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Matter Spin Networks and Gauge Invariance at Planck Scales
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DOI：10.1007/s10773-026-06304-6.png)
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Abstract
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Local gauge invariance is a pivot of modern physics of elementary particles. Standard model of elementary particles is a non-abelian gauge theory with the gauge group G=SU(3)circle times SU(2)circle times U(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{G=SU(3)\otimes SU(2)\otimes U(1)}$$\end{document}, that is the theory is constructed to be invariant under a gauge transformation psi(x)-> U(x)psi(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\psi (x) \rightarrow U(x) \psi (x)}$$\end{document}, where U(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{U(x)}$$\end{document} is a representation of G acting on matter fields psi(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\psi (x)}$$\end{document} at a point x is an element of M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{x\in \mathcal {M}}$$\end{document} of some continuous differentiable manifold. The problem with extrapolation of gauge theories down to the Planck scales is that there is no continuous differentiable manifold in quantum gravity regime, and hence, neither the value of field at a point, nor a covariant derivative are defined in a usual way. We do such extrapolation assuming that our spacetime is a combinatorial spacetime, the points of which are the interaction vertices of matter fields. The vertices are connected to each other by matter particles. The gauge invariance then becomes a symmetry with respect to independent internal rotations of all fermion lines in the edges of the matter spin network graph. There is no need to introduce special gauge fields in this picture, since they are expressed in terms of fermion fields.
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Quantum gravity
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