Gauge invariance confusions: symmetry vs redundancy, active vs passive

[Demystifier]

Symmetry transformations in physics can be either passive or active. Symmetries in field theory can be either global or local. But only the local ones, the so called gauge symmetries, are fundamental. Except that local transformations cannot be active (despite the fact that diffeomorphisms are active local transformations), so local symmetries are not symmetries at all, but just redundancies. Which doesn't mean that gauge invariance does not have physical consequences.

If you are confused, so am I. A lot of this confusion is reduced in the insightful paper by Schwichtenberg https://arxiv.org/abs/1901.10420 , but a lot still needs to be clarified. What are your thoughts?

 

[vanhees71]

I fully agree. Then you have the sloppy physicists' slang to add to the confusion. Almost all textbooks, including the best in the field (Weinberg ;-)) call the "Higgs mechanism" "spontaneous symmetry breaking" though it's clear that a local gauge symmetry cannot be spontaneously broken. What's spontaneously broken are some associated global symmetries, but the very point of the Higgs mechanism in contradistinction to real spontaneous symmetry breaking (of global) symmetries is that you don't get massless Nambu-Goldstone modes but massive gauge bosons. The would-be Goldstone modes are thus providing the additional longitudinal polarization field-degree of freedom of a massive vs. a massless vector field, which is of course very welcome in the electroweak standard model.

It's also of course important for superconductivity, where you have an in-medium Higgs mechanism rather than a spontaneous symmetry breaking too.

 

[stevendaryl]

Yes, it is a little mysterious how gauge symmetry works. In QM, the phase of the wave function has no physical meaning, so the transformation $\psi \rightarrow \psi e^{-i\chi}$, when $\chi$ is a constant, does nothing at all. But this thing of no consequence has all these amazing consequences, such as conservation of charge and (when you let $\chi$ be a function, rather than a constant) electromagnetism.

 

[vanhees71]

The interesting thing is that electromagnetism, for which you need relativity to describe it fully correctly, must be a gauge theory, which follows from the representation theory of the Poincare group for massless vector fields (any theory of massless fields with spin $s \geq 1$ must be gauge theories in a general sense). For the corresponding gauge fields to interact with anything else this implies that it must be through the coupling to conserved currents, which more or less fixes how electromagnetism looks like.

What does not follow necessarily is the extension to more complicated non-Abelian gauge groups, but on the other hand, it's a pretty obvious idea of generalization, i.e., here you use some more complicated non-Abelian symmetry group than the simple multiplication of fields with a phase factor but you use multicomponent fields transforming under some linear representation of a non-Abelian group (most conveniently a compact lie group like SU(N) or SO(N)) and make the symmetry local by introducing gauge-vector fields as connections, defining gauge-covariant derivatives.

The final step in this program of gauge theories is, of course, to make also Poincare symmetry local, which leads, depending on the fields describing everything else than gravitation, either to Einstein's General relativity (leading to a pseudo-Riemannian, Lorentzian manifold as space-time model) or an extension to an Einstein-Cartan theory leading to a space-time with curvature and torsion, when spinor fields are involved.

In this sense one can say that all our fundamental contemporary modern theories are based on the gauge principle.

 

[bhobba]

In this sense, one can say that all our fundamental contemporary modern theories are based on the gauge principle.

The more I learn of physics, the more accurate it becomes. Try to explain it to people; a typical response is how math can be physical? Mathematics reveals its underlying essence. People find it hard to understand this.

Thanks
Bill

 

[strangerep]

The more I learn of physics, the more accurate it becomes. Try to explain it to people; a typical response is how math can be physical?

Heh, ask them to imagine themselves as primitive humans, trying to catch their fleet-footed dinner by throwing a large rock or spear at it. They have to "lead" the target, right? Now explain that, to do so, their primitive brain (an analog computer) is actually solving Newton's equations of motion (in 3D with a gravitational potential).

Mathematics reveals its underlying essence.

Yep, the "essence" of mathematics is the more effective catching of one's dinner.

 

[MathematicalPhysicist]

I fully agree. Then you have the sloppy physicists' slang to add to the confusion. Almost all textbooks, including the best in the field (Weinberg ;-)) call the "Higgs mechanism" "spontaneous symmetry breaking" though it's clear that a local gauge symmetry cannot be spontaneously broken. What's spontaneously broken are some associated global symmetries, but the very point of the Higgs mechanism in contradistinction to real spontaneous symmetry breaking (of global) symmetries is that you don't get massless Nambu-Goldstone modes but massive gauge bosons. The would-be Goldstone modes are thus providing the additional longitudinal polarization field-degree of freedom of a massive vs. a massless vector field, which is of course very welcome in the electroweak standard model.

It's also of course important for superconductivity, where you have an in-medium Higgs mechanism rather than a spontaneous symmetry breaking too.

