In the framework of nonassociative geometry a unified description of continuum and discrete spacetime is proposed. In our approach at the Planck scales the spacetime is described as a so-called diodular discrete structure which at large spacetime scales “looks like” a differentiable manifold. After a brief review of foundations of nonassociative geometry, we discuss the nonassociative smooth and discrete de Sitter spacetimes.

The relationship is established between quantum phase transitions and complex geometric phases for open quantum systems governed by a non-Hermitian effective Hamiltonian with accidental crossing of the eigenvalues. In particular, the geometric phase associated with the ground state of the one-dimensional dissipative Ising model in a transverse magnetic field is evaluated, and it is demonstrated that the related quantum phase transition is of the first order.

We propose a type of the Unruh-DeWitt detector which measures the decoherence of the reduced density matrix of the detector interacting with the massless quantum scalar field. We find that the decoherence decay rates are different in the inertial and accelerated reference frames. We show that the exponential phase decay can be observed for relatively low accelerations, that can significantly improve the conditions for measuring the Unruh effect.