Adopted finite-regulator bosonic 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.

For the fixed model of Adopted finite-regulator polynomial model, use the SI energy-unit convention of Quantum field quantities and compatible SI normalizations. A declared temperature input is a scalar $T>0$ with unit kelvin (K). Use the SI-defining Boltzmann constant $k_B:=1.380649\times {10}^{-23}\mathrm{J}{\mathrm{K}}^{-1}\mathrm{.}$ This exact value is the one stated by the BIPM SI Brochure, 9th edition, version 4.01, §§2.2 and 2.3.1. Define the physical inverse-energy parameter

${\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:=\frac{1}{k_{B}T}[{\mathrm{J}}^{-1}],$

and its dimensionless form

$\widehat{\beta }:=E_{\ast }{\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{.}$

The product $k_{B}T$ has unit joule, and $\widehat{\beta }$ is dimensionless because $E_{\ast }$ has unit joule. Since $k_B>0$ and $T>0$, both inverse parameters are defined and positive. The temperature $T$ is a declared model input; this definition does not infer it from a state or an experiment. It defines a parameter convention only and asserts no dynamical thermalization.

Assume Countable Choice. Use one fixed finite-regulator data set from Adopted finite-regulator polynomial model, and let $h$ be the self-adjoint form realization supplied by Self-adjoint unbounded polynomial interaction at fixed regulator, with domain $D(h)$ and eigenpairs $({\psi }_n,{\lambda }_n)$. Let $E_{\ast }>0$ joules and set $H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:=E_{\ast }h$ on $D(h)$. For physical time $\tau$ in seconds define

$U_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(\tau ):=U(E_{\ast }\tau \mathrm{/}\mathrm{\hslash }),$

where $U(s)$ is the dimensionless-parameter group of Global unitary evolution at fixed finite regulator. Then $U_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is a strongly continuous unitary group, $U_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(0)=I$, $U_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}({\tau }_1+{\tau }_2)=U_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}({\tau }_1)U_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}({\tau }_2)$, and $\mathrm{\parallel }{U}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(\tau )\psi \mathrm{\parallel }=\mathrm{\parallel }\psi \mathrm{\parallel }$ for every $\psi \in H_N$. Its norm derivative exists exactly for $\psi \in D(h)$, and then

$\frac{d}{d\tau }{U}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(\tau )\psi =-\frac{i}{\mathrm{\hslash }}{H}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{U}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(\tau )\psi \mathrm{.}$

The result is a conditional deduction for the selected finite-regulator model. It is not an empirical observation or a continuum-limit theorem.

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the finite-regulator spectral and trace construction. For the fixed model of Adopted finite-regulator polynomial model and a declared temperature input $T>0$, use Finite-regulator temperature and inverse-energy parameter to set ${\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=(k_{B}T{)}^{-1}$ and $\widehat{\beta }=E_{\ast }{\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}>0$. Let $h$ be the model's specified self-adjoint form realization and $H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=E_{\ast }h$. When the finite-mode coercivity hypotheses hold, Finite-regulator Gibbs trace and bounded expectations proves that ${\rho }_{\gamma }$ is positive trace class with finite positive trace for every dimensionless $\gamma >0$. Make its spectral construction explicit: let $({\psi }_k,{\lambda }_k{)}_{k\ge 0}$ be the complete orthonormal eigenbasis and repeated eigenvalues used in that supplier. For $f\in H_N$, define

$G_{\gamma }f:={\sum }_{k\ge 0}{e}^{-\gamma {\lambda }_k}⟨f,{\psi }_k⟩{\psi }_k\mathrm{.}$

