Scalar distribution, covariance, locality and spectrum

The adopted real scalar branch and the field map on ${\mathcal{D}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ are exactly those of [A1] and [F1]. The field is complex-linear in $f$: conjugation of $f$ followed by the conjugate-linear annihilator makes the first term linear, and $K_{\mu }$ and creation are linear in the second term. The field exists on the common dense finite-particle domain by the inspected mathematical construction [F3].

For a Schwartz test, integration by parts transfers $\mathrm{\square }-{\mu }^2$ from $f$ to the shell exponential in $K_{\mu }((\mathrm{\square }-{\mu }^2)f)$. Schwartz decay annuls every boundary term. On the shell, $(\mathrm{\square }-{\mu }^2)e^{i({\omega }_{\mu }{x}^0-\mathbf{k}\cdot \mathbf{x})}=({\omega }_{\mu }^2-\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2-{\mu }^2)e^{i({\omega }_{\mu }{x}^0-\mathbf{k}\cdot \mathbf{x})}=0$ by [F10]. The operator also has real coefficients, so the same equality holds for $\bar{f}$; both creation and annihilation coefficients in ${\mathrm{\Phi }}_{\mu }((\mathrm{\square }-{\mu }^2)f)$ vanish.

The vacuum two-point formula and positivity are [F4]. The proper orthochronous Poincare covariance and invariant vacuum follow from the Fock lift in [F5]. Both use the same shell and pairing conventions as [F1], so no additional transformation convention is introduced.

By [F7], the commutator of real smeared fields is the causal commutator pairing; its value is zero when every support difference is spacelike. [F9] then gives commutation of the bounded Weyl generators for the two regions. Finite sums and products preserve cross-commutation, and operator-norm limits preserve it because multiplication of bounded operators is norm-continuous. Isotony follows from inclusion of the real compact-test generator sets, also as stated in [F9].

Apply [F8] on each symmetric sector with multiplier $\mathrm{\hslash }c{\sum }_{r=1}^n{\omega }_{\mu }({\mathbf{k}}_r)$ and zero vacuum multiplier. This gives the exact self-adjoint domain stated in [F2]. Every multiplier is nonnegative. The translation-spectrum conclusion in [F6] places the joint spectrum in the closed forward cone.

On every sector $n\ge 1$, the energy multiplier is positive almost everywhere: for $\mu >0$ each ${\omega }_{\mu }\ge \mu$, while for $\mu =0$ its only zero is the omitted shell apex. Thus the zero-energy eigenspace is exactly the vacuum sector $\mathbb{C}\mathrm{\Omega }$; its unit vectors are precisely $e^{i\theta }\mathrm{\Omega }$, $\theta \in \mathbb{R}$. If $\mu >0$, all positive-sector multipliers are at least $\mathrm{\hslash }c\mu$. For any $ϵ>0$, choose $r>0$ with $\mathrm{\hslash }c(\sqrt{{\mu }^2+r^2}-\mu )<ϵ$. The ball $\mathrm{\mid }\mathbf{k}\mathrm{\mid }<r$ has finite positive $d{\nu }_{\mu }$ measure by [F10]; its normalized indicator is a one-particle vector with energy in $[\mathrm{\hslash }c\mu ,\mathrm{\hslash }c\sqrt{{\mu }^2+r^2}]$. Hence the positive spectral infimum is $\mathrm{\hslash }c\mu$. For $\mu =0$, every annulus $r\mathrm{/}2<\mathrm{\mid }\mathbf{k}\mathrm{\mid }<r$ has finite positive $d{\nu }_0$ measure. Its normalized indicator has energy in $(0,\mathrm{\hslash }cr)$, so positive spectral values occur arbitrarily close to zero. For $\mu >0$, the unit conversion in the model definition gives $\mathrm{\hslash }c\mu =m_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{S}\mathrm{I}}{c}^2$.