I’ve been spending the last month at the Centre of Quantum Technologies, getting lots of work done. This Friday I’m giving a talk, and you can see the slides now:
• John Baez, Getting to the bottom of Noether’s theorem.
Abstract. In her paper of 1918, Noether’s theorem relating symmetries and conserved quantities was formulated in term of Lagrangian mechanics. But if we want to make the essence of this relation seem as self-evident as possible, we can turn to a formulation in term of Poisson brackets, which generalizes easily to quantum mechanics using commutators. This approach also gives a version of Noether’s theorem for Markov processes. The key question then becomes: when, and why, do observables generate one-parameter groups of transformations? This question sheds light on why complex numbers show up in quantum mechanics.
At 5:30 on Saturday October 6th I’ll talk about this stuff at this workshop in London:
• The Philosophy and Physics of Noether’s Theorems, 5-6 October 2018, Fischer Hall, 1-4 Suffolk Street, London, UK. Organized by Bryan W. Roberts (LSE) and Nicholas Teh (Notre Dame).
This workshop celebrates the 100th anniversary of Noether’s famous paper connecting symmetries to conserved quantities. Her paper actually contains two big theorems. My talk is only about the more famous one, Noether’s first theorem, and I’ll change my talk title to make that clear when I go to London, to avoid getting flak from experts. Her second theorem explains why it’s hard to define energy in general relativity! This is one reason Einstein admired Noether so much.
I’ll also give this talk at DAMTP—the Department of Applied Mathematics and Theoretical Physics, in Cambridge—on Thursday October 4th at 1 pm.
The organizers of London workshop on the philosophy and physics of Noether’s theorems have asked me to write a paper, so my talk can be seen as the first step toward that. My talk doesn’t contain any hard theorems, but the main point—that the complex numbers arise naturally from wanting a correspondence between observables and symmetry generators—can be expressed in some theorems, which I hope to explain in my paper.
Posted by John Baez 
and 
and get a morphism 
In a monoidal category you can also tensor morphisms 
and 
and get a morphism 
This of course relies on your ability to tensor objects. In a symmetric monoidal category you also have 
And of course, there is more to it than this. But this is enough to get started.
to denote the tensor product in our symmetric monoidal category, as usual in chemistry.
so their associator:
and a category of ‘commutative monoidal categories’, which we call 
is a Petri net: that is, a set 
of places, a set 
of transitions, and source and target functions![s, t \colon T \to \mathbb{N}[S]](https://s0.wp.com/latex.php?latex=s%2C+t+%5Ccolon+T+%5Cto+%5Cmathbb%7BN%7D%5BS%5D+&bg=ffffff&fg=333333&s=0 "s, t \colon T \to \mathbb{N}[S]")
![\mathbb{N}[S]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BN%7D%5BS%5D&bg=ffffff&fg=333333&s=0 "\mathbb{N}[S]")
is the underlying set of the free commutative monoid on 
![\mathbb{N}[f] \colon \mathbb{N}[S] \to \mathbb{N}[S']](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BN%7D%5Bf%5D+%5Ccolon+%5Cmathbb%7BN%7D%5BS%5D+%5Cto+%5Cmathbb%7BN%7D%5BS%27%5D&bg=ffffff&fg=333333&s=0 "\mathbb{N}[f] \colon \mathbb{N}[S] \to \mathbb{N}[S']")
![s', t' \colon T' \to \mathbb{N}[S']](https://s0.wp.com/latex.php?latex=s%27%2C+t%27+%5Ccolon+T%27+%5Cto+%5Cmathbb%7BN%7D%5BS%27%5D+&bg=ffffff&fg=333333&s=0 "s', t' \colon T' \to \mathbb{N}[S']")
Explicitly, it’s a 
such that for all objects 
and 
we have
and 
be the category whose objects are commutative monoidal categories and whose morphisms are strict monoidal functors.
