Quantum mechanics with a nonzero quantum correlation time

We propose an extension of quantum mechanics based on the idea that the underlying “quantum noise” has a nonzero, albeit very small, correlation time ${\tau }_c$. The standard (nonrelativistic) Schrödinger equation is recovered to zeroth order in ${\tau }_c$, and the first correction to energy levels is computed. Some consequences are discussed, in particular, the violation of Heisenberg's uncertainty principle and the restoration of locality at short times.

Figure 1

Violation of Heisenberg's uncertainty principle. We plot the minimum value of $\mathrm{\Delta }x\mathrm{\Delta }p/\hslash$ as a function of the ratio $u:=\tau /{\tau }_c$, where $\tau$ is the measurement time. The uncertainty vanishes linearly when $u\to 0$ and saturates to the standard $\hslash$ value for $\tau \gg {\tau }_c$, with a correction going down as $u^{-2}$.