These days in Torino Italy we've been talking a lot about network theory, quantum network theory, complex networks---anything dealing with networks, even the network of networks. We like this stuff.

A big part of this conversation is of course to try to find ways to relate different networks. One of the most interesting developments and topics in this regard is to consider the relation between quantum and stochastic networks. For example, one could consider particles moving on graphs (walks) and let the particles move either by rules from stochastic mechanics, or rules from quantum mechanics. This was done in the network theory series , particularly in part 16 when John related stochastic walks, quantum walks and electrical networks made of resistors.

I wanted to start a conversation related to a further comparison between the quantum and stochastic versions of Noether's theorem. This has of course already been done in posts in the network theory series (see parts 11 and 13!), but this is not to say that more can not be said. We're not sure what else can be said, and so perhaps this thread can serve as a place for ideas/thoughts/questions and developments in this regard.

Just to recall the basic idea, Noether's theorem relates symmetries of generators to conserved quantities.

We call $O$ a symmetry of quantum generator $H$ iff $[O,H]=0$. This is equivalent to the conserved quantity $\langle O\rangle_Q$ not changing in time:

$\frac{d}{dt} \langle O\rangle_Q = 0$

where $\langle O\rangle_Q$ is the expected value of $O$ at time $t$ in some state evolving under $H$. The subscript $Q$ means quantum.

In quantum theory, operators commuting with $H$ and conserved quantities that don't change in time are equivalent concepts. So for every "fish" in quantum theory, we get an exact reflection like this

![Alt text](

What John and Brendan showed is that this perfect symmetry is not the case in stochastic mechanics! Every symmetry in stochastic mechanics given by $[O,H]=0$ of course still leads to a conserved quantity

$\frac{d}{dt} \langle O\rangle_S = 0$

where the subscript $S$ means that the expected value is defined in terms of stochastic mechanics...

but... every conserved quantity does not always lead to a symmetry! So in other words, for every "fish" in stochastic mechanics, you can get something different

![Alt text](

In particular what John and Brendan showed is that you need more than just

$\frac{d}{dt} \langle O\rangle_S = 0$

to give rise to a symmetry such that $[O,H]=0$. For instance, the above equation together with

$\frac{d}{dt} \langle O^2\rangle_S = 0$

is enough to ensure that $[O,H]=0$. So it's a bit subtle, but symmetries and conserved quantities are not always related in such a perfect way as they are in quantum theory.

What else can we say about this? What John and Brendan depicted (in my point of view) exactly shows an interesting difference between quantum and stochastic mechanics, but perhaps there are some specific examples that further illuminate these points.

If anyone can think of anything, or if you have ideas/points you want to discuss, we're very interested here in Torino.