Adopted finite-regulator polynomial model

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) as in Finite-lattice polynomial Hamiltonian data; it supplies the stated Hilbert-space construction and is not a physical measurement assumption. Fix one finite nonempty label set $J$ with $N:=\mathrm{\mid }J\mathrm{\mid }\ge 1$, an enumeration of $J$ by $\{1,\dots ,N\}$, and dimensionless real coordinates $q=(q_1,\dots ,q_N)\in {\mathbb{R}}^N$. Each label indexes one coordinate; no spatial geometry or limit in $N$ is part of these data. The coordinate space, Hilbert space, and polynomial differential expression are those of Finite-lattice polynomial Hamiltonian data: $H_N=L^2({\mathbb{R}}^N,dq;\mathbb{C})$, with inner product linear in its first variable, $D_0=C_c^{\mathrm{\infty }}({\mathbb{R}}^N;\mathbb{C})$, and

$h_{0}u=-{\sum }_{j=1}^N{a}_j{\mathrm{\partial }}_j^{2}u+V(q)u,u\in D_0,$

where $a_j>0$ and $V$ is a real polynomial. Require fixed constants $c>0$ and $C_0\ge 0$ such that

$V(q)\ge c\mathrm{\mid }q{\mathrm{\mid }}^2-C_0(q\in {\mathbb{R}}^N)\mathrm{.}$

For $j=1,\dots ,N$ define dimensionless linear maps $Q_j,P_j:D_0\to D_0$ by $Q_{j}u(q):=q_{j}u(q),P_{j}u(q):=-i{\mathrm{\partial }}_{j}u(q)\mathrm{.}$ For maps preserving $D_0$, write $[A,B]u:=A(Bu)-B(Au)$. For every $u\in D_0$ these maps obey $[Q_i,Q_j]u=0,[P_i,P_j]u=0,[Q_i,P_j]u=i{\delta }_{ij}u,$ where ${\delta }_{ij}=1$ if $i=j$ and $0$ otherwise. In this finite-coordinate model, the mixed identity follows from $-i{q}_i{\mathrm{\partial }}_{j}u+i{\mathrm{\partial }}_j(q_{i}u)=i{\delta }_{ij}u$; the other two vanish by commutativity of coordinate multiplication and mixed derivatives. A bosonic mode means one such coordinate-momentum pair with these commutation relations on $D_0$; this names the algebraic commutation convention and adds no state-space construction beyond $H_N$. A polynomial interaction term means a monomial of total degree at least three in $V$; the model is interacting in this finite-coordinate sense exactly when at least one such term has nonzero coefficient. This terminology asserts no continuum field interaction or scattering construction.

Define $V(Q):D_0\to D_0$ by $(V(Q)u)(q):=V(q)u(q)$. Then $h_{0}u={\sum }_{j=1}^N{a}_j{P}_j^{2}u+V(Q)u(u\in D_0)\mathrm{.}$ All terms here are dimensionless, and the physical energy unit enters only through the scale $E_{\ast }$ below.

For this physical model, take $q_j$, $a_j$, $c$, $C_0$, and the coefficients of $V$ to be dimensionless. Adopt a positive energy scale $E_{\ast }$ in joules as primitive finite-regulator data, and use $\mathrm{\hslash }$ in joule-seconds under Quantum field quantities and compatible SI normalizations. When the mathematical supplier Self-adjoint unbounded polynomial interaction at fixed regulator constructs its specified self-adjoint form realization $h$ of $h_0$, the model Hamiltonian is $H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:=E_{\ast }h$ on that supplied domain. This postulate adopts the finite-regulator polynomial data; it does not assert that the initial differential expression is self-adjoint or choose an operator domain by itself.

The finite label set and polynomial are held fixed. A finite spatial regulator is not thereby Lorentz invariant, and this model asserts no continuum or infinite-volume limit, no convergence of a perturbation series, and no reported observation. Its parameters are selected model inputs, not measured values.