I think John and Brendan's proof is very interesting and valuable.

I also find it a bit depressing, for the following reason:

In quantum mechanics, if we want to look for conserved quantities we know a good way to go about it is to think about observable operators that commute with the Hamiltonian. Even if this is not easy, it is at least clear how we need to go about it.

But John and Brendan showed that if you look for observable operators that commute with the Hamiltonian of a stochastic processes then you'll only find a subset of the conserved quantities for which the higher moments (variance etc.) are also conserved. In fact this subset is so restrictive it means that the observable must take the same value for any states that are connected by the Hamiltonian (i.e. a transition between them is allowed). This means by looking for observable operators that commute with the Hamiltonian we'll only find observables that have the same value in each [connected component](http://en.wikipedia.org/wiki/Connected_component_(graph_theory)) of the system, and these observables are trivially conserved because the probability of being in a connected component don't change in time.

So it basically says: if you want to find the most interesting observed quantities, don't start by looking for commutations. That's the value of the result in my opinion, to show us where we shouldn't look and therefore to guide us to where we should look.

Where is this?...I don't know.

There are a few interesting results from non-equilibrium stochastic mechanics that I feel have some relation to conserved quantities. My favourite one is [Jarzynski's equality](http://pre.aps.org/abstract/PRE/v56/i5/p5018_1), which was proved in the following

* C. Jarzynski, [Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach](http://pre.aps.org/abstract/PRE/v56/i5/p5018_1), Phys. Rev. E **56**, 5018 (1997).

Jarzsynki's result is the following:

$$ \langle \exp(-W/T) \rangle_0 = \exp(-\Delta F /T) $$

which basically says that if I start in a thermal state at temperature $T$ and change some parameter of a system in time, then the expected value of the exponential of the work done $W$ on the system during that change is equal to the exponential of the difference $\Delta F$ in the equilibrium free energies corresponding to the initial and final values of the parameter. For a reasonably general class of evolutions, there is a non-equilibrium quantity whose expected value is entirely determined by equilibrium properties. It doesn't matter how I change the parameter! Slow or fast is ok.

This caused a huge buzz in physics. Experimentalists use it to calculate the free energies of systems, by measuring the work done in many realizations as a parameter is changed. They used to have to do the change in the parameter very slowly because they knew only that $W = \Delta F$ in the infinitely slow limit. But since Jarzynski they can do it quickly and use the relation above.

It strikes me that there might be some way of interpreting this as a conservation of a quantity, and I hope at some point in the future to try and rephrase it in this way. It might then give us some clues about how to generalise the approach to other conserved quantities.

If people are interested in this stuff perhaps I could give a few more details on Jarzynski's equality...

I also find it a bit depressing, for the following reason:

In quantum mechanics, if we want to look for conserved quantities we know a good way to go about it is to think about observable operators that commute with the Hamiltonian. Even if this is not easy, it is at least clear how we need to go about it.

But John and Brendan showed that if you look for observable operators that commute with the Hamiltonian of a stochastic processes then you'll only find a subset of the conserved quantities for which the higher moments (variance etc.) are also conserved. In fact this subset is so restrictive it means that the observable must take the same value for any states that are connected by the Hamiltonian (i.e. a transition between them is allowed). This means by looking for observable operators that commute with the Hamiltonian we'll only find observables that have the same value in each [connected component](http://en.wikipedia.org/wiki/Connected_component_(graph_theory)) of the system, and these observables are trivially conserved because the probability of being in a connected component don't change in time.

So it basically says: if you want to find the most interesting observed quantities, don't start by looking for commutations. That's the value of the result in my opinion, to show us where we shouldn't look and therefore to guide us to where we should look.

Where is this?...I don't know.

There are a few interesting results from non-equilibrium stochastic mechanics that I feel have some relation to conserved quantities. My favourite one is [Jarzynski's equality](http://pre.aps.org/abstract/PRE/v56/i5/p5018_1), which was proved in the following

* C. Jarzynski, [Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach](http://pre.aps.org/abstract/PRE/v56/i5/p5018_1), Phys. Rev. E **56**, 5018 (1997).

Jarzsynki's result is the following:

$$ \langle \exp(-W/T) \rangle_0 = \exp(-\Delta F /T) $$

which basically says that if I start in a thermal state at temperature $T$ and change some parameter of a system in time, then the expected value of the exponential of the work done $W$ on the system during that change is equal to the exponential of the difference $\Delta F$ in the equilibrium free energies corresponding to the initial and final values of the parameter. For a reasonably general class of evolutions, there is a non-equilibrium quantity whose expected value is entirely determined by equilibrium properties. It doesn't matter how I change the parameter! Slow or fast is ok.

This caused a huge buzz in physics. Experimentalists use it to calculate the free energies of systems, by measuring the work done in many realizations as a parameter is changed. They used to have to do the change in the parameter very slowly because they knew only that $W = \Delta F$ in the infinitely slow limit. But since Jarzynski they can do it quickly and use the relation above.

It strikes me that there might be some way of interpreting this as a conservation of a quantity, and I hope at some point in the future to try and rephrase it in this way. It might then give us some clues about how to generalise the approach to other conserved quantities.

If people are interested in this stuff perhaps I could give a few more details on Jarzynski's equality...