Finite coercive regulated polynomial model

Fix a finite integer $r\ge 1$, positive coefficients $a_j$, a real polynomial $V$ on ${\mathbb{R}}^r$, and constants $c>0,C_0\ge 0$ such that $V(q)\ge c\mathrm{\mid }q{\mathrm{\mid }}^2-C_0$. Use the QFT supplier's form realization of Finite-lattice polynomial Hamiltonian data and Self-adjoint unbounded polynomial interaction at fixed regulator on $L^2({\mathbb{R}}^r,dq)$, and denote its self-adjoint operator by $h$. This item's regulated polynomial model is the tuple of these finite data, the selected form realization, and an energy scale $E_{\ast }>0$ joules, with $H=E_{\ast }h$. The coordinate $q$ and coefficients are dimensionless rescaled model parameters unless another dimensional map is explicitly supplied. The class includes $V(q)=\mathrm{\mid }q{\mathrm{\mid }}^2$ as well as potentials with higher-degree terms or terms coupling coordinates; neither higher-degree terms nor coupling is required. No limit in $r$ is included.

For each $b>0$, Finite-regulator Gibbs trace and bounded expectations proves $\mathrm{Tr}(e^{-bh})<\mathrm{\infty }$ and positivity. Thus for physical $\beta >0$ in joules${}^{-1}$, $b=\beta E_{\ast }>0$ and ${\rho }_{\beta }=\frac{e^{-\beta H}}{Z_{\beta }},Z_{\beta }=\mathrm{Tr}(e^{-\beta H})\in (0,\mathrm{\infty })$ is a trace-one positive trace-class density. Its interpretation as a normal quantum measurement state uses the current batch-36 draft carrier Normal-state quantum measurement interpretation, which adopts a trace-class density on a specified separable Hilbert space and the PVM Born law ${\mathbb{P}}_{\rho }\{A\in C\}=\mathrm{Tr}(\rho E_A(C))$. This interface matches the mathematical state constructed here. The mathematical probability and bounded trace-pairing inputs now have exact canonical suppliers through the trace-state calculus. The normal-state interpretation remains an adopted physical premise; source adoption is separate from native readiness and empirical validation. The QFT mathematical trace theorem remains the mathematical supplier.

This selected finite-regulator realization does not assert essential self-adjointness of the bare compact-support differential expression, a unique realization among all self-adjoint extensions, regulator removal, or an interacting continuum theory.