HEP/Astro Seminar, Wednesday 29 October 1997

Quantum Spins and Quantum Links: From Antiferromagnets to QCD

Uwe-Jens Wiese (MIT)

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Abstract

Two-dimensional antiferromagnets are the precursor insulators of
high-temperature superconductors. At low temperatures $T$ the $(2+1)$-d
antiferromagnetic spin 1/2 quantum Heisenberg model reduces to the 2-d
classical $O(3)$ nonlinear $\sigma$-model, which is in many respects similar
to QCD in four dimensions. In particular, in the limit $T \rightarrow 0$ the
correlation length $\xi \propto \exp(2 \pi \rho_s/T)$ diverges due to
asymptotic freedom. Consequently, the extent $\beta=1/T$ of the Euclidean time
direction vanishes in units of $\xi$, and the system undergoes dimensional
reduction. In complete analogy, 4-d QCD is obtained via dimensional reduction
of a $(4+1)$-d quantum link model. Quantum links are the gauge analogs of
quantum spins. Like the link variables in Wilson's formulation of lattice QCD,
quantum links are $3 \times 3$-matrices. However, their elements are
non-commuting operators acting in a finite Hilbert space. The quantum link
formulation of QCD is promising both from an analytic and from a numerical
point of view. Also other field theories arise naturally from the {\em
dimensional} reduction of {\em discrete} variables. The resulting
non-perturbative {\em D-theory} formulation of quantum field theory shares
several interesting features with M-theory --- the non-perturbative formulation
of string theory.

3:30pm, Smith Lab 4079