Exact real scalar free model

Assume Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), as in the inspected Fock construction.

Select the real-scalar branch of the QFT-F01 physical adoption Specified bosonic scalar and CAR Dirac model adoption: one real spin-zero species with bosonic statistics and the zero-particle vacuum. For a smooth real classical field ${\varphi }_{\mathrm{n}}:{\mathbb{R}}^4\to \mathbb{R}$ of canonical length degree $-1$, that postulate adopts the local action density represented by $\frac{S_{\varphi }}{\mathrm{\hslash }}=-\frac{1}{2}{\int }_{{\mathbb{R}}^4}({\eta }^{ab}{\mathrm{\partial }}_a{\varphi }_{\mathrm{n}}{\mathrm{\partial }}_b{\varphi }_{\mathrm{n}}+{\mu }^2{\varphi }_{\mathrm{n}}^2)d^{4}x\mathrm{.}$ The total action is defined by this integral when its density is integrable; it need not converge for an arbitrary smooth global field. For a real $h\in C_c^{\mathrm{\infty }}({\mathbb{R}}^4)$ and $ϵ\in \mathbb{R}$, the normalized action difference under ${\varphi }_{\mathrm{n}}\mapsto {\varphi }_{\mathrm{n}}+ϵh$ is defined by integrating the difference of the displayed densities. That difference is smooth and supported in $\mathrm{supp}h$, so its integral is finite even when neither total action exists. Only these compactly supported variations are used, as in Specified bosonic scalar and CAR Dirac model adoption. The species, statistics, state, and local action density are physical model inputs. The present item fixes the exact flat-spacetime free Fock model to which the later deductions apply.

Use the oriented and time-oriented Minkowski spacetime, signature $\eta =\mathrm{diag}(-1,1,1,1)$, and length coordinates $x^0=ct$ and $\mathbf{x}\in {\mathbb{R}}^3$ from Quantum field quantities and compatible SI normalizations. Write $\mathrm{\square }=-{\mathrm{\partial }}_0^2+{\sum }_{j=1}^3{\mathrm{\partial }}_j^2$. Choose an inverse-length parameter $\mu \ge 0$. When $\mu >0$, the corresponding SI rest mass is $m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{S}\mathrm{I}}=\mathrm{\hslash }\mu \mathrm{/}c>0$; $\mu =0$ denotes the massless branch and assigns no positive rest mass. For $\mathbf{k}\in {\mathbb{R}}^3$, put ${\omega }_{\mu }(\mathbf{k})=\sqrt{\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2+{\mu }^2}$, with units ${\mathrm{m}}^{-1}$.

Let $H_{\mu }^{+}$, $d{\nu }_{\mu }$, $H_{\mu }=L^2(H_{\mu }^{+},d{\nu }_{\mu };\mathbb{C})$, and $K_{\mu }:\mathcal{S}({\mathbb{R}}^4)\to H_{\mu }$ be the future mass shell, measure, one-particle Hilbert space, and Fourier restriction of Scalar future mass shell, invariant measure and test restriction. In the massless case the null apex is omitted; it has measure zero. The shell measure is $d{\nu }_{\mu }=d^3\mathbf{k}\mathrm{/}((2\pi {)}^{3}2{\omega }_{\mu }(\mathbf{k}))$.

Use the symmetric bosonic Fock space ${\mathcal{F}}_s(H_{\mu })$, vacuum $\mathrm{\Omega }=(1,0,0,\dots )$, and dense finite-particle domain ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ from Schwartz/distribution spaces and Hilbert Fock conventions. The Fock pairing is linear in its second variable. Creation $a^{†}(h)$ is linear in $h$, while annihilation $a(h)$ is conjugate-linear in $h$. For $f\in \mathcal{S}({\mathbb{R}}^4)$ define the complex-linear smeared scalar field on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ by

${\mathrm{\Phi }}_{\mu }(f)=a(K_{\mu }\bar{f})+a^{†}(K_{\mu }f)\mathrm{.}$

For real $f$, this is the symmetric scalar field whose self-adjoint closure and bounded Weyl functions are constructed in Smeared free scalar field and its Poincare representation. A smeared field is a domain-qualified operator-valued distribution; the item introduces no pointwise operator ${\mathrm{\Phi }}_{\mu }(x)$.

