Elimination of divergence for the problem of a particle in a scalar quantum field
Abstract
The problem of the interaction of a particle with a scalar quantum field is considered. The use of perturbation theory in this problem leads to ultraviolet divergence in the calculation of the ground state energy, for the renormalisation of which it is necessary to use an indefinite parameter – momentum cutoff. The work describes an iteration scheme for calculating the observed characteristics of the system, which allows to go beyond the perturbation theory. The dependence of the ground state energy on the coupling constant was found and it is shown that it does not contain divergence, but it has a logarithmic singularity in the limit, when the coupling constant of the particle with the field tends to zero. Such a function cannot be represented as a power series over the coupling constant, which explains the inapplicability of the standard perturbation theory. The result obtained is of fundamental importance for quantum field theory, since it shows that the momentum cutoff, which is used for renormalisation when calculating physical quantities, is determined by the parameters of the system, and the divergences are due to the presence of a singularity in the dependence of these quantities on the coupling constant.
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