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.
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