I wouldn't call a physicist take on such a hard topic the best in the field...

How did David Hilbert say:" Physics is becoming too difficult for the physicists. "
https://mathshistory.st-andrews.ac.uk/Biographies/Hilbert/quotations/

 

[vanhees71]

Well yes, and the result are weird but mathematically brillant ideas based on pure thought with no relation to reality. Ironically the history of why we call this class of theories "gauge theories" is a prime example. The name comes from an early attempt by Weyl to unify electricity and magnetism by gauging the scale invariance of the free gravitational field. Immediately Einstein in a polite and Pauli in a much less polite way told him that this contradicts the empirical fact that length scales of bodies don't depend on their "electrodynamical history", but Weyl was hard to convince.

Nevertheless the principle idea was of course very fruitful, i.e., to make global symmetries local. Weyl just applied it to the wrong group.

 

[Nullstein]

Gauge transformations can be both active and passive. Active gauge transformations are principal fiber bundle automorphisms, whereas passive gauge transformations are changes of local trivializations. And there is also a passive version of diffeomorphisms: Coordinate changes. The physical relevance comes from the active versions, because requiring invariance under them restricts theory space.

After having skimmed the paper, I wouldn't say it clarifies anything. Why not just teach the fiber bundle formalism? Then everything becomes quite obvious.

 

[MathematicalPhysicist]

Well yes, and the result are weird but mathematically brillant ideas based on pure thought with no relation to reality. Ironically the history of why we call this class of theories "gauge theories" is a prime example. The name comes from an early attempt by Weyl to unify electricity and magnetism by gauging the scale invariance of the free gravitational field. Immediately Einstein in a polite and Pauli in a much less polite way told him that this contradicts the empirical fact that length scales of bodies don't depend on their "electrodynamical history", but Weyl was hard to convince.

Nevertheless the principle idea was of course very fruitful, i.e., to make global symmetries local. Weyl just applied it to the wrong group.

For me there's no real distinction between theoretical physics and pure maths (hey Witten won the Fields medal in maths).
For example I had taken a few years ago an advanced course in electromagnetism, we had a question in the exercise on Hodge star (you can find it in MSE).
As Galileo put it :"the language of nature is mathematics".

 

[Fra]

For me there's no real distinction between theoretical physics and pure maths

Hmmm I think this is part of the problem. I don't think the problem is to understnad the difference between passive or active transformations, etc. They key fuzz which I am not sure the paper clarifies is:

When and why do the "book keepers" become dynamical actors? And what is the origin and explanation of the symmetries?

I will just note that striking similarity here between this discussion and the deeper physical principle of "observer equivalence". And how one has a similarly paradoxal situation where one sometimes think of the observer as a gauge, which some try to get rid of. There is also a tension here between the observer centered measurement theory, and the principle of observer democracy, and what should be considered observables. For example, when and why is an observer a dynamical actor? As we know, QM as it stands does NOT consider the observer as part of the dynamics in a full sense. I think this has everything to do with the discussions as well. And it has conceptual components that the paper does not clear to me.

/Fredrik

 

[vanhees71]

Well, this example by Schwichtenberg doesn't help me to understand gauge theories at all. It's funny to make an analogy which is more complicated than what you want to explain ;-)).

What he seems to mean to describe is the usual heuristical procedure that you start with some Lagrangian with a global symmetry (e.g., SU(N) isospin symmetry). Then you make this symmetry local by introducing an appropriate connection, the SU(N) gauge vector fields. To close the system you also add a gauge-invariant kinetic term. The one with the lowest number of derivatives is the usual one using the curvature of the connection, which in physical terms is nothing else than the field-strengths tensors $F_{\mu \nu}^a$, from which the lowest order scalar (both Lorentz and gauge scalar) you can build is $F_{\mu \nu}^a F^{a \mu \nu}$, and indeed that's what you add with some factors to get the usual conventions to the Lagrangian to make the gauge field a dynamical field.

 

[Fra]

But its exactly the strange success of the heuristic procedure that begs a better explanation. The main haze isn't coming from understanding how the terms popping out from requiring a local invariance, and how they can be countered/explained by a new field. What is the logic behind requiring the global symmetry to be come local? It's all too nice to be a coincidence or beeing incomprehensible I think. I can't see Schwichtenberg's paper addressed this much?

I have in mind some kind of explanation or insight connecting to the foundations of QM and relational nature of physical law. The succes of the scheme seems to tells us something about the similarity between "measurements" and "interactions", about encoding of information and the structure of matter, and specifically what happens when systems interact; after all, it's similarly funny how QM describes the evolution of "isolated systems", but to make a measurement on something implies to enlarge it. You have to interact with and create some communication channel. Once we understand say the constructing inductive steps here, perhaps we can see how all interactions should unify. There is also a tricky difference between cosmological and subatomic descriptions in this context, when it comes to scientific corroboration and producing statistics. This all means that we should lookg for a deeper understanding of the constructing principles. The heuristic procedures may otherwise misguide us because we don't undersand why it works.

