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I am reading "Gauging What's Real" by Richard Healey and the author argues for formulating electrodynamics/QED on loop space/the holonomy group, so that the real objects described by the theory are the Wilson Loops or Dirac phase factors ${ e }^{ \frac { ie }{ \hbar } \oint _{ \gamma }^{ }{ { A }_{ \mu }d{ x }^{ \mu } } }$.

I was wondering if it is possible to formulate Quantum Field Theory (or maybe even Quantum Mechanics) on loop spaces, so that the quantum field/ wave functions take loops as arguments. (I am especially interested if one could formulate Quantum Mechanics in this way).

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  • $\begingroup$ You can pick up any book on loop quantum gravity, which would have an introduction on loop representation of gauge theories. $\endgroup$
    – MadMax
    Jan 25, 2021 at 21:42

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Quantum mechanics on the loop space is a rather convenient way to think about 1+1d sigma models. It's one way to understand the Kac-Moody symmetry of WZW CFTs for example, whose field is a map from spacetime to a Lie group. Witten famously used SUSY quantum mechanics on loop space to give a formulation of Floer theory of symplectic manifolds.

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  • $\begingroup$ Thanks for the answer. The paper seems quite technical. Is there an easy way to write e.g. a wave function $\Psi(x,t)$ depending on space and time coordinates as a wave function depending on" loop coordinates" $\Psi (\gamma , t)$ $\endgroup$
    – NicAG
    Jan 26, 2021 at 11:02

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