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Stream: community: general

Topic: Noether's theorem

John Baez (Jun 30 2020 at 00:50):

I have a new paper on Noether's theorem, and here is a blog article summarizing it:

• Getting to the bottom of Noether's theorem.

John Baez (Jul 01 2020 at 18:34):

If anyone has questions, please ask! So far nobody in the world has asked me anything about this paper.

John Baez (Jul 01 2020 at 18:34):

In fact nobody in the entire universe.

Fabrizio Genovese (Jul 01 2020 at 18:55):

John Baez said:

If anyone has questions, please ask! So far nobody in the world has asked me anything about this paper.

Maybe only slightly related, but I've often heard physicists say "symmetries of the space are more important/interesting than space itself". I get that Noether theorem's relates symmetries with conserved quantities, so there's this for sure. But is there more? Can you shed some light on this?

John Baez (Jul 01 2020 at 18:55):

Is there more to what, exactly? This idea that symmetries of the space are more interesting than space itself?

John Baez (Jul 01 2020 at 18:56):

I've never heard anyone say that, but if they did, I imagine it's because those symmetries give conserved quantities, which are useful regardless of the "details" of what space "is".

John Baez (Jul 01 2020 at 18:57):

One interesting thing is that in our best model of space, general relativity, spacetime has absolutely no symmetries except approximate ones: the universe is lumpy.

John Baez (Jul 01 2020 at 18:57):

It also has "gauge symmetries", meaning diffeomorphisms, which let you rename points but don't actually "do" anything.

Fabrizio Genovese (Jul 01 2020 at 18:58):

John Baez said:

Is there more to what, exactly? This idea that symmetries of the space are more interesting than space itself?

So, I know this very vaguely but for instance netwon laws give us a group of transformations, the galilean ones, and a space where newton laws hold must be somehow invariant wrt to this transformations I guess. For relativity there's another one, which I guess it's called the Poincare group?

John Baez (Jul 01 2020 at 18:58):

Right!

Fabrizio Genovese (Jul 01 2020 at 18:58):

So the way I understood this, handwavily, is that you come up with some physical laws and get these groups of transformations, and then search for spaces that are invariant wrt those?

Fabrizio Genovese (Jul 01 2020 at 18:59):

(come up with some physical laws = work out how to make experimental observations fit a given theory if they don't for instance)

John Baez (Jul 01 2020 at 18:59):

The Poincare group has the same dimension as the Galilei group, namely 10: 3 for spatial translations, 3 for rotations, 1 for time translation, and 3 for "boosts", which change you into a moving frame of reference.

John Baez (Jul 01 2020 at 18:59):

The Poincare group "contracts" to the Galilei group as $c \to \infty$ (the speed of light approaches infinity).

Fabrizio Genovese (Jul 01 2020 at 19:00):

So I don't really know how a physicists works. Where do they start from? The space? The symmetries? The conserved quantities? The laws of physics?

Fabrizio Genovese (Jul 01 2020 at 19:00):

But I guess the euclidean space is not invariant wrt lorentz transformations, right?

John Baez (Jul 01 2020 at 19:00):

Fabrizio Genovese said:

So the way I understood this, handwavily, is that you come up with some physical laws and get these groups of transformations, and then search for spaces that are invariant wrt those?

You can do whatever you want. In practice people did things like discover Maxwell's equations - some physical laws - and only later realize they were not invariant under the Galilei group, but instead the Poincare group.

Fabrizio Genovese (Jul 01 2020 at 19:01):

Oh, ok

John Baez (Jul 01 2020 at 19:01):

Fabrizio Genovese said:

So I don't really know how a physicists works. Where do they start from? The space? The symmetries? The conserved quantities? The laws of physics?

You can really do whatever you want!

Fabrizio Genovese (Jul 01 2020 at 19:01):

Oh wait, so

Fabrizio Genovese (Jul 01 2020 at 19:01):

Maxwell eqs are invariant wrt Poincarè group. So now we have two possible groups of transformations, Galiei and Poincarè

John Baez (Jul 01 2020 at 19:01):

Right.

John Baez (Jul 01 2020 at 19:02):

This "conflict" is what Einstein resolved, by throwing out the Galilei group entirely.

