The path-ordered exponential from which the Wilson loop is traced is, crudely,

$$ \prod (I+ A_\alpha dx^\alpha) = \mathcal{P}\,\mathrm{exp}(i \oint A_\alpha dx^\alpha )$$

which returns a matrix $\mathbf{W}$ in the Lie group in question. (This should be the Aharonov-Bohm phase for U(1).) Since this applies to closed loops, it equivalently maps surfaces to Lie group elements based only on their boundaries, and so $\mathbf{W}$ is a conserved quantity with a net flux of 0 through a closed surface (the limit of an infinitely small loop).

But this looks suspiciously like the derivation of the Faraday 2-form, where $ \mathbf{F} = \mathrm{d}\mathbf{A} $ (implying a loop-integral relationship between $\mathbf{A}$ and $\mathbf{F}$) and so $\mathrm{d}\mathbf{F}=\mathrm{d}^2\mathbf{A}=0$ (conservation of the loop flux), and thus $\mathrm{d}{\star}\mathbf{F}=\mathbf{J}$ (4-current is the dual of the loop flux) and $\mathrm{d}^2{\star}\mathbf{F}=\mathrm{d}\mathbf{J}=0$ (conservation of that dual).

Question: Is the POE value $\mathbf{W}$ the field value? Could it be treated as a 2-form? Does its dual correspond to a useful 4-current / Could you work out another conserved quantity (as $\mathrm{d}^2{\star}\mathbf{F}=\mathrm{d}\mathbf{J}=0$) from its dual?

Follow-up Question: In lattice QFT, do the links always have to have values $ M = I + i A_{\alpha} \delta x^\alpha$ (corresponding to the differential segments of the path-ordered exponential)?

Follow-up Question 2: What relationship is impled between 2 nodes separated by a path of links whose integral equals some Lie group element $g$? Say for SU(3) whose nodes are in $C^3$; are the 2 node values related by $g$?

(Plz don't bite the newcomer.)

  • 1
    $\begingroup$ Recall that the matrix-valued Wilson loop in (non-abelian) lattice QFT is not gauge invariant, so calling it "a conserved quantity" seems suspect. (Its trace is the gauge-invariant object.) The form $M = I + igA_{\alpha} \delta x^{\alpha} + \dots$ is needed in order for typical lattice actions based on Wilson loops to correctly reproduce tr$F_{\mu\nu} F^{\mu\nu}$ in the continuum limit. It might be possible to design alternative lattice actions that don't require links in this form, but I'm not aware of anything like that. Your 2-form looks abelian; should it have a wedge-product term? $\endgroup$ – David Schaich Dec 15 '18 at 13:48

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