Interacting fermions on non-commutative spaces: Exactly solvable quantum field theories in 2n+1 dimensions

• I present a novel class of exactly solvable quantum field theories. They describe non-relativistic fermions on even-dimensional flat space, coupled to a constant external magnetic field and a four point interaction defined with the Groenewold-Moyal star product. Using Hamiltonian quantization and a suitable regularization, I show that these models have a dynamical symmetry corresponding to gl(infinity) circle plus gl(infinity) at the special points Btheta = I and Btheta = -I, where B and theta are the matrices defining the magnetic field and the star product, respectively. I construct all eigenvalues and eigenstates, of the many-body Hamiltonian at these special points. I argue that this solution cannot be obtained by any mean-field theory, i.e., the models describe correlated fermions. I also mention other possible interpretations of these models in solid state physics.

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