Analytic domination with quadratic form type estimates and nondegeneracy of ground states in quantum field theory

Abstract:

We present a theorem concerning the analytic domination by a semi-bounded selfadjoint operator H of another linear operator A which requires only the quadratic form type estimates \[ \left \| {{H^{ - 1/2}}({{({\text {ad}}\;A)}^n}H){H^{ - 1/2}}u} \right \| \leq {c_n}\left \| u \right \|\] instead of the norm estimates \[ \left \| {{{({\text {ad}}\;A)}^n}Hu} \right \| \leq {c_n}\left \| {Hu} \right \|\] usually required for this type of theorem. We call the new estimates “quadratic form type", since they are sometimes equivalent to \[ |({({\text {ad}}\;A)^n}Hu,u)| \leq {c_n}|(Hu,u)|.\] The theorem is then applied with H the Hamiltonian for the spatially cutoff boson field model with real, bounded below, even ordered polynomial self-interaction in one space dimension and $A = \pi (g)$, the conjugate momentum to the free field. When the underlying Hilbert space of this model is represented as ${L^2}(Q,dq)$ where dq is a probability measure on Q, the spectrum of the von Neumann algebra generated by bounded functions of certain field operators, then ${e^{ - tH}}$ maximizes support in the sense that ${e^{ - tH}}f$ is nonzero almost everywhere whenever f is not identically zero.