"Gauge invariance is not just mathematical redundancy; it is an indication of the relational character of fundamental observables in physics."
-- Carlo Rovelli, https://arxiv.org/abs/1308.5599

/Fredrik

 

[samalkhaiat]

the insightful paper by Schwichtenberg ... What are your thoughts?

If this guy works for me, I sack him.

 

[bhobba]

For me, there's no real distinction between theoretical physics and pure maths (hey, Witten won the Fields medal in maths).

There is a difference in emphasis. I did an applied math degree, but we had a few pure math subjects like Linear Algebra and Analysis. You still did some applications in the pure math subjects and some theory in the applied subjects. For example, we had Analysis A and B. Analysis A did the theory of Lebesque Integration and Hilbert Spaces - but there was some applied stuff around the fact with Lebesque Integration, you can always reverse integrals etc. which makes Fourier Theory easier (Distribution Theory makes it easier still so I am not sure of its value). In Analysis B, which was about several applications of Hilbert Spaces to optimisation and a few other things, we proved the Krein-Millman theorem from which the optimisation techniques of Operations Research fell out naturally. Strangely we did not touch on QM, which would have been straightforward to cover once you had Hilbert Spaces down pat. Mathematical Physics was not particularly popular - most applied math guys wanted to be Actuaries, with a few like me also doing Computer Science. One guy wanted to be, of all things, a librarian. We still had several courses on it if you were interested, like Methods Of Mathematical Physics A and B - but nobody ever did them, so the math department dropped them.

Thanks
Bill

 

[MathematicalPhysicist]

There is a difference in emphasis. I did an applied math degree, but we had a few pure math subjects like Linear Algebra and Analysis. You still did some applications in the pure math subjects and some theory in the applied subjects. For example, we had Analysis A and B. Analysis A did the theory of Lebesque Integration and Hilbert Spaces - but there was some applied stuff around the fact with Lebesque Integration, you can always reverse integrals etc. which makes Fourier Theory easier (Distribution Theory makes it easier still so I am not sure of its value). In Analysis B, which was about several applications of Hilbert Spaces to optimisation and a few other things, we proved the Krein-Millman theorem from which the optimisation techniques of Operations Research fell out naturally. Strangely we did not touch on QM, which would have been straightforward to cover once you had Hilbert Spaces down pat. Mathematical Physics was not particularly popular - most applied math guys wanted to be Actuaries, with a few like me also doing Computer Science. One guy wanted to be, of all things, a librarian. We still had several courses on it if you were interested, like Methods Of Mathematical Physics A and B - but nobody ever did them, so the math department dropped them.

Thanks
Bill

Well I also completed a BSc. in combined math and physics and I think I know what I say. Besides the labs in which you use some statistics which you don't really rigorously grasp why and if it also works. For that you need to take a mathematical statistics course, unfortunately I could find the appropriate time slots to take it not in my UG and not in my Graduate studies, though at some point I wanted but they didn't let me since I had taken too many courses...
But the only difference between maths and physics is the labs, if you don't bother with experiments you can invent what ever you wish, imagination is more important than knowledge.

 

[vanhees71]

Well, this well-known saying by Einstein is disproven by himself. From the moment on he left his solid foundation in phenomenology, he didn't achieve much more of his miraculous insights he had between 1901-1920.

 

[vanhees71]

But its exactly the strange success of the heuristic procedure that begs a better explanation. The main haze isn't coming from understanding how the terms popping out from requiring a local invariance, and how they can be countered/explained by a new field. What is the logic behind requiring the global symmetry to be come local? It's all too nice to be a coincidence or beeing incomprehensible I think. I can't see Schwichtenberg's paper addressed this much?

I think for a relativistic physicist the idea of locality is very natural. I'm not aware of any successful non-local formulation of relativistic physics. The natural language for relativistic dynamics is field theory, i.e., the locality of interactions. So to make a global symmetry local is a pretty obvious heuristic step. Already the free massless fields with spin $s \geq 1$ occur to be most naturally realized as Abelian gauge fields just from the point of view of representation theory of the Poincare group, at least if you take it for granted that there are no continuous polarization-like internal degrees of freedom.

Of course, Yang and Mills could not anticipate the surprising properties like asymptotic freedom and confinement of non-Abelian gauge theories in 1956, but these turned out to be the key for success in building the Standard Model in the mid 1960ies to mid 1970ies.

 

[Demystifier]

If this guy works for me, I sack him.

I think I understand why do you say that, but this guy also has some serious technical papers:
https://link.springer.com/content/pdf/10.1140/epjc/s10052-019-6878-1.pdf
https://arxiv.org/abs/1704.04219

 

[strangerep]

If this guy works for me, I sack him.

