For negative coupling, the renormalized non-relativistic and semi-relativistic Nelson semigroups are positivity improving with respect to the Fröhlich cone at every total momentum, proven via Feynman-Kac functional integration.
On the semigroup generated by the renormalized Nelson Hamiltonian
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abstract
Let us consider the renormalized Nelson model at a fixed total momentum $P$: $H_{\mathrm{ren}}(P)$; The Hamiltonian $H_{\mathrm{ren}}(P)$ is defined through an infinite energy renormalization. We prove that $e^{-\beta H_{\mathrm{ren}}(P)}$ is positivity improving for all $P\in \mathbb{R}^3$ and $\beta >0$ in the Fock representation.
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On the Ergodicity of Renormalized Translation-Invariant Nelson-Type Semigroups
For negative coupling, the renormalized non-relativistic and semi-relativistic Nelson semigroups are positivity improving with respect to the Fröhlich cone at every total momentum, proven via Feynman-Kac functional integration.