Fabrizio Genovese (Jul 01 2020 at 19:02):

so I guess that Einstein realized that to unify the two either you come up with a theory of light that is invariant wrt Galilei group, or you come up with a theory of mechanics that is invariant wrt Poincarè group. I guess he went for the second one

Fabrizio Genovese (Jul 01 2020 at 19:02):

Yeah ok, this is nice

Fabrizio Genovese (Jul 01 2020 at 19:03):

So has anyone any idea of what group we need to use to unify QM and relativity?

Fabrizio Genovese (Jul 01 2020 at 19:03):

Or are we still searching for one?

John Baez (Jul 01 2020 at 19:03):

Einstein didn't actually talk in the language of group theory. He talked about transformations of spacetime.

John Baez (Jul 01 2020 at 19:03):

The idea of an abstract group hadn't made its way into physics back then.

John Baez (Jul 01 2020 at 19:04):

Nowadays a physicist could start by finding a group they like and then think about a spacetime it acts on it, and laws invariant under it.

John Baez (Jul 01 2020 at 19:05):

Fabrizio Genovese said:

So has anyone any idea of what group we need to use to unify QM and relativity?

If you mean special relativity, it's the Poincare group. If you mean general relativity, nobody knows what the hell is going on (though everyone will tell you their own favorite theory).

Fabrizio Genovese (Jul 01 2020 at 19:05):

This sounds like the closest thing to "make up your own physical world" I can imagine :D

John Baez (Jul 01 2020 at 19:05):

Yes, it makes it easy to write papers.

Fabrizio Genovese (Jul 01 2020 at 19:06):

John Baez said:

Fabrizio Genovese said:

So has anyone any idea of what group we need to use to unify QM and relativity?

If you mean special relativity, it's the Poincare group. If you mean general relativity, nobody knows what the hell is going on (though everyone will tell you their own favorite theory).

So, may it be that one of the problem to put everything together is that this nice invariant transformations/group thing doesn't really scale up to general relativity?

John Baez (Jul 01 2020 at 19:06):

I think it's a pretty safe bet that to unify QM and general relativity we need more than just a new group.

Fabrizio Genovese (Jul 01 2020 at 19:06):

Yeah, I was about to say that the only possible solution I can imagine is that either one reworks GR completely to make it fit into the group picture or one comes up with a generalization of group that does the job

Fabrizio Genovese (Jul 01 2020 at 19:07):

I mean, if people tried for ~100 years and didn't succeed probably a group thing for GR is not really going to happen

John Baez (Jul 01 2020 at 19:07):

General relativity already completely blows the old group philosophy out of the water, because it has an enormous infinite-dimensional group of "gauge symmetries", namely diffeomorphisms that relabel points of spacetime, which are usually considered "purely mathematical".

Fabrizio Genovese (Jul 01 2020 at 19:07):

Mhhh

Fabrizio Genovese (Jul 01 2020 at 19:07):

Why does one need relabeling of points?

John Baez (Jul 01 2020 at 19:07):

I'd say we're still struggling to understand what that really means, though the math is well-understood by now (umm, for some of us).

John Baez (Jul 01 2020 at 19:08):

We "need" relabelling of points because we "need" to use coordinates to do anything, but there's no "best" coordinate system, so we should use all of them.

John Baez (Jul 01 2020 at 19:09):

"Need" means "well, this is how it seems now - we seem to need it".

Fabrizio Genovese (Jul 01 2020 at 19:09):

I meant, why is this not a problem in special relativity?

Fabrizio Genovese (Jul 01 2020 at 19:09):

Because what you say seems useful to me also for just cinematics

John Baez (Jul 01 2020 at 19:10):

In special relativity there's a 10-dimensional space of "best" coordinate systems, the "inertial frames". These are the coordinate systems in which an object with no force on it moves at constant velocity.

John Baez (Jul 01 2020 at 19:10):

So there's not one best coordinate system, but there's a bunch, and the Poincare group acts (freely and transitively) on this bunch.

John Baez (Jul 01 2020 at 19:10):

"Inertial frames" are a big deal in Newtonian mechanics and special relativity, but in general relativity that concept dies.

John Baez (Jul 01 2020 at 19:11):

People transitioning from special to general relativity sometimes don't get this.

