Experiments in a Zeno paradox in stochastic mechanics (Rev #3)

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.

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.

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, for a normalised state $\alpha + \beta + \gamma = 1$
- In quantum mechanics, for a normalised state $|\alpha|^2 + |\beta|^2 + |\gamma|^2 = 1$

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.

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 bra**c**ket 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)$\langle z^k, z^k \rangle = k!$

We can then describe a measurement outcome by acting on a state by a projector. Consider the operator $P_k$ such that

(15)$P_k z^k = z^k$

and for $k\neq m$

(16)$P_k z^m = 0$

acting on a stochastic or quantum state amounts to acting with a projector, and then rescaling the coefficient of the remaining $z^k$ term, to be unity. That is, given $\psi = \sum_n c_n z^n$

(17)$P_k \psi = c_k z^k \mapsto z^k$

Consider a system starting in state $z^k$ and being acted on by the evolution generated by

(18)$U_t = e^{-t H}$

We will consider making a measurement which we denote by the appropriate action of the projector $P_k$. We will then evolve our system for some time $t=s$ under $U_t$. We will repeat this as follows.

(19)$(P_k U_s P_k) ... (P_k U_s P_k)(P_k U_s P_k)(P_k U_s P_k)\z^k = (P_k U_s P_k)^N z^k$

Now we will consider the time interval $s$ to be $s = t\N$ and we will determine what happens for large values of $N$. That is, we will repeat this procedure quickly and for a long time and see how the state evolves.

**Exercise**. Show that in the limit $N\rightarrow \infinity$, $(P_k N_{t/N} P_k)^N \rightarrow \exp(-t \langle H \rangle_k) P_k$ where $\langle H \rangle_k := \langle z^k, H z^k\rangle$

From the exercise, it follows that the state of the system evolves as

(20)$\psi(t) = \exp(-t \langle H \rangle_k) z^k$

and so corresponds to the same physical state. This is the so called, Zeno paradox, but this time we’re considering stochastic instead of quantum mechanics, which is typically considered.