The Azimuth Project Experiments in a Zeno paradox in stochastic mechanics (Rev #1)

For those of you following the Network Theory Series, we’ve been comparing quantum mechanics with its stochastic twin. Stochastic mechanics is very close to quantum mechanics, with some notable differences. Today we’re going to talk about some more similarities, and save the differences for later.

There is an old result in quantum theory, called Zeno’s paradox, or the watchpot effect. A quantum system that is constantly measured appears to be frozen in time. In other words, quantum evolution is totally halted if we try to watch it.

Evolution in Stochastic Mechanics

In stochastic mechanics, the master equation is given as

(1)$\frac{d}{dt} (e^{-tH}\psi) = 0$

with solution for all $t$ starting at $t = 0$ as

(2)$\psi(t) = e^{-t H}\psi(0)$

In the present post, we will be concerned with the operator $e^{-t H}$ for small times. In this limit, we can use Taylor’s expansion to arrive at

(3)$e^{-t H} \approx 1 - t H + O(t^2)$

So if we know the state of the system at time zero, and we want to know the state of the system after a short time passes, we can use this formula

(4)$\psi(t) \approx \psi(0) - t H \psi(0)$

First, notice that we will have to renormalise our state, so that probabilities sum to one. The other thing is that this equation is wrong! However, it is wrong with error $O(t^2)$, and for small times $t$, this is negligible.

Measurement in Stochastic Mechanics

In prior posts, we’ve repeatedly used a prevalent trick from quantum theory, called a formal series. Here is how it works. We write our state as

(5)$\psi = \sum_n c_n z^n$

and interpret it as follows. We are considering occurrences of ‘things’ of type $z$. In stochastic mechanics, $c_n$ is a probability of finding $n$ of these things. In quantum mechanics, $c_n$ is a complex amplitude. The modulus of $c_n$ in the quantum world, gives the probability.

Here is how this works in practice. Say we have some state, and that this state is not evolving in time. Let’s pick a state to make things more concrete.

(6)$\phi = \alpha z^k + \beta z^l + \gamma z^m$
• In stochastic mechanics, $\alpha + \beta + \gamma = 1$
• In quantum mechanics, $|\alpha|^2 + |\beta|^2 + |\gamma|^2 = 1$

If we measure the state $\phi$ in stochastic mechanics, we can find the system in either state $z^k$, state $z^l$ or state $z^m$ with respective probabilities $\alpha$, $\beta$, or $\gamma$. Let’s say we find the system in state $z^k$. Then we become completely certain of the state of our system, and so the probabilities $\beta$ and $\gamma$ vanish and the new state becomes

(7)$\phi \rightarrow \phi' = z^k$

This same sort of scenario holds for quantum mechanics. This is called a projective measurement. This is a postulate of quantum mechanics often attributed to von Neumann.

Constructing a hilbert space

Before we go any further, we need to think about turning the vector space formed by $\{z^n|n=0,...,n\}$ into a full fledged Hilbert space, by adding an inner product. We can then evaluate quantities such as

(8)$\langle z^k, z^k \rangle$

This inner product of vectors evaluates to a number. Recall that in Part X John started using creation ($a^+$) and annihilation ($a^-$) operators. These are defined as

(9)$a^+ = z$
(10)$a^- = \frac{\partial}{\partial z}$

so one is multiplication by $z$ and the other takes the derivative. For reasons that we won’t explain now (but are in common use in quantum theory), taking the so called dagger ($\dagger$) or adjoint of an $a$ works as follows

(11)$(a^\pm)^\dagger = a^\mp$

Now we note that

(12)$\langle z^k, z^k \rangle = \langle z^k, a^+z^{k-1} \rangle = \langle a^- z^k, z^{k-1} \rangle$

This works because taking an $a^+$ from one side of the bracket to the other, amounts to taking a dagger, which changes $a^+$ to $a^-$. We then have that

(13)$\langle a^- z^k, z^{k-1} \rangle = \langle z^{k-1}, z^{k-1} \rangle$

If we continue this procedure, we find

(14)$\langle z^k, z^k \rangle = k!$