John Baez (Jul 01 2020 at 19:11):

They have to realize: "all that cool stuff you learned in special relativity - it's all wrong, just an approximation".

John Baez (Jul 01 2020 at 19:12):

This shows that Einstein was a genius: he was able to jump ahead twice and create two radical new paradigms.

John Baez (Jul 01 2020 at 19:12):

(While inventing quantum mechanics on the side.)

John Baez (Jul 01 2020 at 19:13):

Noether proved 2 "Noether theorems" for Einstein while she was working for Hilbert.

John Baez (Jul 01 2020 at 19:13):

The first, which is what my paper is about, is suited to theories with a group of symmetries, like special relativity.

John Baez (Jul 01 2020 at 19:14):

The second is suited to general relativity, and that's the theorem Einstein really needed.

Fabrizio Genovese (Jul 01 2020 at 19:14):

Interesting

Fabrizio Genovese (Jul 01 2020 at 19:15):

but still, I guess the group thingy breaks down also in newtonian mechanics, right?

Fabrizio Genovese (Jul 01 2020 at 19:15):

Inertial frames are the ones where newton laws hold, so if there are forces going around does this concept work anyway?

Fabrizio Genovese (Jul 01 2020 at 19:15):

i mean I get that in newtonian mechanics things are easier since forces do not warp the space itself, but i don't know how much is this important in this context

Dan Doel (Jul 01 2020 at 19:16):

I'm no expert but, isn't another way to state the point that GR makes accelerating reference frames just as good as the inertial ones, because accelerating frames are equivalent to frames with 'fake' gravitational fields. But then you want to know about the 'real' gravitational fields, which are curvature that can't be removed by changing the acceleration of your frame.

John Baez (Jul 01 2020 at 19:16):

Group theory is fundamental to Newtonian mechanics and special relativity; it's just a different group.

John Baez (Jul 01 2020 at 19:16):

There's a 10-dimensional group acting on a 10-dimensional space of "inertial frames".

John Baez (Jul 01 2020 at 19:16):

The laws have a simple form when expressed using an inertial frame.

Dan Doel (Jul 01 2020 at 19:17):

And does the 'relabeling' correspond to changing the acceleration of the frame? I'm not sure about that.

John Baez (Jul 01 2020 at 19:17):

Yeah, it can.

John Baez (Jul 01 2020 at 19:17):

But there's no such thing as an "inertial frame" in general relativity - as you say, the curvature of spacetime screws that up.

John Baez (Jul 01 2020 at 19:18):

So there's usually no "unaccelerated frames".

John Baez (Jul 01 2020 at 19:18):

There are just lots of coordinate systems, an infinite-dimensional space of them acted on by an infinite-dimensional group.

John Baez (Jul 01 2020 at 19:19):

And the laws of physics must be written to take the same form in all coordinate systems.

John Baez (Jul 01 2020 at 19:19):

This is what Einstein called "general covariance".

John Baez (Jul 01 2020 at 19:19):

In short, we give up on the idea of a finite-dimensional space of "good" coordinate systems.

John Baez (Jul 01 2020 at 19:19):

Hence the term general relativity.

John Baez (Jul 01 2020 at 19:20):

An awesome leap!

John Baez (Jul 01 2020 at 19:20):

But very reasonable in retrospect.

John Baez (Jul 01 2020 at 19:22):

Unfortunately nobody knows how to blend this with quantum theory.

Fabrizio Genovese (Jul 01 2020 at 19:24):

At least I hope there are good leads

John Baez (Jul 01 2020 at 19:29):

Yes, I think general relativity and quantum theory are not just amazing at making accurate predictions: they have contain good insights about the world, and we need to combine those insights (and maybe some other stuff).

John Baez (Jul 01 2020 at 19:29):

People tend to fiddle around with equations instead of thinking. It's very hard to hold a bunch of insights in ones mind at the same time.

