Quantum Anti-Zeno Paradox

We establish an exact differential equation for the operator describing time-dependent measurements continuous in time and obtain a series solution. Suppose the projection operator $E\left(t\right)=U\left(t\right){\mathrm{EU}}^{†}\left(t\right)$ is measured continuously from $t=0\mathrm{}$ to $T$, where $E$ is a projector leaving the initial state unchanged and $U\left(t\right)$ a unitary operator obeying $U\left(0\right)=1\mathrm{}$. We prove that the probability of always finding $E\left(t\right)=1\mathrm{}$ from $t=0\mathrm{}$ to $T$ is unity. If $U\left(t\right)\ne 1$, the watched kettle is sure to “boil.”