Let Q be the phase space of the Yang-Mils field. We introduce a dynamical system with a quadratic Hamiltonian H defined on a neighbourhood U of the Yang-Mills zero energy point in a certain subspace P of Q. At low energies the Yang-Mills Hamiltonian restricted to U can be regarded as a perturbation of H. The definitions of H and of P use the Morse-Palais lemma applied to the potential energy term of the Yang-Mills Hamiltonian. This forces to introduce a certain cut-off parameter m in the definition of P. As a consequence, the secondary quantized Hamiltonian H has mass gap m which suggests that the quantization of the restriction of the Yang-Mills Hamiltonian to P should have mass gap m and should provide a proper gauge invariant quantum system describing massive quantum fields.

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