Fabrizio Genovese (Jul 01 2020 at 19:34):

I think this is one of the main problems

Fabrizio Genovese (Jul 01 2020 at 19:35):

The cool thing about relativity and QM is that it really starts with insights and mental experiments. One can explain it starting from philosophical considerations. Einstein was famous for being able to explain things this way

Fabrizio Genovese (Jul 01 2020 at 19:35):

I don't think it's the same with superstring theory, say. I feel it's more of a "this is where the maths has taken us" kind of thing

Fabrizio Genovese (Jul 01 2020 at 19:36):

But I don't know how far can one go by just trusting the maths

John Baez (Jul 01 2020 at 19:39):

Indeed, math is great once you have a theory of physics and want to do things with it. But I don't think math is a substitute for physical insight when creating a theory.

John Baez (Jul 01 2020 at 19:41):

So, I think people will be stuck on fundamental physics for a while longer, and lose interest in it, and then some people will slowly, painfully, try to think. (Of course some people already are. But not most, and it's a very slow process.)

John Baez (Jul 01 2020 at 19:43):

My paper on Noether's theorem is a small attempt to think. It uses math, but there's no really new math in it. Instead it's about what does it mean about physics we like theories where Noether's theorem holds?

John Baez (Jul 01 2020 at 19:43):

And interestingly this turns out to be connected to why we like using the complex numbers in quantum mechanics.

Valeria de Paiva (Jul 02 2020 at 02:33):

John Baez said:

And interestingly this turns out to be connected to why we like using the complex numbers in quantum mechanics.

I will bite. Why do you like using the complex numbers in quantum mechanics? does it matter that they're one of the normed algebras of Hurwitz theorem https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)?

(=_=) (Jul 02 2020 at 08:12):

While engaging in the discussion at #practice: applied ct > Manin--Marcolli, I came across this paper by Combe and Manin, where they claim that certain statistical manifolds carry the structure of "F-manifolds" (weak Frobenius manifolds). They recall certain results of E. B. Vinberg on homogeneous convex cones and Jordan algebras come up there. It seems to me that it may be relevant for this discussion as well.

John Baez (Jul 02 2020 at 21:20):

I don't know anything about weak Frobenius manifolds. But the connection between homogeneous convex cones and (formally real) Jordan algebras is fundamental to how Jordan algebras come up in quantum mechanics. It's often wise to think of a formally real Jordan algebra as just a goofy algebraic way of studying a homogeneous convex cone.

John Baez (Jul 02 2020 at 21:21):

The book by Farhaut and Koranyi Analysis on Symmetric Cones has a quite efficient proof that any homogeneous convex cone gives you a Jordan algebra... but I'm still struggling to get to the idea behind that proof, because it seems like a bunch of elegant unmotivated algebraic manipulations to me.

John Baez (Jul 02 2020 at 21:22):

I feel that if I understood the proof more deeply I could maybe do something that I can't do now. (I'm not sure what.)

John Baez (Jul 02 2020 at 21:24):

Here's a review of Farhaut and Koranyi's book.

John Baez (Jul 02 2020 at 21:31):

Valeria de Paiva said:

John Baez said:

And interestingly this turns out to be connected to why we like using the complex numbers in quantum mechanics.

I will bite. Why do you like using the complex numbers in quantum mechanics? does it matter that they're one of the normed algebras of Hurwitz theorem https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)?

The main reason I like using them is that they seem to describe our universe. The big question is why. What about our universe calls for them?

John Baez (Jul 02 2020 at 21:31):

There's a connection to the 4 normed division algebras. When Jordan, von Neumann and Wigner tried to isolate the features that "observables" should have in quantum mechanics, they decided these should form a "formally real Jordan algebra". They classified these algebras, at least in the finite-dimensional case, and found there were 5 kinds. Self-adjoint matrices with real, complex and quaternion entries are 3 of the kinds. Self-adjoint matrices with octonion entries only work if the matrices are $3 \times 3$ in size or smaller! And then there's a fifth kind.

Of all these different kinds, only the self-adjoint matrices with complex entries obey a version of Noether's theorem! That's what my paper tries to think about.

John Baez (Jul 02 2020 at 21:33):

If reading the paper seems too tiring, people can read my blog article.

