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added to Noether theorem a brief paragraph on the symplectic/Hamiltonian Noether theorem
added a tad more References.
Just a comment: It always seems strange to find one theorem of someone, X, called X’s theorem when they proved lots of important theorems!
In this case it seems to be entirely standard, though, and hence I think no confusion will arise. Sometimes people say “Noether’s first theorem”, for emphasis, and I have added that to the entry now.
In fact I added the following in the Idea-section:
What is commonly called Noether’s theorem or Noether’s first theorem is a theorem due to Emmy Noether (Noether 1918) which makes precise and asserts that to every continuous symmetry of a Lagrangian physical system (prequantum field theory) corresponds equivalently a conservation law stating the conservation of a charge (conserved current).
For instance the time-translation invariance of a physical system equivalently means that the quantity of energy is conserved, and the space-translation invariant of a physical system means that momentum is preserved.
The original and still most common formulation of the theorem is in terms of variational calculus applied to a local action functional. A modern version of this formulated properly in terms of the variational bicomplex we discuss below in
There is another formulation of the same physical content, but using the formalism of symplectic geometry of phase spaces. In this formulation of physics the relation between symmetries and charges/conserved currents happens to be built deep into the formalism in terms of Hamiltonian flows generated by the Poisson bracket with a Hamiltonian function. Accordingly, in this powerful formalism Noether’s theorem becomes almost a tautology. This we discuss in
I was not suggesting there was confusion here, just commenting on the strangeness of the use in general, but I like your addition.
Is there a version of Noether’s theorem in higher symplectic geometry, i.e., of what you have in ’Hamiltonian/symplectic vection – In terms of moment maps’? (Do you mean ’version’ rather than ’vection’?)
What happens to the conservation principles under quantization?
David,
oops, yes, I meant “version” (I didn’t even know “vection”, had to look it up now :-). Fixed now.Thanks for catching this.
Is there a version of Noether’s theorem in higher symplectic geometry,
Yes, so this is of course what I am secretly looking at behind the scenes.
So, yes, the traditional symplectic version immediately generalizes to a higher symplectic version and actually unifies with the Lagrangian version there.
So the thing is that for $\mathbf{Fields}$ a moduli stack of fields, a local Lagrangian on it for a higher prequantum theory in dimension $n$ is a map
$\exp(i S) : \mathbf{Fields} \to \mathbf{B}^n U(1)_{conn} \,.$This $n$-connection on the space of fields is the Lagrangian. For instance over a test maifold $U$ the above map sends a field configuration $\phi$ on $U$ to an $n$-form $L(\phi)$ on $U$, and this is the Lagrangian.
Now, a “Noether symmetry” of this Lagrangian is an equivalence of the fields $\mathbf{Fields} \stackrel{\simeq}{\to} \mathbf{Fields}$ that preserves the Lagrangian up to a gauge transformation (“up to an exact term”) hence up to a homotopy in
$\array{ \mathbf{Fields} &&\stackrel{\simeq}{\to}&& \mathbf{Fields} \\ & {}_{\mathllap{\exp(i S)}}\searrow &\swArrow_{H}& \swarrow_{\mathrlap{\exp(i S)}} \\ && \mathbf{B}^n U(1)_{conn} } \,.$Here $H$ is effectively the “Hamiltonia”/conserved charge corresponding to the symmetry. If $\mathbf{Fields}$ is a smooth manifold, then this is a plain Hamiltonian form.
The $n$-group of such symmetries is the quantomorphism n-group $QuantMorph(exp(i S))$ of the Lagrangian/local action functional. Hence a $G$-symmetry of the Lagrangian is an $\infty$-group homomorphism
$G \to QuantMorp(\exp(i S))$and as for the traditional “symplectic Noether theorem”, but now refined to current forms, this automatically says that the symmetry comes with conserved currents.
Excellent. Sounds like it’s ready to be rendered in type theoretic terms.
By the way, are you attending this workshop Quantum Mathematics and Computation?
Particular attention will be devoted to questions concerning the theory and mathematical application of local topological quantum field theory, and an investigation of how modern mathematical perspectives on quantum theory could be advanced through an understanding of the geometry of type theory.
Can’t think of anyone better qualified to talk there.
Sounds like it’s ready to be rendered in type theoretic terms.
Oh for sure. That’s what Higher geometric prequantum theory (schreiber) is about. But it’s true that we didn’t amplify the relation to Noether’s theorem much. At one point we just remark about conserved currents. I’ll expand on that now.
By the way, are you attending this workshop Quantum Mathematics and Computation?
Hm, no, hadn’t heard of the event until you mentioned it. Thanks for the pointer.
Yes, maybe we should talk Noether charges etc.
Tamarkin was saying in his formalism for BV and higher geometry Noether theorem is kind of upside down. Not that symmetries are basic and then conservation laws from them, but the conservation laws are fundamental and the symmetries appear defined in some cases from those (maybe I was not precise).
