What are we trying to do()?

Interventions in a Hamiltonian world

If you are reading this on a phone, watch out: there will be some equations and unfortunately Substack’s app butchers equations at times.

Let’s take a Hamiltonian H. We can write our equations of motion as

\(\dot{Q} = \frac{\partial H}{\partial P}, \qquad \dot{P} = - \frac{\partial H}{\partial Q}\)

Now we want to “implement” Pearl’s notion of do(Q = q), that is we want to force the coordinate Q to a constant value q but subject to the constraint of respecting the laws of (classical) mechanics. This comes in handy if we want to know, for instance, the causal effect of Q on some other variable S. But can we do() that?

A way to proceed is to look for an additional term G to be added to our Hamiltonian to get a new one, H + G. We want it to push Q towards q, so in the equations of motion we should have

\(\dot{Q} = \frac{\partial H}{\partial P} - k(Q - q)\)

Basically when Q > q the additional term is negative, forcing Q back towards q. Conversely, when Q < q, it is positive, pushing Q up towards q. Now this term is due to the additional Hamiltonian G, so

\(- k(Q - q) = \frac{\partial G}{\partial P}\)

hence

\(G = -kP(Q-q)\)

but this means that the equation for P will also be modified

\(\dot{P} = kP - \frac{\partial H}{\partial Q}\)

As a result, if the additional term dominates, P will explode exponentially.

If we want to confine the coordinate Q tightly around q, then we need k to be big. As a result, the conjugate momentum P will be dominated by the additional kP term and blow up, either to positive or negative infinity.

Is do(Q=q) a model of the measurement process in quantum mechanics?