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