The vector series converges in $H_N$ for every $f$, and this is the action of the supplier's operator ${\rho }_{\gamma }$, obtained there as the operator-norm limit of its finite-rank truncations. In this item the notation $e^{-\gamma h}$ means this explicitly defined operator $G_{\gamma }$. Since $H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{\psi }_k=E_{\ast }{\lambda }_k{\psi }_k$, define $e^{-{\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{H}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}$ on $H_N$ by the corresponding series $e^{-{\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{H}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}f:={\sum }_{k\ge 0}{e}^{-{\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{E}_{\ast }{\lambda }_k}⟨f,{\psi }_k⟩{\psi }_k$. Since $\widehat{\beta }=E_{\ast }{\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, its coefficients equal those of $G_{\widehat{\beta }}$ term by term, and both series converge in $H_N$. Therefore, as operators on $H_N$, $e^{-{\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{H}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}=G_{\widehat{\beta }}=e^{-\widehat{\beta }h}$. The Gibbs-trace supplier consequently applies to exactly the operator in the state assignment below. Adopt the normalized operator

${\rho }_T:=\frac{e^{-{\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{H}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}}{Z_T}=\frac{e^{-\widehat{\beta }h}}{Z_T},Z_T:=\mathrm{Tr}(e^{-{\beta }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{H}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}})=\mathrm{Tr}(e^{-\widehat{\beta }h})$

as this finite-regulator model's canonical equilibrium state. This is a physical state assignment, not a deduction that the dynamics reach equilibrium. It asserts no infinite-volume trace, no zero-temperature state, and no empirical observation.

Assume Countable Choice and the finite-regulator coercivity hypotheses of Finite-regulator Gibbs trace and bounded expectations. Use the model state of Adopted finite-regulator equilibrium state and the inverse-energy parameters of Finite-regulator temperature and inverse-energy parameter. Let ${\alpha }_j:=\sqrt{a_{j}c}$, $j=1,\dots ,N$, for the dimensionless kinetic coefficients and coercivity constant in the model, and let $C_0\ge 0$ be its potential lower-bound shift. Then

$0<Z_T={\sum }_{k\ge 0}{e}^{-\widehat{\beta }{\lambda }_k}\le e^{\widehat{\beta }{C}_0}{\prod }_{j=1}^N\frac{e^{-\widehat{\beta }{\alpha }_j}}{1-e^{-2\widehat{\beta }{\alpha }_j}}<\mathrm{\infty },$

where ${\lambda }_k$ are the eigenvalues of $h$. The adopted state ${\rho }_T$ is a positive trace-class operator with trace one. For every bounded linear operator $B:H_N\to H_N$,

$\mathrm{Tr}({\rho }_{T}B)=Z_T^{-1}{\sum }_{k\ge 0}{e}^{-\widehat{\beta }{\lambda }_k}⟨B{\psi }_k,{\psi }_k⟩,$

and the series is absolutely convergent. If $E_a$ is an effect in a supplied normalized finite instrument of Physical interpretation of a supplied constructed quantum model, this trace is its adopted canonical model probability. These are conditional finite-regulator predictions, not measured outcomes or a continuum-limit claim.

Assume Countable Choice and use one fixed one-mode regulator of Adopted finite-regulator polynomial model with dimensionless $a>0$, $r>0$, and $\lambda \ge 0$. Here $r$ is a polynomial coefficient, not an SI rest mass. Take $V_{\lambda }(q)=\frac{r^2}{2}{q}^2+\lambda q^4$ on $H_1=L^2(\mathbb{R},dq;\mathbb{C})$, with the form realization from Self-adjoint unbounded polynomial interaction at fixed regulator. Let ${ϵ}_g(\lambda )$ be its least eigenvalue. The physical ground-energy prediction for the adopted scale $E_{\ast }>0$ joules is $E_g^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(\lambda ):=E_{\ast }{ϵ}_g(\lambda )$. Then

$0\le E_g^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(\lambda )-E_g^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(0)\le \frac{3E_{\ast }a\lambda }{2r^2}\mathrm{.}$

For $\lambda >0$, the potential has a nonzero degree-four term and is interacting in the finite-coordinate sense defined by Adopted finite-regulator polynomial model. This conclusion makes no claim about field scattering or a continuum field theory. The inequality is an exact fixed-regulator model deduction; it supplies neither an observed energy nor a perturbation-series or continuum-limit claim.