  It was previously argued that the phenomenon of quantum gravitational
decoherence described by the Wheeler-DeWitt equation is responsible for the
emergence of the arrow of time. Here we show that the characteristic
spatio-temporal scales of quantum gravitational decoherence are typically
logarithmically larger than a characteristic curvature radius $R^{-1/2}$ of the
background space-time with a factor under the logarithm proportional to
$M_{P}^{2}/R$. This largeness is a direct consequence of the fact that gravity
is a non-renormalizable theory, as the corresponding effective field theory is
nearly decoupled from matter degrees of freedom in the physical limit
$M_{P}\to\infty$. Therefore, as such, quantum gravitational decoherence is too
ineffective to guarantee the emergence of the arrow of time at scales of
physical interest. We argue that the emergence of the arrow of time is directly
related to the nature and properties of physical observer.

General Relativity and Quantum Cosmology