Autor: A. Jaster | Datum: 2026/05/24 | DOI: 10.5281/zenodo.20368388
Building on the causal-Machian emergent time theory [Jaster2026a] and its Planck-scale regularisation [Jaster2026b], we examine the implications for quantum field theory in an exploratory and phenomenological spirit. All field actions are evaluated on the physical clock metric ĝμν; the factor √Θ in the Lagrangian density follows from the identity √−ĝ = √−q · √Θ and is geometrically derived. We work in signature (−,+,+,+) with the convention ℒ ¹ −1⁄2 ĝμν ∂μφ ∂νφ. Throughout the paper Θ is treated as a fixed semiclassical background in the fixed-Θ approximation; the self-consistent variation generates non-local additional terms and is an open task. Two variants are analysed. In the base model a single cosmological causal field Θ ≈ 1 governs all current-epoch physics; all Standard Model predictions that depend only on the local metric are reproduced to the precision of 1−Θ ≈ e−1060 (preferred-frame effects from UHμ are not included in this estimate). In the scale-separation model Θ = Θcosm · Θloc, a local causal field Θloc at the scale mloc−1 is introduced as an approximation valid to 𝒪(HEW/mH) ∼ 10−17 and may provide candidate ingredients for a modified electroweak phase transition and for the Sakharov conditions; quantitative predictions require calculations not yet performed. The scale-separation model introduces no new dimensionful parameter, but a dimensionless coupling ζ (Section~3.3) remains undetermined. The scalar Θ is a semiclassical constraint functional requiring no independent quantisation within the present framework.
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\begin{document}
% =============================================================================
\begin{center}
{\LARGE\bfseries Quantum Field Theory Implications\\[4pt]
of the Emergent Causal Time Field}\\[1.2em]
{\large A.~Jaster}\\[0.4em]
{\normalsize \today}
\end{center}
\vspace{0.8em}
% ── Abstract ──────────────────────────────────────────────────────────────────
\begin{abstract}
Building on the causal-Machian emergent time theory \citep{Jaster2026a}
and its Planck-scale regularisation \citep{Jaster2026b}, we examine
the implications for quantum field theory in an exploratory and
phenomenological spirit. All field actions are evaluated on the
physical clock metric $\hat{g}_{\mu\nu}$; the factor $\sqrt{\Theta}$
in the Lagrangian density follows from the identity
$\sqrt{-\hat{g}}=\sqrt{-q}\cdot\sqrt{\Theta}$ and is geometrically
derived. We work in signature $(-,+,+,+)$ with the convention
$\mathcal{L}\supset-\frac{1}{2}\hat{g}^{\mu\nu}\partial_\mu\phi\,
\partial_\nu\phi$. Throughout the paper $\Theta$ is treated as a
fixed semiclassical background in the fixed-$\Theta$ approximation;
the self-consistent variation generates non-local additional terms and
is an open task. Two variants are analysed. In the \emph{base model}
a single cosmological causal field $\Theta\approx1$ governs all
current-epoch physics; all Standard Model predictions that depend only
on the local metric are reproduced to the precision of
$1-\Theta\approx e^{-10^{60}}$ (preferred-frame effects from
$U_H^\mu$ are not included in this estimate). In the \emph{scale-separation
model} $\Theta=\Tcosm\cdot\Tloc$, a local causal field $\Tloc$ at
the scale $m_{\rm loc}^{-1}$ is introduced as an approximation valid
to $\mathcal{O}(H_{\rm EW}/m_H)\sim10^{-17}$ and may provide
candidate ingredients for a modified electroweak phase transition and
for the Sakharov conditions; quantitative predictions require
calculations not yet performed. The scale-separation model introduces
no new dimensionful parameter, but a dimensionless coupling $\zeta$
(Section~\ref{sec:cpbary}) remains undetermined. The scalar $\Theta$
is a semiclassical constraint functional requiring no independent
quantisation within the present framework.
\end{abstract}
\vspace{1em}
% =============================================================================
\section{The Model: Actions, Equations of Motion, and Quantisation}
\label{sec:model}
% =============================================================================
\subsection{Geometric Foundations and Sign Convention}
\label{sec:geometry}
We work throughout with metric signature $(-,+,+,+)$ and adopt the
following sign convention for the scalar field action:
\begin{equation}
\mathcal{L}_\phi
= -\tfrac{1}{2}\,\hat{g}^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi
- V(\phi).