Nikolaj Kuntner (Jul 04 2020 at 09:55):

Regarding the make-up-spacetimes discussion you guys had a few days ago, there's a fun Wikipedia page listing over 50 of classical (non-quantum) metric theories of gravity (i.e. alla general relativity)
https://en.wikipedia.org/wiki/Alternatives_to_general_relativity
Many of them have long Wikipedia articles themselves, e.g.,
https://en.wikipedia.org/wiki/Nordstr%C3%B6m%27s_theory_of_gravitation

Nikolaj Kuntner (Jul 04 2020 at 10:07):

John, I haven't read your article on Noether yet, but I'll get to it in the next 2 weeks.
There's a book to be released only in 2021 (but with a 10yo arxiv version online) by Arnold Neumaier - who lives in Vienna like me, although I have never met him - he's a mathematician formally working in optimization - trying to introduce classical and quantum and QM from a as-symmetric-as-possible Lie algebra perspective
https://arxiv.org/pdf/0810.1019.pdf
A glimpse of it is in another book by him that was released this year, for which he probably sidestepped the above for a while
https://www.degruyter.com/view/title/561158
That's a more philosophical minded text, but with one third on "fully formal" axiomatizations of QM. A second third is on trying to establish a lot of QM math in "coherent spaces", which is a weaker setup of vector spaces with emphasize (or restriction to) just the inner product <foo|bar>. And the third third is really a critique of Borns rule or it's interpretation in Copenhagen, Multiverses and all that. At the risk of being coarse, he's basically arguing against the interpretation of physical values being the eigenvalue and instead pulls up a reading that centers the "physical" on the expectations themselves.

Nikolaj Kuntner (Jul 04 2020 at 10:08):

Much of that book is also on the arxiv.

John Baez (Jul 04 2020 at 16:44):

@Nikolaj Kuntner - thanks! I know Arnold Neumaier; once we were going to work on a project together on the math of the hydrogen atom, but I got too busy with other things. I like his ideas. Thanks for pointing me to those books.

Alastair Grant-Stuart (Jul 22 2020 at 16:20):

John Baez said:

I have a new paper on Noether's theorem, and here is a blog article summarizing it:

• Getting to the bottom of Noether's theorem.

I've never encountered Jordan algebras before reading this paper, or the Jordan algebra approach to quantum mechanics. Historically, what became of Jordan's attempt to axiomatise QM this way? If there's no actual no-go result saying that this approach is bad, then this paper makes me want to look into it further.

In particular: I've never felt particularly comfortable about the "inverse temperature is imaginary time" ($\beta \sim it$) idea -- it's always seemed to me like a coincidence of the mathematical formalism rather than something physically meaningful (please correct me if I'm wrong about this!). But: the theorem of Alfsen & Schultz quoted in this paper says that the complex $\text{C}^*$-algebraic formulation of QM is recovered from a Jordan algebra formulation exactly when you have a dynamical correspondence, and @John Baez observes that a dynamical correspondence builds in both Noether's theorem and $\beta \sim it$. So it seems like if I wanted to avoid $\beta \sim it$, I should avoid complex $\text{C}^*$-algebras and think only in Jordan algebras.

I suppose this would need an awful lot of QM to be reformulated though. Relevant here: I guess to retain Noether's theorem we'd still need a map between observables (the Jordan algebra) and generators (its derivations) -- just not a full dynamical correspondence, otherwise we'd end up back at $\text{C}^*$-algebras, complex numbers and $\beta \sim it$.

John Baez (Jul 22 2020 at 22:08):

Historically, what became of Jordan's attempt to axiomatise QM this way?

Well, people worked on it a lot. The famous Jordan--von Neumann--Wigner paper classified the finite-dimensional formally real Jordan algebras (where "formal reality" is a condition necessary for a Jordan algebra to describe observables in quantum mechanics), and my paper on Noether's theorem recalls that classification. This paper made a lot of progress on the infinite-dimensional case:

• Irving Segal, Postulates for general quantum mechanics, Ann. Math. 48 (1947), 930-948.

though I'm not sure he says the words "Jordan algebra" very much. The usual modern approach uses "JB-algebras", short for Jordan--Banach algebras. A good overview is this book, kindly made open-access by one of its authors:

• H. Hanche-Olsen and E. Størmer, Jordan Operator Algebras, Pitman, 1984.

John Baez (Jul 22 2020 at 22:13):

In particular: I've never felt particularly comfortable about the "inverse temperature is imaginary time" (β∼it) idea -- it's always seemed to me like a coincidence of the mathematical formalism rather than something physically meaningful (please correct me if I'm wrong about this!).

