further have spontaneous "corner theories" induced by the non-trivial properties of the boundary theory in the presence of corners. Or in the parlance of condensed matter physics, it might have a second-order boundary theory (the "ordinary" boundary theory is called a first-order boundary theory; in general, an $n$-th order boundary theory lives on pieces of the boundary of co-dimension $n$). See e.g. this talk on higher-order topological insulators for a related discussion of higher-order boundary theories in the context of a different example of bulk-boundary correspondence (induced by non-trivial topological invariants of the bulk theory compared to the outside vacuum).

8. Nov 7, 2017

### Demystifier

In this correspondence, the boundary term is uniquely determined by the bulk term, but the bulk term is not uniquely determined by the boundary term. In this sense, the correspondence is not an equivalence (duality). Do you agree?

9. Nov 8, 2017

### no-ir

Yes, I do. In particular, the boundary term is only sensitive to the normal component of the bulk $l$ at the boundary (the scalar $n_\mu l^\mu$, where $n$ is the boundary normal), so adding to the action any bulk term with $n_\mu l^\mu = 0$ at the boundary (e.g. a bulk term with a trivial $l = 0$) does not change the boundary theory, which means that the bulk action is not uniquely determined by the boundary term (bulk theories are "richer"). The inference goes the other way: if you have a known bulk theory you can deduce from it the boundary theory (which might or might not be trivial).

It is also not a duality in the sense that here the bulk and the boundary terms are part of the same action and describe the dynamics of the same field simultaneously. They are not independent ways of looking at this dynamics, but only coupled together describe the full dynamics.

What usually "saves" you in condensed matter physics, such that you can argue that only the boundary term is relevant near the boundary and only the bulk term is relevant in the bulk, is if the bulk theory is gapped (loosely, but incorrectly: "it has mass"), while the boundary theory is gapless (or at least has a smaller gap). Then for energies inside the bulk gap the wavefunctions (or the field, in general) become localized at the boundaries and decay exponentially with distance towards the bulk, as they are in the forbidden energy range for existing inside the bulk. For quick decay (large gap) the dominant part of the action in this energy range is the boundary term as it is independent of the bulk decay rate, while the bulk contribution from these wavefunctions decreases towards zero as the decay becomes quicker. We can thus, to a good approximation, describe the states with energies inside the bulk gap using only the boundary term of the action.

This indeed happens, e.g., in topological insulators, where the bulk is insulating (gapped) and the boundary is conductive (gapless), or in certain quantum wires where the bulk is gapped while the boundary (the two endpoints of the wire) are Majorana zero modes with degenerate energy (the two Majorana zero modes behave as Majorana fermions with respect to themselves, but are non-abelian anyons under mutual exchange, making them useful for quantum computing).

Of course, if the decay towards the bulk is not sufficiently fast, or if opposite boundaries are brought closer together than the characteristic decay length, a pure boundary theory is no longer a good approximation. In the example of a short quantum wire: when the wavefunctions of the two Majorana modes start to significantly overlap in the bulk they hybridize and break their degeneracy, ceasing to behave as non-abelian anyons under mutual exchange.

So in short:
- the bulk term determines the boundary term but not the other way around (bulk theories are "richer"),
- the correspondence is not a duality as both terms are coupled in the same action,
- but under certain conditions (gapped bulk theory, states in the bulk gap) you can still approximate the dynamics by a pure boundary term,
- which is nice, as the boundary theory can be more exciting that the bulk one (e.g. emergent Majoranas and non-abelian anyons).

Last edited: Nov 8, 2017
10. Nov 8, 2017

### Demystifier

I have argued that it is actually so in all bulk/boundary-correspondence theories such as AdS/CFT.
https://arxiv.org/abs/1507.00591

11. Nov 9, 2017

### jakob1111

All this disagreement and confusion about the status of "gauge symmetry" is really puzzling. So many smart people say things that are simply not true, at least not in general. In addition to the guys mentioned above, other prominent example would be Arkani-Hamed, who also likes to stress that gauge symmetries are not physical and merely redundancies, c.f. https://arxiv.org/abs/1612.02797 or Guidice, who in his last paper writes: "Gauge symmetry is the statement that certain degrees of freedom do not exist in the theory. This is why gauge symmetry corresponds only to as a redundancy of the theory description".