For future redirects: under Noether’s theorem algebraic geometers mean Noether normalization theorem, quite fundamental in algebraic geometry and commutative algebra (e.g. the Hilbert Nullstellensatz being a corrolary).
added a paragraph with the Simple schematic idea.
have added one more brief pargraph at the end of Simple schematic idea on the more general case where the symmetry leaves the Lagrangian invariant only up to an exact term.
@Jim: Okay, I have edded YKS’s book to the References
@Zoran: not sure I understand the thing about upside-down. The point of the theorem is that both are equivalent to each other, neither is more fundamental. Maybe in the usual Lagrangian formulation this is not 100% explicit, but in the Hamiltonian formulation it is fully manifest.
So, Tamarkin said that in his formalism of “hrestomaties” in BV, every conservation law gave some symmetry but not every symmetry came that way, so that it was not truly 1-1. Upside down because deriving conservation from a symmetry, i.e. producing the inverse did not exist in full generality which he allowed for the higher symmetries and conservation laws. Maybe an artifact of the theory, of course. Though the theory was based on quite nice homotopical resolutions, what looked canonical. I might have hand written notes somewhere.
Hrestomaty (chrestomaty ?) is certain object in homotopical algebra in the language of pseudotensor categories (a version of color operads used by Beilinson-Drinfeld school) which in his published preprint is something like what he calls there system. But the symmetries like what is essentially $L_\infty$ kind of thing are another special case. Thus the physical system and the symmetries are objects belonging to the same general class of (c)hrestomaties.
He gave lectures on this in Paris in 2004 (series of about 6 sessions) with examples like derived Hamiltonian reduction, Noether theorems, quantization of nonlinear Poisson sigma model, and its reductions (which included topological A-model and topological B-model) etc. It was amazing project; he got Hopf algebras of renormalization as functions of the torsor of equivalences between different homotopical resolutions for example (probably Connes-Kreimer Hopf algebras are a special case, but he did not go into combinatorial details, though it was very likely). He also explained the role of Green functions in producing short exact sequences needed to obtain the homotopical resolutions by splicing. Goncharov was in the lectures and he suspected that he could do this over other fields (not only complex numbers). But Tamarkin was not happy with the progress of the project and abandoned it. There is a preprint on the free field theory case on the net (before he started using the terminology of hrestomaties), and the rest has never appeared as a preprint. I asked him later, he said he considered it too difficult project for himself and that he thinks that the approach by Costello is better. Still I feel pity some of his discoveries and examples never appeared in print. He also used lots of techniques from Beilinson-Drinfeld work in action what is quite nice.
I took from some lectures but not all, but it is not easy to find them. I thnk Goncharov took more careful notes.
arXiv:math/0312219 Title: A formalism for the renormalization procedure Authors: Dmitry E. Tamarkin
has the first part of the work, the rest unfortunately never appeared in a form of a preprint. n particular derived symplectic reduction, and the study of nonlinear Poisson sigma model (and ist relations to topological A and B-model) are not treated in the first part. Chrestomaty is a general variant of what is in this first preprint defined as “system”.
Thanks for the pointer. Such as not to keep this a secret, I have added it here.
I have added to the References at Noether theorem the following
A formalization of (aspects of) the Noether theorem in parametric dependent type theory is discussed in
- Robert Atkey, From Parametricity to Conservation Laws, via Noether’s Theorem, talk at Principles of Programming Languages (POPL) 2014 (web, pdf slides
pdf link does not work
I had fixed and expanded in the entry:
- Robert Atkey, From Parametricity to Conservation Laws, via Noether’s Theorem, talk at Principles of Programming Languages (POPL) 2014 (pdf article, web slides, pdf slides)
Looking through the article, I see that the parametric type theory is used to prove (as a “free theorem”) that a given (“Lagrangian”) function has a given invariance. But it seems to me now that neither Noether’s theorem, nor conservation laws nor conserved quantities are claimed to be derived by means of any type theory. Instead, it seems that once the invariance of a function is established via parametricity, there is an appeal to the standard textbook discussion of Noether’s theorem to claim that “now a conservation law follows”. But this step itself is not being formalized in the text.
That’s at least my impression from reading the text. If anyone here has further insight, please let me know.
Therefore I have changed in the entry “formalizes the Noether theorem” to something like “formalizes the invariance of a (Lagrangian) function”.
