Exact space-time symmetry conservation and automatic mesh refinement for classical lattice field theory

Abstract

The breaking of space-time symmetries and the non-conservation of the associated Noether charges constitutes a central artifact in lattice field theory. In prior work we have shown how to overcome this limitation for classical actions describing point particle motion, using the world-line formalism of general relativity. The key is to treat coordinate maps (from an abstract parameter space into space-time) as dynamical and dependent degrees of freedom, which remain continuous after discretization of the underlying parameter space. Here we present latest results where we construct a reparameterization invariant classical action for scalar fields, which features dynamical coordinate maps. We highlight the following achievements of our approach: 1) global space-time symmetries remain intact after discretization and the associated Noether charges remain exactly preserved 2) coordinate maps adapt to the dynamics of the scalar field leading to adaptive grid resolution guided by the symmetries.

Figures (5)

• Figure 1: Lattice spacing induced symmetry breaking and its effects. See main text for discussion.
• Figure 2: Comparison of the (top) conventional formulation of field theory where fields $\phi(t,x)$ lives in space and time and discretization breaks spacetime symmetries. (bottom) Sketch of our novel formalism with dynamical coordinate maps $t(\tau,\sigma)$ and $x(\tau,\sigma)$ in addition to the fields $\phi(\tau,\sigma)$. All degreed of freedom live in an abstract parameter space, whose discretization leaves spacetime symmetries intact (from Rothkopf:2024hxi).
• Figure 3: (blue) Spectrum of the naive SBP121 finite difference operator on a $N=32$ grid. Note that two zero modes, one physical, one unphysical occur. (red) Spectrum of the regularized SBP operator is devoid of zero modes and the physical constant function is represented by the eigenvalue unity (from Rothkopf:2022zfb).
• Figure 4: (left) Scalar wave propagation from the critical point of the IBVP action based on \ref{['eq:bvpEdisc']} on a $N_\sigma=48$ and $N_\tau=60$ grid. (right) $\tau$ derivative $\dot t_{\rm cl}[\tau,\sigma]$ showing adaptive mesh refinement (from Rothkopf:2024hxi).
• Figure 5: Exact conservation of the Noether charge associated with time translation symmetry (from Rothkopf:2024hxi).