\label{eq:conv}
\end{equation}
With this convention the kinetic energy in the non-relativistic limit
is $+\frac{1}{2}(\partial_t\phi)^2$ (positive-definite), and the
Klein-Gordon equation reads $\hat\Box\phi - V'(\phi)=0$. All sign
choices in this section are consistent with \eqref{eq:conv}.
The physical clock metric of \citet{Jaster2026a} is the disformal
modification
\begin{equation}
\hat{g}_{\mu\nu} = q_{\mu\nu}
+ \frac{1-\Theta}{c^{2}}\,U_{H\mu}\,U_{H\nu},
\qquad
\hat{g}^{\mu\nu} = q^{\mu\nu}
- \frac{1-\Theta}{\Theta c^{2}}\,U_H^\mu\,U_H^\nu,
\label{eq:ghat}
\end{equation}
where $q_{\mu\nu}$ is the auxiliary FLRW-normalised reference geometry
($q_{\mu\nu}U_H^\mu U_H^\nu=-c^2$) and $\Theta(x)=S_\infty(\Omega_{H,\lP}
/\Omega_{{\rm ref},\lP})\in[0,1]$. The matrix determinant lemma gives
the exact identity
\begin{equation}
\sqrt{-\hat{g}} = \sqrt{-q}\cdot\sqrt{\Theta}.
\label{eq:det}
\end{equation}
The factor $\sqrt{\Theta}$ in field actions is therefore a geometric
consequence of \eqref{eq:ghat}, not an independent postulate.
\paragraph{Fixed-$\Theta$ approximation.}
Throughout this paper $\Theta(x)$ is treated as a prescribed
semiclassical background functional and is not varied when deriving
equations of motion. The self-consistent variation --- in which
$\delta\Theta/\delta\phi\neq0$ because $\Theta$ is a functional of
$T_{\mu\nu}[\phi]$ --- generates non-local additional terms and is
an open task (Section~\ref{sec:open}, item~1).
\subsection{Field Actions}
\label{sec:action}
Under the fixed-$\Theta$ approximation all matter fields propagate on
$\hat{g}_{\mu\nu}$. For a real scalar field $\phi$:
\begin{equation}
S_\phi = \int\sqrt{-q}\,\sqrt{\Theta}\,\Bigl(
-\tfrac{1}{2}\,\hat{g}^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi
- V(\phi)\Bigr)\dint^4 x.
\label{eq:Sphi}
\end{equation}
The Yang-Mills action for a gauge field with field strength $F_{\mu\nu}^a$:
\begin{equation}
S_{YM} = -\frac{1}{4}\int\sqrt{-q}\,\sqrt{\Theta}\,
\hat{g}^{\mu\alpha}\hat{g}^{\nu\beta}
F_{\mu\nu}^a F_{\alpha\beta}^a\,\dint^4 x.
\label{eq:SYM}
\end{equation}
At the level of the classical fixed-$\Theta$ action the gauge
principle is preserved: $\Theta$ is a Lorentz scalar that is
gauge-invariant by construction. BRST consistency, gauge fixing,
ghosts, and anomalies at the quantum level are open tasks
(Section~\ref{sec:open}, item~12).
The Dirac action, with vierbein $\hat{e}^\mu{}_a$ (satisfying
$\hat{e}^\mu{}_a\hat{e}^\nu{}_b\eta^{ab}=\hat{g}^{\mu\nu}$) and
$\hat{\gamma}^\mu=\hat{e}^\mu{}_a\gamma^a$:
\begin{equation}
S_D = \int\sqrt{-q}\,\sqrt{\Theta}\,
\bar\psi\bigl(i\hat{\gamma}^\mu\hat{D}_\mu - m\bigr)\psi
\,\dint^4 x,
\label{eq:SD}
\end{equation}
with Clifford algebra $\{\hat{\gamma}^\mu,\hat{\gamma}^\nu\}
=2\hat{g}^{\mu\nu}$. The explicit construction of the disformal
vierbein and spin connection is an open task
(Section~\ref{sec:open}, item~2).