Without a careful and precise definition of what they mean by "gauge symmetry" these statements simply do not have any meaning. This is really the main problem that causes all this confusion: people talk about gauge symmetry without defining exactly what they mean by that word.

For some the global group is a subgroup of the local $U(1)$ gauge symmetry. This is possible if you define all transformation of the form $e^{i \alpha_a (x) T_a}$, with arbitrary functions $\alpha_a (x)$ as local gauge transformations. Global symmetry is then a special case where the function $\alpha(x)$ that parametrizes the local transformation happens to be constant. This is a naive definition that is repeated in many textbooks and believed by most students. With this definition, gauge symmetry is, of course, not just a redundancy but physical. It's physical effects are the conservation of electrical charge, the masslessness of the photon, the non-trivial QCD vacuum etc. Therefore, with this naive definition, the statements of Witten and Schwartz quoted above do not make sense. However, it's hard to tell what they really mean. Apparently they do not use this naive definition, but they do not specify any other definition. This is a problem because there is no canonical more precise definition.

As soon as you no longer want to use the naive definition, you run into a big problem: apparently there are as many other definitions as there are authors.
• For example, Urs prefers the notion "gauge symmetry" for the compactly supported symmetries, and "gauge-parameterized gauge symmetry". for all other.
• Other authors, like Strominger, call the "compactly supported symmetries" "trivial gauge symmetries". The group of all gauge transformations modulo the trivial ones is then called "asymptotic gauge group".
• I'm pretty sure that, at least some, of the "Generalized Global Symmetries" by Davide Gaiotto, Anton Kapustin, Nathan Seiberg, Brian Willett are just another incarnation of Strominger's asymptotic symmetries.
• For another even different definition see, e.g. https://arxiv.org/abs/1405.5532, where the local symmetry is defined as a collection of infinitely many global ones. However, the difference between these global gauge transformations and the "real" global ones is that the correct global gauge group is realized linearly, while the others are not and therefore broken. (Gauge bosons are then the Goldstones of this symmetry breaking.)
(I could add lots of other examples to this list).

So to summarize:
1. Talking about the meaning of gauge symmetries makes absolutely no sense unless you specify precisely what you mean.
2. There is a real need for some kind of dictionary that translates between all these different approaches to make the definition of gauge symmetries more precise.

12. Nov 9, 2017

### Urs Schreiber

Exactly.

Actually in the above discussion I did say "gauge symmetry" for "gauge-parameterized gauge symmetry", since that is really what we mean when we say "gauge theory" (as opposed to when we speak more generally about Lagrangian field theories).

But the difference between these definitions, while important for the fine print, is not actually relevant for just seeing that there is a problem at all, that it is in general wrong to say (or even to think) that "gauge symmetry is just a redundancy": Simply consider something like Chern-Simons theory on a closed, hence compact, 3-manifold. Then the condition of "compact support" becomes automatic, and hence then no matter which definition is used, one concludes that there is more than one gauge transformation relating any field configuration and/or state, and hence the space of configurations or states modulo gauge symmetries is a groupoid or stack with non-trivial isotropy, and this is more information than the naive quotient space which reflects the "is just a redundancy" idea.

13. Nov 9, 2017

### Fra

We can see that several approach this from the mathematical theory side, and make excellent contributions here! Regardless of our main areas, I think most of us has experience with both the mathematics, logic or applied mathematics side as well as physics side and some other life sciences, and has observed that the fields sometimes requires different mindsets or approaches. My experience is that alot of mathematicians that work on applied physiscs, do so with a personal motivation slightly different that some physicists. Physicists are admittedely more sloppy and informal, or philosophical so they can focus on what they are building, rather than "respecting" the tools they use. But occasionally physicists happend to actually deform the tools and create new tools, without thinking about it. Some mathematicians feel frustration about these attitudes and feels like they have to take responsibility and make this properly. I just know from from personal relations. You can also feel this yourself, whenever you deep dive into matehematics, and proof thinking where you need to trace it all to axioms vs the sometimes more free philosophical creativity that is required to UNDERSTAND soem things in physics. Or to create for yourself what we called "mental picture".

