Counting Degrees of Freedom: A Method Applicable from Scalars to f(Q) Gravity and Beyond

Abstract

We present a clear, step-by-step method for counting degrees of freedom and identifying constraints in general field theories. This approach, grounded in the works of Einstein, Hilbert, Cartan, Kuranishi, and, more recently, Seiler, is neither Lagrangian nor Hamiltonian in nature. Instead, it applies directly to the field equations. We offer a transparent physical interpretation of the method, establish new results, and explore a broad set of examples to demonstrate its power and generality. Notably, we apply the method to $f(\mathbb{Q})$ gravity, where traditional techniques such as the Dirac-Bergmann algorithm are ineffective, and obtain seven propagating degrees of freedom.