Action-Driven Flows for Causal Variational Principles
Felix Finster, Franz Gmeineder
TL;DR
This work presents action-driven, non-smooth flows for causal variational principles by extending De Giorgi's minimizing movements to a non-convex, measure-valued setting. To overcome non-convergence in the non-convex case, the authors introduce a penalization parameter $\xi>0$ that yields convergent, Hölder continuous measure flows and allows for quantitative control of the flow length, with explicit bounds tied to the initial action. The framework is developed first in the compact setting, then generalized to finite-dimensional causal action principles, using moment measures to manage non-compactness and enabling a causal fermion systems interpretation. In the infinite-dimensional regime, a filtration by finite-dimensional subspaces provides a constructive pathway to a global flow that monotonically decreases the causal action and yields (approximately) satisfying Euler–Lagrange equations, highlighting potential numerical and analytical applications in fundamental physics.
Abstract
We introduce action-driven flows for causal variational principles, being a class of non-convex variational problems emanating from applications in fundamental physics. In the compact setting, Hölder continuous curves of measures are constructed by using the method of minimizing movements. As is illustrated in examples, these curves will in general not have a limit point, due to the non-convexity of the action. This leads us to introducing a novel penalization which ensures the existence of a limit point, giving rise to approximate solutions of the Euler-Lagrange equations. The methods and results are adapted and generalized to the causal action principle in the finite-dimensional case. As an application, we construct a flow of measures for causal fermion systems in the infinite-dimensional situation.