Borel Summability of the Ground-State Energy in Spatially Cutoff<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>

Abstract

We show that the ground-state energy for a Hamiltonian ${H}_{0}+\ensuremath{\int}g(x):{\ensuremath{\phi}}^{4}(x):dx$ ($g\ensuremath{\in}{L}^{1}\ensuremath{\cap}{L}^{2}$; $g>~0$; ${H}_{0}=\mathrm{free}$ Hamiltonian for Bose particle of mass $m$ in space-time of two dimensions) may be determined from the Feynman perturbation series by the method of Borel summability. This demonstrates that summability methods can be applicable to divergent series in systems with a continuous infinity of degrees of freedom.

Locations

• Physical Review Letters - View
• CaltechAUTHORS (California Institute of Technology) - View - PDF