D-theory: field theory via dimensional reduction of discrete variables

Abstract

A new non-perturbative approach to quantum field theory -D-theory -is proposed, in which continuous classical fields are replaced by discrete quantized variables which undergo dimensional reduction. The 2-d classical O(3) model emerges from the (2 + 1)-d quantum Heisenberg model formulated in terms of quantum spins. Dimensional reduction is demonstrated explicitly by simulating correlation lengths up to 350,000 lattice spacings using a loop cluster algorithm. In the framework of D-theory, gauge theories are formulated in terms of quantum links -the gauge analogs of quantum spins. Quantum links are parallel transporter matrices whose elements are non-commuting operators. They can be expressed as bilinears of anticommuting fermion constituents. In quantum link models dimensional reduction to four dimensions occurs, due to the presence of a 5-d Coulomb phase, whose existence is confirmed by detailed simulations using standard lattice gauge theory. Using Shamir's variant of Kaplan's fermion proposal, in quantum link QCD quarks appear as edge states of a 5-d slab. This naturally protects their chiral symmetries without fine-tuning. The first efficient cluster algorithm for a gauge theory with a continuous gauge group is formulated for the U (1) quantum link model. Improved estimators for Wilson loops are constructed, and dimensional reduction to ordinary lattice QED is verified numerically.

Meron-cluster simulation of a chiral phase transition with staggered fermions

“…A natural arena for applications of the meron-cluster algorithm are quantum link models [14] which are used in the D-theory formulation of QCD [15,16,17]. In D-theory continuous classical fields arise from the dimensional reduction of discrete quantum variables.…”

Green’s functions from quantum cluster algorithms

“…At present, this is used to investigate the physics of the U(1) quantum link model. The results of this study will be published elsewhere [8].…”

QCD as a quantum link model

“…Due to the discrete nature of the Hilbert space, one can even work directly in the continuum for the extra Euclidean direction [11]. Indeed for a U(1) quantum link model a cluster algorithm operating in the continuum has already been constructed [18]. It is plausible that cluster algorithms can also be constructed for non-Abelian quantum link models, which would allow high-precision simulations of gauge theories.…”