We propose a new framework for matrix theories that are equivalent to field theories on a toroidal spacetime. The correspondence is accomplished via infinite Toeplitz matrices whose entries match the field degrees of freedom on an energy-momentum lattice, thereby replacing the background geometry with matrix indices. These matrix theories can then be embedded into the purely cubic action of a single matrix and combined into a common universality class. We reconstruct the Standard Model action in this framework and discuss its extensions within the same class.

We consider the possibility that, in the semiclassical Einstein equation for cosmological spacetimes, gravity is sourced by the amount of stress-energy that is above that of the instantaneous ground state. For this possibility to be consistent, the Bianchi identities must continue to hold. This is nontrivial because it means that the ground state expectation value of the stress-energy tensor must be covariantly conserved in spite of the fact that the ground state is generally a different state at different times. We prove that this consistency condition does hold. As a consequence, we find that the vacuum stress-energy which is above the instantaneous ground state does not renormalize the cosmological constant, as long as the instantaneous ground states and the instantaneous adiabatic vacua exist.

The modular spaces are a family of polarizations of the Hilbert space that are based on Aharonov's modular variables and carry a rich geometric structure. We construct here, step by step, a Feynman path integral for the quantum harmonic oscillator in a modular polarization. This modular path integral is endowed with novel features such as a new action, winding modes, and an Aharonov-Bohm phase. Its saddle points are sequences of superposition states and they carry a non-classical concept of locality in alignment with the understanding of quantum reference frames. The action found in the modular path integral can be understood as living on a compact phase space and it possesses a new set of symmetries. Finally, we propose a prescription analogous to the Legendre transform, which can be applied generally to the Hamiltonian of a variety of physical systems to produce similar modular actions.

Vaccinating individuals with more exposure to others can be disproportionately effective, in theory, but identifying these individuals is difficult and has long prevented implementation of such strategies. Here, we propose how the technology underlying digital contact tracing could be harnessed to boost vaccine coverage among these individuals. In order to assess the impact of this “hot-spotting” proposal we model the spread of disease using percolation theory, a collection of analytical techniques from statistical physics. Furthermore, we introduce a novel measure which we call the efficiency, defined as the percentage decrease in the reproduction number per percentage of the population vaccinated. We find that optimal implementations of the proposal can achieve herd immunity with as little as half as many vaccine doses as a non-targeted strategy, and is attractive even for relatively low rates of app usage.