Anyway, what i wanted to say here, is that as at the core of these discussions are a bunch of the open problems in physics, and such things can not be phrased merely as a axiomatic or mathematical terms. Its not like the question here is like, howto prove a conjecture theory from some axioms. To take the logical perspective i think it more has to do with either extenting the axioms on which theory are built, without adding inconsistencies, butit might well end up so that we need to replace some axioms!

To me the observation is this: Gauge theory and various symmetry princiiples has obviously been extremelt successful, and is at the heart of modernt physics. Why is this? Yet there seems this procedure seems to have hard time to solve some of the current open problems. Why is this?

Can we find a different angle or twist to this successful procedure, that helps us forward? That the question i have in mind when reading this thread.

That said, it is of course important to once things are mature, axiomatise and clean up the theory. I think axiomatising theories often really helps to understand the core of the theory (the axioms), you can then ponder on the one by one though by mapping the axioms to physical postulats like sometimes is done in QM for example. I think Urs insight thread about QFT is awesome work and great contributions on here.

I just feel that it is easy to sterilize discussions by insisting on the axiomatic style approach in the phase where one ponders about possible new schemas or paradigms?

I think if we can have both in parallelll that is the best of both worlds?

/Fredrik

14. Nov 9, 2017

### jakob1111

Fredrik,

reading your comment about different "mindsets" I was immediately reminded of the following quote by Tony Zee:

"Indeed, a Fields Medalist once told me that top mathematicians secretly think like physicists and after they work out the broad outline of a proof they then dress it up with epsilons and deltas. I have no idea if this is true only for one, for many, or for all Fields Medalists. I suspect that it is true for many."

Oftentimes, to make huge steps forward you need to be a bit sloppy. Only if you do incremental research you can do everything rigorous all the time. Nevertheless, before you try a huge leap forward you should have a firm understanding of the current theory.

I think the answer to this question is well known. Gauge symmetries appear because we want to describe particles with spin using fields. Particle transform according to little groups, while fields are representations of the Poincare group. Hence, fields carry too many degrees of freedom. These superfluous degrees of freedom are what we call gauge symmetry. While gauge symmetry certainly can't solve all the open problems, they can indeed solve a lot of them. If you replace the standard model symmetry with a simple group like SO(10), you get almost automatically an explanation for:

• the different strength of the standard model forces
• the tiny masses of the neutrinos
• the baryon asymmetry.
So, the explanatory power of gauge symmetry is still not exhausted. If one day proton decay is observed, lots of problems of the standard model vanish automatically.

However, of course, for other problems you have to look elsewhere.

I'm a sloppy physicist by heart. However, there are certain topics where a bit more rigor would be tremendously helpful. Gauge symmetry is probably the best example. The main problem, as already mentioned above, is that those people who try to work things out more rigorously are often not able to communicate in a way that "normal" - whatever that means - physicists can understand.

So I would like to add that we not only need both worlds, but also translators who are capable of mediating between the worlds.

15. Nov 9, 2017

### Fra

I pretty much agree with what out you said! I just felt i wanted to throw that out.
On this part though, i do not quite find your answer satisfactory. Its not that what you write is wrong, and maybe its because I secretly have something else in mind. What you write here is still living within a context with alot of baggge, alot of which is not conceptually clear to me at least.

/Fredrik

16. Nov 9, 2017

### star apple

Most interesting paragraph in witten paper is the following:

"We can see the relation between gauge symmetry and global symmetry in another way if we imagine whether physics as we know it could one day be derived from something much deeper – maybe unimaginably deeper than we now have. Maybe the spacetime we experience and the particles and ﬁelds in it are all “emergent” from something much deeper."

If gauge symmetry is emergent.. What could be the properties or characteristic of this more fundamental field by extrapolation (do you still call it field?) that create our gauge symmetry? What do Witten and other genius think about this? Since gauge symmetry is connected directly to the wave function.. does it mean the more fundamental nongauge primary field (or whatever) is not based on wave function (or QFT)?

