  This paper is a generalization of earlier papers [Nucl. Phys. B 884, 344
(2014) (arXiv:1312.2759) and JHEP 6, 63 (2015) (arXiv:1401.2488)]. We
generalize the idea of quantum clock time to quantum spacetime reference frame
via physical realization of a reference system by quantum rulers and clocks.
Omitting the internal degrees of freedom (such as spins) of the physical rulers
and clocks, only considering their metric properties, the spacetime reference
frame is described by a bosonic non-linear sigma model (NLSM). We study the
quantum behavior of the system under approximations, and obtain (1) a
cosmological constant valued $(2/\pi)\rho_{c0}$ ($\rho_{c0}$ the critical
density at near current epoch) which is very close to the observations; (2) an
effective Einstein-Hilbert term; (3) the ratio of variance to mean-squared of
spacetime interval tends to a universal constant $2/\pi$ in the infrared
region. This effect is testable by observing a linear dependence between the
inherent quantum variance and mean-squared of the redshifts from cosmic distant
spectral lines. The proportionality is expected to be the observed percentage
of the dark energy. The equivalence principle is also generalized to the
quantum level.
