*guest post by Ville Bergholm*

In 1915 Emmy Noether discovered an important connection between the symmetries of a system and its conserved quantities. Her result has become a staple of modern physics and is known as Noether’s theorem.

The theorem and its generalizations have found particularly wide use in quantum theory. Those of you following the Network Theory series here on Azimuth might recall Part 11 where John Baez and Brendan Fong proved a version of Noether’s theorem for stochastic systems. Their result is now published here:

• John Baez and Brendan Fong, A Noether theorem for stochastic mechanics, *J. Math. Phys.* 54:013301 (2013).

One goal of the network theory series here on Azimuth has been to merge ideas appearing in quantum theory with other disciplines. John and Brendan proved their stochastic version of Noether’s theorem by exploiting ‘stochastic mechanics’ which was formulated in the network theory series to mathematically resemble quantum theory. Their result, which we will outline below, was different than what would be expected in quantum theory, so it is interesting to try to figure out *why*.

Recently Jacob Biamonte, Mauro Faccin and myself have been working to try to get to the bottom of these differences. What we’ve done is prove a version of Noether’s theorem for Dirichlet operators. As you may recall from Parts 16 and 20 of the network theory series, these are the operators that generate both stochastic *and* quantum processes. In the language of the series, they lie in the intersection of stochastic and quantum mechanics. So, they are a subclass of the infinitesimal stochastic operators considered in John and Brendan’s work.

The extra structure of Dirichlet operators—compared with the wider class of infinitesimal stochastic operators—provided a handle for us to dig a little deeper into understanding the intersection of these two theories. By the end of this article, astute readers will be able to prove that Dirichlet operators generate doubly stochastic processes.

Before we get into the details of our proof, let’s recall first how conservation laws work in quantum mechanics, and then contrast this with what John and Brendan discovered for stochastic systems. (For a more detailed comparison between the stochastic and quantum versions of the theorem, see Part 13 of the network theory series.)

### The quantum case

I’ll assume you’re familiar with quantum theory, but let’s start with a few reminders.

In standard quantum theory, when we have a closed system with states, the unitary time evolution of a state is generated by a self-adjoint matrix called the Hamiltonian. In other words, satisfies Schrödinger’s equation:

The state of a system starting off at time zero in the state and evolving for a time is then given by

The observable properties of a quantum system are associated with self-adjoint operators. In the state the expected value of the observable associated to a self-adjoint operator is

This expected value is constant in time for all states if and only if commutes with the Hamiltonian :

In this case we say is a ‘conserved quantity’. The fact that we have two equivalent conditions for this is a quantum version of Noether’s theorem!

### The stochastic case

In stochastic mechanics, the story changes a bit. Now a state is a probability distribution: a vector with nonnegative components that sum to 1. Schrödinger’s equation gets replaced by the **master equation**:

If we start with a probability distribution at time zero and evolve it according to this equation, at any later time have

We want this always be a probability distribution. To ensure that this is so, the Hamiltonian must be **infinitesimal stochastic**: that is, a real-valued matrix where the off-diagonal entries are nonnegative and the entries of each column sum to zero. It no longer needs to be self-adjoint!

When is infinitesimal stochastic, the operators map the set of probability distributions to itself whenever and we call this family of operators a **continuous-time Markov process**, or more precisely a **Markov semigroup**.

In stochastic mechanics, we say an **observable** is a real diagonal matrix, and its expected value is given by

where is the vector built from the diagonal entries of More concretely,

where is the th component of the vector

Here is a version of Noether’s theorem for stochastic mechanics:

**Noether’s Theorem for Markov Processes (Baez–Fong).** Suppose is an infinitesimal stochastic operator and is an observable. Then

if and only if

and

for all and all obeying the master equation. █

So, just as in quantum mechanics, whenever the expected value of will be conserved:

for any and all However, John and Brendan saw that—unlike in quantum mechanics—you need more than just the expectation value of the observable to be constant to obtain the equation You really need both

together with

for all initial data to be sure that

So it’s a bit subtle, but symmetries and conserved quantities have a rather different relationship than they do in quantum theory.

### A Noether theorem for Dirichlet operators

But what if the infinitesimal generator of our Markov semigroup is *also self-adjoint?* In other words, what if is both an infinitesimal stochastic matrix but also its own transpose: ? Then it’s called a **Dirichlet operator**… and we found that in this case, we get a stochastic version of Noether’s theorem that more closely resembles the usual quantum one:

**Noether’s Theorem for Dirichlet Operators.** If is a Dirichlet operator and is an observable, then

**Proof.** The direction is easy to show, and it follows from John and Brendan’s theorem. The point is to show the direction. Since is self-adjoint, we may use a spectral decomposition:

where are an orthonormal basis of eigenvectors, and are the corresponding eigenvalues. We then have:

where the third equivalence is due to the vectors being linearly independent. For any infinitesimal stochastic operator the corresponding transition graph consists of connected components iff we can reorder (permute) the states of the system such that becomes block-diagonal with blocks. Now it is easy to see that the kernel of is spanned by eigenvectors, one for each block. Since is also symmetric, the elements of each such vector can be chosen to be ones within the block and zeros outside it. Consequently

implies that we can choose the basis of eigenvectors of to be the vectors which implies

Alternatively,

implies that

where we have used the above sequence of equivalences backwards. Now, using John and Brendan’s original proof, we can obtain █

In summary, by restricting ourselves to the intersection of quantum and stochastic generators, we have found a version of Noether’s theorem for stochastic mechanics that looks formally just like the quantum version! However, this simplification comes at a cost. We find that the only observables whose expected value remains constant with time are those of the very restricted type described above, where the observable has the same value in every state in a connected component.

### Puzzles

Suppose we have a graph whose graph Laplacian matrix generates a Markov semigroup as follows:

**Puzzle 1.** Suppose that also so that is a Dirichlet operator and hence generates a 1-parameter unitary group. Show that the indegree and outdegree of any node of our graph must be equal. Graphs with this property are called **balanced**.

**Puzzle 2.** Suppose that is doubly stochastic Markov semigroup, meaning that for all each row and each column of sums to 1:

and all the matrix entries are nonnegative. Show that the Hamiltonian obeys

and all the off-diagonal entries of are nonnegative. Show the converse is also true.

**Puzzle 3.** Prove that any doubly stochastic Markov semigroup is of the form where is the graph Laplacian of a balanced graph.

**Puzzle 4.** Let be a possibly time-dependent observable, and write for its expected value with respect to some initial state evolving according to the master equation. Show that

This is a stochastic version of the Ehrenfest theorem.