## Network Theory and Discrete Calculus – Noether’s Theorem

This post is part of a series

As stated in the Introduction, one of the motivations for this series is to work in parallel with John Baez’ series on network theory to highlight some applications of discrete calculus. In this post, I reformulate some of the material in Part 11 pertaining to Noether’s theorem.

### The State-Time Graph

The directed graphs associated with discrete stochastic mechanics are described in the post The Discrete Master Equation, where the simple state-time graph example below was presented

Conceptually, the thing to keep in mind is that any transition from one state to another requires a time step. Therefore a transition from node to node is more precisely a transition from node to node .

On a state-time graph, a discrete 0-form can be written as

and a discrete 1-form can be written as

### The Master Equation

The master equation for discrete stochastic mechanics can be expressed simply as

where is a discrete 0-form representing the state at all times with

and is a discrete 1-form representing transition probabilities with

for all . When expanded into components, the master equation becomes

### Observables and Expectations

A general discrete 0-form on a state-time graph is defined over all states and all time. However, occasionally, we would like to consider a discrete 0-form defined over all states at a specific point in time. To facilitate this in a component-free manner, denote

so the identity can be expressed as

The discrete 0-form is a projection that projects a general discrete 0-form to a discrete 0-form defined only at time . For instance, given a discrete 0-form , let

so that

In discrete stochastic mechanics, an observable is nothing more than a discrete 0-form

The expectation of an observable with respect to a state is given by

where was defined in a previous post. Note:

### Some Commutators

In preparation for the discrete Noether’s theorem, note that

and

For these commutators to vanish, we must have

This implies if and only if is constant on each connected component of the state-time graph.

### Constant Expectations

In this section, we determine the conditions under which the expectation of an observable is constant in time, i.e.

for all . This is a fairly straightforward application of the discrete master equation, i.e.

indicating the condition we’re looking for is

### Noether’s Theorem

In this section, we demonstrate that when both and are constant in time, this implies

which, in turn, implies . To do this, we first expand

The condition for this trace to vanish is the same as the condition for the commutators above to vanish, i.e.

Expanding the trace further results in

Summing over and when and are constants results in

while summing and in the third term results in

by definition of the transition 1-form. Consequently, when and are constants, it follows that

Finally, this implies if and only if and are constant in time.

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