In this talk, I intend to show a new formalism that should solve the famous many-body problem of quantum mechanics with a complexity varying as a function of n⁴ where n is the finite number of states. There was already an object, the reduced two-body density matrix (2-RDM) that had this complexity but for it to be a quantum state with N electrons, a gigantic number of conditions must be taken into account.

This formalism should considerably reduce the computation time avoiding the exponential complexity of the current exact methods. One can also ask the question of the interest of quantum computers if n⁴ parameters suffice to describe an intricate quantum experimental state.

Based on anti-commutativity relations, we show that we can construct a class of mathematical objects that are isomorphic to many-body wavefunctions but have the advantage of being compactable. Thanks to this new formalism, we show that the 2-RDM of the wavefunctions solutions of a two-body interaction Hamiltonian can be represented exactly thanks to this new set of mathematical objects. We will give also a new geometric interpretation of the 2-RDM in the space of these new mathematical objects.

We will develop the Lagrangian, the corresponding Euler-Lagrange equations and illustrate with numerical examples.