Define $d{\mathrm{\Gamma }}_s({\omega }_{\mu })$ on the $n$-particle sector as multiplication by ${\sum }_{r=1}^n{\omega }_{\mu }({\mathbf{k}}_r)$, with value zero on the vacuum sector. The exact maximal domain is

$D(d{\mathrm{\Gamma }}_s({\omega }_{\mu }))=\{\mathrm{\Psi }:{\sum }_{n\ge 1}{\int }_{(H_{\mu }^{+}{)}^n}{\mid {\sum }_{r=1}^n{\omega }_{\mu }({\mathbf{k}}_r)\mid }^2\mathrm{\mid }{\mathrm{\Psi }}_n({\mathbf{k}}_1,\dots ,{\mathbf{k}}_n){\mathrm{\mid }}^{2}d{\nu }_{\mu }^n<\mathrm{\infty }\}\mathrm{.}$

The physical Hamiltonian of this selected model is the vacuum-subtracted self-adjoint operator $H_{\mu }^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=\mathrm{\hslash }cd{\mathrm{\Gamma }}_s({\omega }_{\mu })$ on that domain, as defined by the multiplier construction of Second-quantized multipliers, joint spectral measure and exact domains. It has units of joules and $H_{\mu }^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\mathrm{\Omega }=0$. No infinite zero-point-energy sum is part of this definition. The canonical scalar engineering dimension is ${\mathrm{m}}^{-1}$; a spacetime test function of unit ${\mathrm{m}}^{-3}$ makes the smeared field dimensionless, consistently with Quantum field quantities and compatible SI normalizations.

Assume the scalar model adoption planned in Specified bosonic scalar and CAR Dirac model adoption and Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Let $\mu \ge 0$ and use the model, shell, field, vacuum, and units of Exact real scalar free model and Scalar future mass shell, invariant measure and test restriction. These are conditional model deductions, with no empirical premise.

For every Schwartz test $f$, ${\mathrm{\Phi }}_{\mu }(f)$ is defined on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$; for real $f$ it is symmetric and essentially self-adjoint there. Distributionally, ${\mathrm{\Phi }}_{\mu }((\mathrm{\square }-{\mu }^2)f)=0$. The Fock lift of the proper orthochronous Poincare action is unitary, leaves $\mathrm{\Omega }$ invariant, and implements covariance of the smeared field. For real Schwartz tests the vacuum two-point form is $⟨\mathrm{\Omega },{\mathrm{\Phi }}_{\mu }(f){\mathrm{\Phi }}_{\mu }(g)\mathrm{\Omega }⟩=⟨K_{\mu }f,K_{\mu }g{⟩}_2$, and it is nonnegative when $g=f$. The joint spectrum of spacetime translations is contained in the closed forward cone.

For real compactly supported smooth tests with pairwise spacelike-separated supports, the smeared fields commute on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$. The bounded local Weyl algebras are isotonic and algebras for spacelike-separated regions commute. The physical Hamiltonian is self-adjoint on the exact multiplier domain in Exact real scalar free model, is nonnegative, and has zero-energy eigenspace $\ker H_{\mu }^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=\mathbb{C}\mathrm{\Omega }$. Thus its normalized zero-energy vectors are $e^{i\theta }\mathrm{\Omega }$ for $\theta \in \mathbb{R}$. If $\mu >0$, the infimum of its positive spectrum is $\mathrm{\hslash }c\mu =m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{S}\mathrm{I}}{c}^2$. If $\mu =0$, its positive spectrum has no gap above zero.

Assume the scalar model adoption planned in Specified bosonic scalar and CAR Dirac model adoption, Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), and the exact model of Exact real scalar free model. For real Schwartz tests $f_1,\dots ,f_n$, the vacuum expectation of the ordered product of smeared fields is zero when $n$ is odd. When $n$ is even, it is the sum over pairings of products of the corresponding two-point functions $⟨K_{\mu }{f}_r,K_{\mu }{f}_s{⟩}_2$ for $r<s$ in each pair. For a real Schwartz test $f$, the bounded Weyl operator has vacuum expectation $⟨\mathrm{\Omega },W_{\mu }(f)\mathrm{\Omega }⟩=\exp (-\mathrm{\parallel }{K}_{\mu }f{\mathrm{\parallel }}_2^2\mathrm{/}2)$.

The time-ordered two-point carrier is the tempered distribution $H_{\mu }$ defined by the partial spatial Fourier formula of Tempered time-ordered scalar two-point carrier. It agrees with the ordered Wightman distributions away from equal-time coincidence and satisfies $(\mathrm{\square }-{\mu }^2)H_{\mu }=i{\delta }_0$. The formula fixes its extension at coincidence and includes no additional prefactor $i$. These are conditional deductions for the selected free model.