Wavelet-Based Representations of Quantum Field Theory*

Here we present current results from our investigations into wavelet-based representations of quantum field theory. Specifically, we develop representations of one-dimensional free field theories for fermions and scalar bosons using the Daubechies wavelets, which are desirable due to their compact support and vanishing moments. We reproduce entanglement area laws with a resolution-dependent cutoff and generalize to fractal sets.

The ground states of these one-dimensional free field theories have a holographic dual representation in terms of multiscale wavelet degrees of freedom. We show how an emergent geometry can be inferred from the scaling of mutual information between wavelet degrees of freedom in the bulk. At the critical point, the bulk has an anti-de-Sitter geometry with radius of curvature that depends on the Daubechies wavelet index.

Our work has implications for resource-theory-based approaches to quantum field theory as well as applications to the development of quantum algorithms for simulating quantum field theory.