Gauge theories, which describe forces and particles in terms of fields, are the foundation of modern particle physics. Different configurations of these fields can have no impact on related observable quantities, a behavior known as gauge invariance. Ensuring gauge invariance often requires additional independent variables. These degrees of freedom, however, are often partially redundant, rendering gauge theories notoriously difficult. What’s more, only the subset of gauge-invariant states is physically relevant. For the simplest case of one temporal and one spatial dimension, the gauge degrees of freedom are not truly independent and can, in principle, be integrated out, making this situation simpler to work with. Despite this long known fact, it has been used in practice only for Abelian theories, the simplest case in which gauge fields interact with matter but not among themselves. More complex, but also more interesting, is the non-Abelian scenario, which underlies many theories describing particle physics and fundamental forces. Here we show how the integration of gauge degrees of freedom in one spatial dimension can actually be extended to the lattice formulation of a non-Abelian SU(2) gauge theory.

We develop a basis for the physical subspace that is completely general and can be used with any numerical or analytical method. This formulation is especially well suited to address the model with tensor networks, a tool originally developed in the context of quantum information theory. Using this method, we numerically explore the low-lying spectrum of the model and the effect of truncating the gauge degrees of freedom to a small finite dimension.

This approach allows us to study the entanglement entropy in the ground state and its scaling towards the continuum limit, opening up new possibilities to characterize lattice gauge theories. Additionally, our formulation might be suited for designing quantum simulators.