Emergent Newtonian dynamics and the geometric origin of mass

Abstract

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 relating it to the non-equal time correlation functions in equilibrium or alternatively expressing it through dressing 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.

28 + 14 pages, 4 figures

References in corpus (9)

• Quantum critical scaling of the geometric tensors
• Supplementary Information to the paper ``Breakdown of the adiabatic limit in low dimensional gapless systems''
• Breakdown of the adiabatic limit in low dimensional gapless systems
• The geometry of thermodynamic control
• Scattering theory of current-induced forces in mesoscopic systems
• Quantum response of dephasing open systems
• Scattering theory of adiabatic reaction forces due to out-of-equilibrium quantum environments
• Geometric magnetism in open quantum systems
• Dynamics of thermalization and decoherence of a nanoscale system

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• Coarse Grained Quantum Dynamics
• Nonadiabatic Dynamics of a Dissipative Two-level System
• Divergences in open quantum systems
• Dynamic trapping near a quantum critical point
• The Moving Born-Oppenheimer Approximation
• Stochastic Thermodynamics at the Quantum-Classical Boundary: A Self-Consistent Framework Based on Adiabatic-Response Theory