Emergent Newtonian dynamics and the geometric origin of mass

• Derive the macroscopic Newton’s equation from the microscopic many-particle Schrödinger’s equation.
• Deep connection between geometry and dynamics.
• Geometrical interpretation of the mass of macroscopic object as deformation of Hilbert space.
• Microscopic expression for mass and friction tensors.

We consider a set of macroscopic (classical) degrees of freedom coupled to an arbitrary many-particle Hamiltonian system, quantum or classical. These degrees of freedom can represent positions of objects in space, their angles, shape distortions, magnetization, currents and so on. Expanding their dynamics near the adiabatic limit we find the emergent Newton’s second law (force is equal to the mass times acceleration) with an extra dissipative term. In systems with broken time reversal symmetry there is an additional Coriolis type force proportional to the Berry curvature. We give the microscopic definition of the mass tensor. The mass tensor is related to the non-equal time correlation functions in equilibrium and describes the dressing of the slow degree of freedom by virtual excitations in the system. In the classical (high-temperature) limit the mass tensor is given by the product of the inverse temperature and the Fubini–Study metric tensor determining the natural distance between the eigenstates of the Hamiltonian. For free particles this result reduces to the conventional definition of mass. This finding shows that any mass, at least in the classical limit, emerges from the distortions of the Hilbert space highlighting deep connections between any motion (not necessarily in space) and geometry. We illustrate our findings with four simple examples.