\subsection{Equation of Motion for the Scalar Field}
\label{sec:eom}
Varying \eqref{eq:Sphi} with respect to $\phi$ under the fixed-$\Theta$
approximation gives:
\begin{equation}
\boxed{
\frac{1}{\sqrt{-\hat{g}}}\,\partial_\mu\!\Bigl(
\sqrt{-\hat{g}}\,\hat{g}^{\mu\nu}\partial_\nu\phi\Bigr)
- V'(\phi) = 0.
}
\label{eq:KG}
\end{equation}
\textit{Consistency check:} Varying $S_\phi$ with the convention
\eqref{eq:conv} and using $\sqrt{-\hat{g}}=\sqrt{-q}\sqrt{\Theta}$
gives directly $\hat\Box\phi - V'(\phi)=0$, i.e.\ equation~\eqref{eq:KG},
where $\hat\Box$ denotes the d'Alembertian of the clock metric:
\begin{equation}
\hat\Box\phi
= \frac{1}{\sqrt{-\hat{g}}}\,
\partial_\mu\!\Bigl(\sqrt{-\hat{g}}\,\hat{g}^{\mu\nu}\partial_\nu\phi\Bigr).
\label{eq:Box}
\end{equation}
For $V(\phi)=\frac{1}{2}m^2\phi^2$ the equation of motion becomes
$(\hat\Box-m^2)\phi=0$. In the GR sector ($\Theta=1$,
$\hat{g}_{\mu\nu}=q_{\mu\nu}$) and in the flat-space limit,
$\hat\Box\to-\partial_t^2+\nabla^2$. \checkmark
No expansion in $\nabla\Theta$ or $\nabla U_H^\mu$ is given because
the full disformal structure involves both; a partial expansion would
be incomplete. In the GR sector ($\Theta=1$,
$\hat{g}_{\mu\nu}=q_{\mu\nu}$) equation~\eqref{eq:KG} reduces to
the standard Klein-Gordon equation.~\checkmark
\subsection{Semiclassical Treatment of $\Theta$}
\label{sec:semiclassical}
In the quantised theory $\Theta$ is evaluated on the expectation
value of the energy-momentum tensor:
\begin{equation}
\Theta(x)
= S_\infty\!\!\left[
\frac{\Omega_H\!\bigl[\langle\hat{T}_{\mu\nu}\rangle\bigr]}
{\Omega_{\rm ref}}
\right].
\label{eq:Thetasemic}
\end{equation}
This is a \emph{semiclassical constraint functional}: generated by
the quantised matter sector and not itself an independently quantised
operator. Within the semiclassical fixed-$\Theta$ framework of this
paper no separate quantisation of $\Theta$ is required; whether this
remains true in the fully self-consistent quantum theory is open
(Section~\ref{sec:open}, item~3).
\subsection{UV Structure}
\label{sec:UV}
The $C^\infty$ boundary kernel $\mathcal{B}_{\lP}$ of
\citet{Jaster2026b} is smooth at the causal boundary and does not
require Wick rotation for its regularisation. Its superpolynomial
Fourier decay in the weak-field limit provides UV suppression of
causal-boundary gradient contributions at scales $\lesssim\lP$; full
UV-finiteness of the quantised theory requires an independent analysis.
% =============================================================================
\section{Scale Separation: $\Theta = \Tcosm\cdot\Tloc$}
\label{sec:scalesep}
% =============================================================================
\subsection{Status of the Factorisation}
\label{sec:status}
The scale-separation ansatz
\begin{equation}
\Theta(x)
= \Tcosm(x)\cdot\Tloc(x)
+ \mathcal{O}(H_{\rm EW}/m_H)
\label{eq:product}
\end{equation}
is an approximation scheme, not an identity derived from the base
model. At the electroweak epoch, $H_{\rm EW}/m_H\approx
m_H/m_P\approx10^{-17}$, so the error is negligible. The exact RG
derivation is an open task (Section~\ref{sec:open}, item~4).
