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Kenneth Wilson and Lattice QCD

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Abstract

We discuss the physics and computation of lattice QCD, a space-time lattice formulation of quantum chromodynamics, and Kenneth Wilson’s seminal role in its development. We start with the fundamental issue of confinement of quarks in the theory of the strong interactions, and discuss how lattice QCD provides a framework for understanding this phenomenon. A conceptual issue with lattice QCD is a conflict of space-time lattice with chiral symmetry of quarks. We discuss how this problem is resolved. Since lattice QCD is a non-linear quantum dynamical system with infinite degrees of freedom, quantities which are analytically calculable are limited. On the other hand, it provides an ideal case of massively parallel numerical computations. We review the long and distinguished history of parallel-architecture supercomputers designed and built for lattice QCD. We discuss algorithmic developments, in particular the difficulties posed by the fermionic nature of quarks, and their resolution. The triad of efforts toward better understanding of physics, better algorithms, and more powerful supercomputers have produced major breakthroughs in our understanding of the strong interactions. We review the salient results of this effort in understanding the hadron spectrum, the Cabibbo–Kobayashi–Maskawa matrix elements and CP violation, and quark-gluon plasma at high temperatures. We conclude with a brief summary and a future perspective.

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Notes

  1. Contemporary research toward lattice gauge theory was described by Wilson in his plenary talk at 1983 Lepton Photon Symposium [2], and in more detail in his historical talk at 2004 International Symposium on Lattice Field Theory at FNAL [3].

  2. Probably his most famous numerical work is a renormalization group solution of the Kondo problem [57]. The numerical rigor he maintained for this work has become legendary. For the universal ratio of the two temperatures \(T_K\) and \(T_0\) characterizing the high and low temperature scales, he obtained \(W/(4\pi )=(4\pi )^{-1}T_K/T_0=0.1032(5)\). Six years later, an exact solution by the Bethe ansatz yielded \(W/(4\pi )=0.102676\ldots \) [58], verifying the Wilson’s number to the fourth digit within the estimated error!

  3. Strictly speaking, this identity requires positivity of the Hermitian part of matrix \(D\). We shall not go into this technical detail and mention only that this can be guaranteed.

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Acknowledgments

I would like to thank Sinya Aoki, Norman Christ, Carleton De Tar, Zoltan Fodor, Shoji Hashimoto, Yoichi Iwasaki, Kazuyuki Kanaya, Frithjof Karsch, Andreas Kronfeld, Martin Lüscher, and Paul Mackenzie for valuable comments on the manuscript.

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    Akira Ukawa

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Ukawa, A. Kenneth Wilson and Lattice QCD. J Stat Phys 160, 1081–1124 (2015). https://doi.org/10.1007/s10955-015-1197-x

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