I have trouble believing such an immensely productive idea in physics is a "coincidence" - the whole connection between quantum field theory and statistical mechanics is based on this, and thus the use of the same techniques (renormalization, etc.) in both subjects.

I agree that it's still mysterious, which is why I keep studying it from many different angles. I was really excited to see that it's built into any JB-algebra, and plays a role in the work of Alfsen and Shultz.

Alastair Grant-Stuart (Jul 24 2020 at 21:05):

Thanks for the references -- I have much reading to do now!

I have trouble believing such an immensely productive idea in physics is a "coincidence" - the whole connection between quantum field theory and statistical mechanics is based on this, and thus the use of the same techniques (renormalization, etc.) in both subjects.

I think that's roughly my point, although I may have expressed it poorly and in overly-skeptical-sounding terms -- sorry. The relation between statistical mechanics and QFT seems insufficiently explained (in a physics sense) by just saying that inverse temperature of one is imaginary time of the other.

So one way to look for a better explanation would be to work in a formalism without the "coincidence" that thermal states and time evolution are related by swapping out a $\beta$ for an $it$ in their matrix representations. One could see how much of QM can be recovered without $\beta \sim it$ or its precursor in the formalism of interest (e.g. the $[\psi_a,\psi_b] = -[\delta_a,\delta_b]$ property of the dynamical correspondence on a JB-algebra, in the notation of your paper).

Points of failure in the formalism might then give some insight. Hence seeing if the dynamical correspondence notion can be loosened to "forget $\beta \sim it$ while preserving Noether's theorem" seems relevant.

I was really excited to see that it's built into any JB-algebra, and plays a role in the work of Alfsen and Shultz

From the paper I gathered that a given JB-algebra could potentially be equipped with many different dynamical correspondences. Any such dynamical correspondence then gives a way to create a complex $\text{C}^*$-algebra from the JB-algebra. So: is there any sensible choice of dynamical correspondences on all JB-algebras that give a functor from a category of JB-algebras to a category of $\text{C}^*$-algebras?

I'm not sure exactly how much use that would be. I've seen various ideas about the functoriality of quantization, for instance, the conclusions of which depend heavily on one's definition of quantization. If one were to look for a functorial sort of JB-algebra-quantization, then composition with a "dynamical correspondence functor" would automatically give a functorial quantization in the usual C*-algebra formulation.

Nikolaj Kuntner (Jul 25 2020 at 10:13):

Something I find confusing when passing between Euler-Lagrange differential equation and the Stationary action principle dS=0 in terms of the integral is how exactly initial conditions translate. Some related notes and a confused question at the end:

• First observation is that for any given points (t1, q1), (t2, q2) in a space M, potentially punctured or other topological complication, an extremal solution may not exist (e.g. free particle action and there being hole in the to-be straight line path).
• It's better the points q1 and q2 are close and for very close, (q2-q1)/(t2-t1) should be just about the starting velocity.
• But for a differential equation, one generally provides initial position and velocity as data and then locally the differential equation will always give you a solution - but the stationarity condition seems more restrictive. It seems that as soon as I tweak the initial velocity for a problem posed in the dS=0 sense, this non-fitting initial velocity will rule out that there exists a stationary path, since with a velocity point in a wrong direction, I can never end up at an also chosen end-point (t2, q2).

Nikolaj Kuntner (Jul 25 2020 at 10:21):

Also, as a follow-up to our Galilean group representation theory / Levy-Leblond chat, I seared for literature and probably the most comprehensive text out there seems to be
https://www.cambridge.org/core/books/lie-groups-lie-algebras-cohomology-and-some-applications-in-physics/B570D04EC2EAA2A2C21BA23F245D2457
(which also covers prequantization stuff, as an aside)
I've also come across this Master thesis which does a good job of summarizing relevant group cohomology results and then talks about projective representations
https://www.math.ru.nl/~landsman/Nesta.pdf
I invited the student to this server, maybe he's around.
His advisor has a formal mathematical physics type book in that style on the QM framework online here
https://link.springer.com/content/pdf/10.1007%2F978-3-319-51777-3.pdf