We consider the two basic theories of the standard physics: (1) the theory of relativity, as a four-dimensional time-space description of our universe, with the invariance of the time-space interval which leads to the Lorentz transformation, and (2) quantum mechanics, based on the Schrödinger equation, with a solution called wave function. However, we find that this equation is very unsatisfactory, describing fully unlocalized free particles, any initially localized particle rapidly spreading in space, in disagreement with the fundamental Hamilton equations. In agreement with the Hamilton equations, we describe a quantum particle by propagation wave functions in the coordinate and momentum spaces, with the Lagrangian in the time dependent phase instead of the Hamiltonian coming from the Schrödinger equation. With the relativistic Lagrangian, we obtain wave functions describing invariant distributions propagating in space.
Collapse models possibly suggest the need for a better understanding of the structure of space-time. We argue that physical space, and space-time, are emergent features of the Universe, which arise as a result of dynamical collapse of the wave function. The starting point for this argument is the observation that classical time is external to quantum theory, and there ought to exist an equivalent reformulation which does not refer to classical time. We propose such a reformulation, based on a non-commutative special relativity. In the spirit of Trace Dynamics, the reformulation is arrived at, as a statistical thermodynamics of an underlying classical dynamics in which matter and non-commuting space-time degrees of freedom are matrices obeying arbitrary commutation relations. Inevitable statistical fluctuations around equilibrium can explain the emergence of classical matter fields and classical space-time, in the limit in which the universe is dominated by macroscopic objects. The underlying non-commutative structure of space-time also helps understand better the peculiar nature of quantum non-locality, where the effect of wave-function collapse in entangled systems is felt across space-like separations.