17. Nov 9, 2017

### Fra

Yes this is the key. I can only guess, but the probable idea that fits in string thery is bulk dimensionality is emegent from boundaries. And there new symmetries form. This need to be phrased without starting from 4d continium spacetime baggage.

Ironically if you read my post#5 the two problems are initimatly related :)

The connection is motivated by

Gsuge equivalence ~ observer equivalence

In the laws of physics should be the same to all observers. This is easy to agree with but if you thimk again about the physicsl inferences look like... you may see (or at least i do) that this should be understood as a vision (or equilibrium point) NOT as as logical constraint.

Another way: observer equivalence is not a fundamental constraint in evolving law - it is merely an attractor.

/Fredrik

18. Nov 10, 2017

### star apple

Ah.. ok. i'll read in more details the papers of Urs and Demystifier. Besides boundaries.. no other candidate? how about not related to string theory?

If this is important. How come no other physicists worrying about this. And looking at archives and over the years you seemed to be the only one mentioning it and because you use language that is getting more complex.. I wonder if other physicists here can get a basic of what you were describing so hope you write a paper that gives fundamental and basic introduction to it starting with general relativity, qm and how the observers vary amongst them. Witten should worry about this if it's really important. Or maybe they are using another language, what is the language and jargon they use? I'm asking this because f feel what you were saying is important about the role of observers in GR and QM and how even the observer role in them are not compatible. Thanks.

19. Nov 10, 2017

### Fra

Conceptually this is not dependent on string theory. Its just that just to relate to other ideas to look for common denominators, rather than just find discriminators(which is usually easier but less constructive) i mentioned it. I think many research programs have merits! So why not try to see what we can learn from all of them?

And of course the bulk/boundary ideas, and specifically the holographic principle while in principle again having nothing todo with string theory, has probably is most explicit example in Ads/CFT. So associatiing to strings is natural.

Also interpreting what Witten says, in terms of string also seems natural.

(But myself does not work on string theory, but i still enjoy wittens ponderings of course)
While alot of people have been pondering over the observer and measurement problem over the years, i agree that the specific thinking i have in mind seems to be sparsely represented out there.

I have wondered why as well. One easy answer is course that i am the only one stupid enough to not see its wrong thinking.

Another answer is that i understand the resistance in this directin, because thinking this ways inavoidably LEADS you to the evolving law view. And this
strips us from many of existing tools.
Thats the remote idea of course.

My ambition is to work this out, but due to the fact that this represents non-mainstream ideas there is no context where this really fits. This means partial results would appear completely ad hoc or disconnected to physics. The closest place where the ideas might fit is into AI research, but that is still the wrong place i feelI. It takes too much energy to try to "sell things" before they are done (thats the american way of marketing). I am better of using that energy to make progress, and once its done, there are no selling costs.

So I want to make a nontrivial prediction or postdicition before i will even consider publishing anything. Unfortunately thats close to an unrealistic goal for one man, that also have a regular job. There is one advantage though and that is that the slowly grown crystals are often more perfect than the fast growin ones. I am not in a hurry and the ride is enjoyable meanwhile!

/Fredrik

20. Nov 10, 2017

### star apple

The only nongauge field is the higgs field. Is there a way to create a universe where electromagnetism doesn't come from gauge freedom where phase is the U(1).. or should all strings landscape or even smolin different black hole/universe with different laws of physics always have to reduce to gauge symmetry like U(1) of electromagnetism? What do you think? and Why? Why can't electromagnetism be like the Higgs field that is fundamental and doesn't come from gauge symmetry?

Also for that thing more fundamental or primary than gauge symmetry (which makes this emergy).. does it have to always occur in high energy (small scale)? Because electromagnetism is low energy and so can U(1) itself can derive from more fundamental nongauge low energy stuff or is this not possible because U(1) is always part of the Electroweak SU(2)XU(1) so whatever is more primary (than gauge symmetry) always have to be based on Electroweak and not merely on U(1)?

What is the connection of the incompatibility of different observers in GR and GM to evolving law (stripping us of many of existing tools)? At least Smolin mentions it so at least you have company or it's based on an authority.