Quantum Yang-Mills theory: An overview of a program

Abstract

We present an overview of a programme to understand the low-energy physics of quantum Yang-Mills theory from a quantum-information perspective. Our setting is that of the hamiltonian formulation of pure Yang-Mills theory in the temporal gauge on the lattice. Firstly, inspired by recent constructions for Z/2Z lattice gauge theory, in particular, Kitaev's toric code, we describe the gauge-invariant sector of hilbert space by introducing a primitive quantum gate: the quantum parallel transporter. We then develop a nonabelian generalisation of laplace interpolation to present an ansatz for the ground state of pure Yang-Mills theory which interpolates between the weak-and strong-coupling RG fixed points. The resulting state acquires the structure of a tensor network, namely, a multiscale entanglement renormalisation ansatz, and allows for the efficient computation of local observables and Wilson loops. Various refinements of the tensor network are discussed leading to several generalisations. Finally, the continuum limit of our ansatz as the lattice regulator is removed is then described. This paper is intended as an abstract for an ongoing programme: there are still many open problems.

Fusion basis for lattice gauge theory and loop quantum gravity

“…Firstly, we will discuss how to use the fusion basis to easily design multi-scale states. It is interesting to compare the tools developed here to the closely related philosophy underlying the introduction of tensor network states, which provide an Ansatz for the ground state of Yang-Mills theories [21]. 5 Secondly, using the muli-scale states, we describe a coarse-graining scheme based on the fusion basis.…”

“…We expect that this will help to design low-energy states for Yang-Mills (lattice) theory, by merging our tools with the techniques developed for this purpose in the context of tensor network states or MERA (multi-scale entanglement renormalization Ansatz, [63]), see [21] and also [64,65] for some recent developments.…”

“…21 We then demand that the lengthwise product should be such that it does not induce any excitation at the 'middle' puncture p 2 . And hence that this product in fact coincides with some (not self-crossing) ribbon operator along = 2 • 1 , directly going from p 1 to p 3 .…”

“…To avoid this, we can first (unitarily) transform the state onto a lattice where all the finer links one wishes to integrate out are given by loops, see e.g. [12,21]. (In [21] such transformations are called 'controlled rotation unitary gates'.)…”

“…21 Later, we will define closed ribbon operators that project onto wave functions with prescribed charges at a given puncture.…”