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I'm wondering what the statement is that one has to prove for the Millenium Problem "Quantum Yang-Mills Theory".

According to the official article, it is required to show that for every simple Lie group G there exists a YM quantum field theory for G with a mass gap. Finding a quantum field theory amounts to finding a Hilbert space H, a representation of the restricted Lorentz group by unitary transformations of H and operator valued tempered distributions $\varphi_1,...,\varphi_m$ which satisfy density conditions, transformation properties under the Poincare-group, (anti-)commutativity of field operators for test functions of space-like separated supports, an asymptotic completeness property and existence of a unique vacuum state, the field operators acting on this vacuum state span (a dense subspace of) H.

A quantum field theory has a mass gap if the spectrum of the energy operator is contained in ${0} \cup [a,\infty)$ for $a>0$.

Now these are straightforward defninitionsdefinitions, but what turns a quantum field theory into a YM theory for a group? I know of classical YM theory which gives a Lagrangian and thus equations of motion for the curvature components (fields). Does that mean that $\varphi_1,...,\varphi_n$ have to satisfy these equations? In what way? I could not find any reference for this. Is this millenium problem a mathematical statement at all?

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# Statement of Millenium Problem: Yang-Mills Theory and Mass Gap

I'm wondering what the statement is that one has to prove for the Millenium Problem "Quantum Yang-Mills Theory".

According to the official article, it is required to show that for every simple Lie group G there exists a YM quantum field theory for G with a mass gap. Finding a quantum field theory amounts to finding a Hilbert space H, a representation of the restricted Lorentz group by unitary transformations of H and operator valued tempered distributions $\varphi_1,...,\varphi_m$ which satisfy density conditions, transformation properties under the Poincare-group, (anti-)commutativity of field operators for test functions of space-like separated supports, an asymptotic completeness property and existence of a unique vacuum state, the field operators acting on this vacuum state span (a dense subspace of) H.

A quantum field theory has a mass gap if the spectrum of the energy operator is contained in ${0} \cup [a,\infty)$ for $a>0$.

Now these are straightforward defninitions, but what turns a quantum field theory into a YM theory for a group? I know of classical YM theory which gives a Lagrangian and thus equations of motion for the curvature components (fields). Does that mean that $\varphi_1,...,\varphi_n$ have to satisfy these equations? In what way? I could not find any reference for this. Is this millenium problem a mathematical statement at all?