I had similarly negative feelings about his book from a few years ago -- even though others have praised it.

 

[strangerep]

Well, this well-known saying by Einstein is disproven by himself.

What are you referring to? "If this guy works for me, I sack him" ?

 

[Demystifier]

What are you referring to. "If this guy works for me, I sack him" ?

It's well known that @vanhees71 does not like to quote previous messages when he replies.

 

[martinbn]

Well, this well-known saying by Einstein is disproven by himself. From the moment on he left his solid foundation in phenomenology, he didn't achieve much more of his miraculous insights he had between 1901-1920.

I agree that he didn't achive much more, but I emphasize the word more. He did achive much after 1920, it is just not much more than what he did before. I am not sure if it was possible. Speial relativity, general relativity, one of the founding fathers of quantum theory and a few other things. How do you do much more than that!

 

[vanhees71]

What are you referring to? "If this guy works for me, I sack him" ?

No I referred to "imagination is more important than knowledge."

I'd also not go so far to sack somebody who tries to do some "outreach work". The only problem I have is that I found the attempt to be didactical in this preprint (unpublished as far as I can see!) failed, because the economy example is more complicated than the physics (maybe it's just because I'm more familiar with physics than theoretical economy).

His book on symmetries is an example for the rule that you shouldn't write textbooks right after getting your first degree in the subject...

 

[atyy]

Amazingly, one of the references cited for the analogy between gauge symmetry and exchange rates is Maldacena!

 

[vanhees71]

So?

 

[atyy]

So?

So that explains why when physicists work on finance they cause the economy to crash, because everything gets sucked into a black hole

 

[Demystifier]

So that explains why when physicists work on finance they cause the economy to crash, because everything gets sucked into a black hole

Usually it's a black hole remaining after a supernova, because the economy crashes after a bubble.
https://www.investopedia.com/terms/b/bubble.asp

 

[Demystifier]

because the economy example is more complicated than the physics (maybe it's just because I'm more familiar with physics than theoretical economy)

I'm not an expert in theoretical economy either, but I found the economy analogy very insightful.

 

[Fra]

The specific example didn't make me spin, but in general I think gaming perspectives, when we have not just a simply dice thrower, but interacting players that encode and update their own maps of the other players are very good and deep analogies, where one can find common mathematics. Such examples can I think be good ways to illustrate both symmetries, as well as the difference between descriptive and guiding probability concepts.

Its how I secrectly thinkg of things as well. But then it's interacting Qbist agents, that has to learn and surviva, or get outcompeted. The "stable" population of players could conceptually correspond to the population of elementary particles, and the interaction rules enoded in the relational expectation the agents has about each other. But the agents decisions does, just like QM does not require conscious observers, not require brains. The idea is that it's all about guided random processes (I mean, given any stance, it's natural to fall forwards; So the agents expectation of the future, determines its rambling direction - this is the meaning of probability in the qbist view as well, it reflects the agents prospensity for actions) And the quest is: How can we understand the emerge of players populating the game, that correspond to the groups and interactions of standard model (and gravity?)

Smolin had also interesting ideas sniffing along these lines, but still very immature.
https://arxiv.org/abs/1205.3707

But to make it really exciting one has to allow for the agents (or active bookkeepers in the xample) to learn and evolve. How this is going to be modeled requires further assumptions. I can't blame anyone for leaving out these details though, but it's what would have made me spin.

There is ALOT of bits and pieces everywhere, different researchers sniffing along these lines, but none I have seen has the big picture.

Edit: See also
Evolutionary game theory using agent-based methods
https://arxiv.org/pdf/1404.0994.pdf, its computational biology gets you in the mood

/Fredrik

 

[MathematicalPhysicist]

Well, this well-known saying by Einstein is disproven by himself. From the moment on he left his solid foundation in phenomenology, he didn't achieve much more of his miraculous insights he had between 1901-1920.

Yeah I know, every genius makes mistakes.
I know I had my fair share of mistakes... :-(

 

[MathematicalPhysicist]

But I keep on trying!

 

[PeterDonis]

Yeah I know, every genius makes mistakes.
I know I had my fair share of mistakes... :-(

Unfortunately, "being a genius" -> "making mistakes" is not logically equivalent to "making mistakes" -> "being a genius". Otherwise every single person on the planet would qualify as a genius.

 

[MathematicalPhysicist]

Unfortunately, "being a genius" -> "making mistakes" is not logically equivalent to "making mistakes" -> "being a genius". Otherwise every single person on the planet would qualify as a genius.

Obviously.
But who defines who is a "genius" in the first place?

 

[Fra]

So?

A taste of the dark age roots of probability theory, when appeal to authority was the major ranking factor.

/Fredrik