We demonstrate the mathematical existence of a meson theory with nonrelativistic nucleons. A system of Schrödinger particles is coupled to a quantized relativistic scalar field. If a cutoff is put on the interaction, we obtain a well‐defined self‐adjoint operator. The solution of the Schrödinger equation diverges as the cutoff tends to infinity, but the divergence amounts merely to a constant infinite phase shift due to the self‐energy of the particles. In the Heisenberg picture, we obtain a solution in the limit of no cutoff. We use a canonical transformation due to Gross to separate the divergent self‐energy term. It is shown that the canonical transformation is implemented by a unitary operator, and that the transformed Hamiltonian, with an infinite constant subtracted, can be interpreted as a self‐adjoint operator.
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September 1964
Research Article|
September 01 1964
Interaction of Nonrelativistic Particles with a Quantized Scalar Field
Edward Nelson
Edward Nelson
Department of Mathematics, Princeton University, Princeton, New Jersey
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J. Math. Phys. 5, 1190–1197 (1964)
Article history
Received:
February 18 1964
Citation
Edward Nelson; Interaction of Nonrelativistic Particles with a Quantized Scalar Field. J. Math. Phys. 1 September 1964; 5 (9): 1190–1197. https://doi.org/10.1063/1.1704225
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