to its underlying Petri net. This Petri net has the set of objects 
as its set of places and the set of morphisms 
as its set of transitions, and![s, t \colon \mathrm{Mor}(C) \to \mathrm{Ob}(C) \hookrightarrow \mathbb{N}[\mathrm{Ob}(C)]](https://s0.wp.com/latex.php?latex=s%2C+t+%5Ccolon+%5Cmathrm%7BMor%7D%28C%29+%5Cto+%5Cmathrm%7BOb%7D%28C%29+%5Chookrightarrow+%5Cmathbb%7BN%7D%5B%5Cmathrm%7BOb%7D%28C%29%5D+&bg=ffffff&fg=333333&s=0 "s, t \colon \mathrm{Mor}(C) \to \mathrm{Ob}(C) \hookrightarrow \mathbb{N}[\mathrm{Ob}(C)]")
is a finite multisubset of 
We can think of this as a way of placing finitely ‘tokens’—little black dots—on the elements of 
![\mathbb{N}[X]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BN%7D%5BX%5D&bg=ffffff&fg=333333&s=0 "\mathbb{N}[X]")
denote the set of markings of the set 
there is a reachability relation![\blacksquare P \subseteq \mathbb{N}[X] \times \mathbb{N}[Y]](https://s0.wp.com/latex.php?latex=%5Cblacksquare+P+%5Csubseteq+%5Cmathbb%7BN%7D%5BX%5D+%5Ctimes+%5Cmathbb%7BN%7D%5BY%5D+&bg=ffffff&fg=333333&s=0 "\blacksquare P \subseteq \mathbb{N}[X] \times \mathbb{N}[Y]")
feed those tokens into the Petri net 
move them around using transitions in 
leaving no tokens behind.
and move them around using transitions in 
that is ‘reachable’ from the original marking of 
extends to a ‘lax double functor’
is a double category having open Petri nets as horizontal 1-cells and 
is a double category having relations as horizontal 1-cells.
as objects,
as vertical 1-morphisms,
This double category has
as horizontal 1-cells from 
![\mathbb{N}(f) \colon \mathbb{N}[X] \to \mathbb{N}[Y]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BN%7D%28f%29+%5Ccolon+%5Cmathbb%7BN%7D%5BX%5D+%5Cto+%5Cmathbb%7BN%7D%5BY%5D&bg=ffffff&fg=333333&s=0 "\mathbb{N}(f) \colon \mathbb{N}[X] \to \mathbb{N}[Y]")
is a really a functor.)![\blacksquare P \colon \mathbb{N}[X] \to \mathbb{N}[Y]](https://s0.wp.com/latex.php?latex=%5Cblacksquare+P+%5Ccolon+%5Cmathbb%7BN%7D%5BX%5D+%5Cto+%5Cmathbb%7BN%7D%5BY%5D&bg=ffffff&fg=333333&s=0 "\blacksquare P \colon \mathbb{N}[X] \to \mathbb{N}[Y]")
is a lax double functor. This means that it does not send a composite open Petri net 
to the composite of the reachability relations for 
So, we do not have
to be this one:
is this:
is a proper subset of 
In 
a token can move all the way from point 1 to point 5. But it does not do so by first moving through 
It has to take a more complicated zig-zag path where it first leaves 
then comes back into 
Various choices of semantics for open Petri nets can be described using symmetric monoidal double functors out of 
for example this:
is to put the pictures together:
we have a new Petri net whose set of places is the disjoint union of those for 
which looks like this:
or 
This is why to get a category, with composition strictly associative, we need to use isomorphism classes of open Petri nets as morphisms. But there are advantages to avoiding this and working with open Petri nets themselves. Basically, it’s better to work with things than mere isomorphism classes of things! If we do this, we obtain not a category but a 
and 
‘objects’, 
and 
‘horizontal 1-cells’, and 
a ‘2-morphism’. We can compose vertical 1-morphisms to get new vertical 1-morphisms and compose horizontal 1-cells to get new horizontal 1-cells. We can compose the 2-morphisms in two ways: horizontally and vertically. (This is just a quick sketch of the ideas, not the full definition.)
with:
