Suppose that we have a system that can be described by $N$ (is a large number!) degree of freedom besides well-known degree of freedom like energy, momentum, spin, etc.. Now scattering processes are invariant under exchanges in these extra degree of freedom; then a $SU(N)$ gauge Group acts locally, such that an $SU(N)$ Yang-Mills theory arises. Can this Yang-Mills theory describe collective phenomena between even a huge number of particles? (I ask that question because I have seen a Statement about $SU(N)$ for large $N$ at the 8th page of http://people.brandeis.edu/~headrick/talks/EntanglementGeometry.pdf)

My ideas:

Suppose that a fermion state $\left|k_\mu,s\right>$ arose causally from the $n$-th Fermion of the same Kind with probability Amplitude $\left|k_\mu,s,n\right>$. If this state was generated causally by all of the Fermions, then we can express it as a linear combination

$$\left|k_\mu,s\right> = \sum_{n=1}^N \alpha_n \left|k_\mu,s,n\right>$$

with specific causal amplitudes $\alpha_n$. Now it is known that we do not know how much this state was contributed by the other Fermions (undistinguishable particles) and therefore we have a symmetry by unitary changing in the basis $\left|\dots,n\right>$. Therefore we have a gauge field that interacts with all of the Fermions, there are $N^2-1$ different Generators of the Group and hence even if there is a simple 2-particle interaction, some aspects of Fermions far apart are taken into account due to high number of gauge field degree of freedom. In some cases, collective behavior can occur. But how does collective behavior between a large number of particles emerge?

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    $\begingroup$ I'm not quite sure what your question is asking, but I can recommend a classic review by Witten that discusses the simplifications that arise in the large-N limit of gauge theories, which may be useful to you. $\endgroup$ Nov 29 '17 at 11:23

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