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A λφ2n field theory without cutoffs

Abstract

We consider a self-interacting scalar boson field in two-dimensional space-time with self-interaction given by an arbitrary Wick polynomial of even degree in the field. It is shown that the field theory can be constructed in a Hilbert space of physical states. The hamiltonian is a positive self-adjoint operator possessing a physical vacuum. The method of proof consists of imposing and then removing three cutoffs: a box cutoff, an ultraviolet cutoff, and a space cutoff. As the first two are removed the resolvents of the cutoff hamiltonians converge uniformly and this leads to the self-adjointness of the spatially cutoff hamiltonian.

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This work was supported by the U.S. Atomic Energy Commission, Contract AT (30-1)-1480.

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Rosen, L. A λφ2n field theory without cutoffs. Commun.Math. Phys. 16, 157–183 (1970). https://doi.org/10.1007/BF01646785

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  • DOI: https://doi.org/10.1007/BF01646785

Keywords

  • Neural Network
  • Physical State
  • Hilbert Space
  • Field Theory
  • Complex System