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Interacting fermions on non-commutative spaces: Exactly solvable quantum field theories in \(2n+1\) dimensions. (English) Zbl 1010.81082

Summary: The author presents 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 by the Groenewold-Moyal star product. Using Hamiltonian quantization and a suitable regularization, he shows that these models have a dynamical symmetry corresponding to \(\text{gl}_{\infty}\oplus\text{gl}_{\infty}\) at the special points \(B\theta=I\) and \(B\theta=-I\), where \(B\) and \(\theta\) are the matrices defining the magnetic field and the star product, respectively. He constructs all eigenvalues and eigenstates of the many-body Hamiltonian at these special points. The author argues that this solution cannot be obtained by any mean-field theory, i.e. the models describe correlated fermions. He also mentions other possible interpretations of these models in solid state physics.

MSC:

81T75 Noncommutative geometry methods in quantum field theory
81S10 Geometry and quantization, symplectic methods
81V70 Many-body theory; quantum Hall effect
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References:

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