Ed Witten on Symmetry and Emergence

[king vitamin]

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.

 

[Demystifier]

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]

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]

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

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]

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, that's 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 isn't 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]

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 modeled as an evolving structure that codes information.

For example if you believe 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, that's 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 can't be the right formalism.

/Fredrik

 

[Urs Schreiber]

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.

 

[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

 

[Urs Schreiber]

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.

 

[Crass_Oscillator]

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.

You could have just said "No."

 

[Urs Schreiber]

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.

 

[Demystifier]

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]

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]

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]

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.

 

[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, that's 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

 

[star apple]

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

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..

 

[no-ir]

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.

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).

 

[Demystifier]

This would thus be one instance of a bulk-boundary correspondence.

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?

 

[no-ir]

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?

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).

 

[Demystifier]

- the bulk term determines the boundary term but not the other way around (bulk theories are "richer"),

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

 

[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.

 

[Urs Schreiber]

Exactly.

For example, Urs prefers the notion "gauge symmetry" for the compactly supported symmetries, and "gauge-parameterized gauge symmetry". for all other.

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.

 

[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 a lot 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 happened 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!

I added fuzzy comments to this thread just to encourage the conceptual understanding.

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

 

[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.

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?

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 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'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.

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

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

 

[Fra]

I pretty much agree with what out you said! I just felt i wanted to throw that out.

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...

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 a lot of baggge, a lot of which is not conceptually clear to me at least.

/Fredrik