Brief introduction to Wilson loops and large N
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EMENTARY PARTICLES AND FIELDS Theory
Brief Introduction to Wilson Loops and Large N Yu. М. Makeenko* Institute of Theoretical and Experimental Physics, Bol’shaya Cheremushkinskaya ul. 25, Moscow, 117218 Russia Received June 29, 2009
Abstract—A brief pedagogical introduction to Wilson loops, lattice gauge theory, and 1/N expansion in QCD is presented. DOI: 10.1134/S106377881005011X
1. INTRODUCTION
Under the gauge transformation
A brief pedagogical introduction to methods used in nonperturbative studies within QCD and within other gauge theories is given. Attention is given primarily to Wilson loops (both on a lattice and in a continuum limit), which play the most important role in modern formulations of gauge theories, and to the 1/N -expansion method. For a more comprehensive treatment of this subject, the interested reader is referred to the textbook of the present author [1], where a detailed list of references can be found. In the present review, I refer to only some classic studies.
1 g.t. Aμ (z) −→ Aμ (z) + ∂μ α(z), e the Abelian phase factor transforms as g.t.
U [Γyx ] −→ eiα(y) U [Γyx ]e−iα(x) .
g.t.
ϕ(x) −→ eiα(x) ϕ(x);
The Abelian phase factor is defined as ⎡ ⎤ ⎢ ⎥ dz μ Aμ (z)⎦ . U [Γyx ] = exp ⎣ie Γyx *
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(1)
(4)
therefore, the phase factor transforms as the product ϕ(y)ϕ† (x); that is, g.t.
2.1. Phase Factors in QED
(3)
Accordingly, a wave function at a point x transforms as
2. WILSON LOOPS Wilson loops are essentially phase factors in Abelian or non-Abelian gauge theories. Wilson loops are observable in the quantum theory (in contrast to the case of classical theory, where only electric- and magnetic-field strengths are observable) owing to the Aharonov–Bohm effect. Wilson loops play a key role in the lattice formulation of gauge theories. Quantum chromodynamics (QCD) can be exhaustively reformulated in terms of Wilson loops in a manifest gaugeinvariant way. It appears that analogs of Wilson loops are of great use in solving various matrix models.
(2)
U [Γyx ] ∼ “ϕ(y)ϕ† (x)”.
(5)
Upon the multiplication by the phase factor, the wave function at the point x transforms as the wave function at the point y: g.t.
U [Γyx ]ϕ(x) ∼ “ϕ(y)”.
(6)
Similarly, we have g.t.
ϕ† (y)U [Γyx ] ∼ “ϕ† (x)”.
(7)
The phase factor plays the role of a parallel transporter in an electromagnetic field; in order to compare the phases of a wave function at points x and y, we must first make a parallel transportation along some contour Γyx from x to y. The result depends on the shape of the contour always, with the exception of the case where Aμ (z) is a pure gauge (that is, where the field strength Fμν (z) vanishes). For spaces that are not simply connected, there are some subtleties (Aharonov–Bohm effect). 878
BRIEF INTRODUCTION TO WILSON LOOPS
2.2. Propagators in an External Fields Let us consider a quantum particle in a classical electromagnetic field. To introduce an electromagnetic field, the derivative ∂μ must be replaced by the covariant derivative (8) ∂μ −→ ∇μ = ∂μ − ieAμ (x) . For
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