I took the liberty of adding to the list of References at Noether theorem the following lines:
A formalization of Noether’s theorem in cohesive homotopy type theory is discussed in sections “2.7 Noether symmetries and equivariant structure” and “3.2 Local observables, conserved currents and higher Poisson bracket homotopy Lie algebras” of
- Urs Schreiber, Classical field theory via Cohesive homotopy types (schreiber) (2013)
I wonder whether there is a sign mistake in the Lagrangian version derivation (and maybe a confusion between $\theta$ and $\Theta$?) - or whether my derivation is wrong. I tried to show the energy conservation from a simple Lagrangian $L = \mathcal{L}(q^a,\dot{q}^a)dt$. The vector field describing time translation should be $v = \sum_{\lambda = 0}^{\infty}q^a_{(\lambda+1)}\bar{\partial}_a^{\lambda}$. Now taking $\Theta$ from $d\Theta = \delta L - E(L)$ I get
$\sigma_v = \mathcal{L}\,, \quad \iota_v\Theta = -\frac{\partial\mathcal{L}}{\partial\dot{q}^a}\dot{q}^a\,, \quad \sigma_v - \iota_v\Theta = \mathcal{L} + \frac{\partial\mathcal{L}}{\partial\dot{q}^a}\dot{q}^a$Well, you see the problem - the Hamiltonian should have a relative $-$ between these terms. I think $\Theta$ right in the top should be defined with the opposite sign. Oh, and it should probably be $\Theta$ instead of $\theta$ in the statement of the Noether theorem?
That may happen that there is a sign error. I’ll try to check tomorrow when I have the leisure. Thanks for the alert.
mhohmann, I have now fixed in the entry the issue with $\theta$ appearing in lower case where it was introduced instead in upper case notation, thanks again.
(for bystanders: we are talking about the paragraphs in the entry starting here)
Regarding the sign:
a sign in the definition of $\Theta$ should not matter (we could have any non-zero prefactor in the definition of $\Theta$ without changing anything but the normalization of $\Theta$) so I wouldn’t think that this solves the isse that you ran into.
I am rather wondering why you have the minus sign in
$\iota_v\Theta = -\frac{\partial\mathcal{L}}{\partial\dot{q}^a}\dot{q}^a$
(in #27). I don’t seem to get that.
Hi Urs, thanks for checking this. I wonder whether we might have different sign conventions somewhere… Let’s see. Here’s my calculation, starting from the Lagrangian $L = \mathcal{L}(q^a,\dot{q}^a)dt$, I find the vertical derivative and Euler-Lagrange-Equations:
$\delta L = \left(\frac{\partial\mathcal{L}}{\partial q^a}\theta^a + \frac{\partial\mathcal{L}}{\partial\dot{q}^a}\dot{\theta}^a\right) \wedge dt\,, \quad E(L) = \left(\frac{\partial\mathcal{L}}{\partial q^a} - D_t\frac{\partial\mathcal{L}}{\partial\dot{q}^a}\right)\theta^a \wedge dt\,.$Here I wrote $\theta^a$ and $\dot{\theta}^a$ for the basic contact one-forms and $D_t$ for the total derivative. Now the next line is where the minus comes into the game, from reordering the wedge product $\theta^a \wedge dt$, since $d(f\theta^a) = D_t f\,dt \wedge \theta^a + f\,dt \wedge \dot{\theta}^a$:
$d\Theta = \delta L - E(L) = \left(\frac{\partial\mathcal{L}}{\partial\dot{q}^a}\dot{\theta}^a + D_t\frac{\partial\mathcal{L}}{\partial\dot{q}^a}\theta^a\right) \wedge dt = - d\left(\frac{\partial\mathcal{L}}{\partial\dot{q}^a}\theta^a\right)\,.$Finally, with the vector field $v = \sum_{\lambda = 0}^{\infty}q^a_{(\lambda+1)}\bar{\partial}_a^{\lambda}$, which has $\iota_v\theta^a = \dot{q}^a$, I get
$\iota_v\Theta = -\frac{\partial\mathcal{L}}{\partial\dot{q}^a}\dot{q}^a\,.$By the way, you can also find this sign inconsistency if you look at the proof of Noether’s theorem at the end of this section. There you conclude that $\delta L + d\Theta = E(L)$, but on top of the section you have $\delta L = E(L) + d\Theta$. I checked the proof (in particular the part $-d\iota_v\Theta = \iota_v d\Theta$) and came to the same result, i.e., Theorem 1 and its proof are consistent among each other, but Proposition 1 is not.
Oh, now I see. You are right. Of course I can have any sign I want where $d\Theta$ is introduced, but then I need to carry it along properly, which I didn’t. Fixed now here. Sorry for causing you this trouble. Thanks for insisting.
Thanks for fixing it! I was a bit unsure where to switch the sign, whether in Proposition 1 or in Theorem 1, since also the presymplectic structure in Definition 1 depends on it, but I guess that can also be defined with either sign, and so it wouldn’t really matter. Now it’s consistent in any case.
I’ve been hearing a bit about Noether’s second theorem, and its interpretation in GR. Is it sufficiently different from her first theorem to warrant a new page?
I know Urs has written on the second theorem here.
I believe there is some question of what it says in GR, i.e., how to understand energy-momentum conservation.
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