Classical World Arising out of Quantum Physics under the Restriction of Coarse-Grained Measurements

Conceptually different from the decoherence program, we present a novel theoretical approach to macroscopic realism and classical physics within quantum theory. It focuses on the limits of observability of quantum effects of macroscopic objects, i.e., on the required precision of our measurement apparatuses such that quantum phenomena can still be observed. First, we demonstrate that for unrestricted measurement accuracy, no classical description is possible for arbitrarily large systems. Then we show for a certain time evolution that under coarse-grained measurements, not only macrorealism but even classical Newtonian laws emerge out of the Schrödinger equation and the projection postulate.

Figure 1

An initial spin-$j$ coherent state $|{\vartheta }_0,{\phi }_0⟩$ precesses into the coherent state $|\vartheta ,\phi ⟩$ at time $t$ under a quantum time evolution. (a) The probability $p(m,t)$ for the outcome $m$ in a measurement of the spin’s $z$-component is given by a Gaussian distribution with width $\sigma$ and mean $\mu$, which can be seen under the magnifying glass of sharp measurements. (b) The measurement resolution $\Delta m$ is finite and subdivides the $2j+1$ possible outcomes into a smaller number of coarse-grained “slots.” If the measurement accuracy is much poorer than the width $\sigma$, i.e., $\Delta m\gg \sqrt{j}$, the sharply peaked Gaussian cannot be distinguished anymore from the Kronecker delta ${\delta }_{\overline{m},\overline{\mu }}$ where $\overline{m}$ is numbering the slots and $\overline{\mu }$ is the slot in which the center $\mu$ of the Gaussian lies. (c) In the limit $j\to \infty$, the slots seem to become infinitely narrow and ${\delta }_{\overline{m},\overline{\mu }}$ becomes the delta function $\delta (\overline{m}-\overline{\mu })$.