\subsection{Local Causal Field}
\label{sec:Tlocdef}
The local causal field $\Tloc$ at scale $\lambda=m_{\rm loc}^{-1}$ is
\begin{equation}
\Tloc(x)
= S_\infty\!\!\left(
\frac{\Omega_{H,\rm loc}(x)}{\Omega_{\rm ref,loc}}
\right),
\label{eq:Tloc}
\end{equation}
with
\begin{equation}
\Omega_{H,\rm loc}(x)
= \int
W_{H,\lP}^{\rm loc}(x,x')\,\mathcal{E}_{H,\rm loc}(x')\,
\sqrt{-q}\,\dint^4 x'.
\label{eq:Omegaloc}
\end{equation}
\paragraph{$q$-foliation-defined local domain and smooth kernel.}
Let $d_q^{\rm sp}(x,x')$ denote the spatial geodesic distance in the
$q_{\mu\nu}$-foliation between $x'$ and the spatial position of $x$,
and let $\Delta T_q = T_q(x)-T_q(x')$. Three smooth conditions define
the local causal neighbourhood: (i)~spatial radius $d_q^{\rm sp}<\lambda$;
(ii)~temporal depth $0<\Delta T_q<\lambda/c$; (iii)~causal past
$d_q^{\rm sp} < c\,\Delta T_q$. Each is encoded by one factor of the
$C^\infty$ saturating function $\varphi$ of \citet{Jaster2026b}. The
local bump function is defined as the product
\begin{equation}
\mathcal{B}_{\lP}^{\rm loc}(x,x')
= \varphi\!\left(\frac{\lambda - d_q^{\rm sp}}{\lP}\right)
\cdot
\varphi\!\left(\frac{\lambda/c - \Delta T_q}{\lP/c}\right)
\cdot
\varphi\!\left(\frac{c\,\Delta T_q - d_q^{\rm sp}}{\lP}\right).
\label{eq:Bloc}
\end{equation}
Since each factor is $C^\infty$ and all derivatives vanish at the
transition point, $\mathcal{B}_{\lP}^{\rm loc}\in C^\infty$ everywhere
$d_q^{\rm sp}$ is smooth (away from caustics;
Section~\ref{sec:open}, item~5). No $\min$-function or sharp
characteristic function appears; the product itself is zero outside
the desired domain, positive inside, and transitions smoothly to zero
at each boundary over a layer of width $\lP$. The locally normalised
kernel is
\begin{equation}
W_{H,\lP}^{\rm loc}(x,x')
= \frac{\mathcal{B}_{\lP}^{\rm loc}(x,x')}
{\mathcal{N}_{\rm loc}(x)},
\qquad
\mathcal{N}_{\rm loc}(x)
= \int\mathcal{B}_{\lP}^{\rm loc}(x,x'')\,
\sqrt{-q(x'')}\,\dint^4 x'',
\label{eq:Wloc}
\end{equation}
normalised independently of the base-theory kernel, and the integral
in \eqref{eq:Omegaloc} runs over all of $\mathcal{M}$ with the support
of $\mathcal{B}_{\lP}^{\rm loc}$ providing the effective restriction.
\paragraph{Gauge-invariant field-selective energy density.}
The energy density $\mathcal{E}_{H,\rm loc}$ is restricted to the
gauge-invariant combinations of the relevant field sector. For the
electroweak sector:
\begin{equation}
\mathcal{E}_{H,\rm loc}^{\rm EW}(x')
= \frac{T_{\mu\nu}^{\rm EW}(x')\,
n_q^\mu(x')\,n_q^\nu(x')}{c^2},
\label{eq:EHlocEW}
\end{equation}
where $T_{\mu\nu}^{\rm EW}$ is constructed from the gauge-invariant
quantities $(D_\mu H)^\dagger D^\mu H$ and $H^\dagger H$ (not the
bare field $\phi$, which is not gauge-invariant). Its covariant
derivation from a unified constraint action is an open task
(Section~\ref{sec:open}, item~6).
\paragraph{Local reference density.}
The reference density is the broken-phase equilibrium energy density
at the electroweak scale. At tree level:
\begin{equation}
\Omega_{\rm ref,loc}^{\rm EW}
\sim m_H^2\,v^2
\quad ([E^4]\text{ in natural units}).
\label{eq:OrefH}
\end{equation}
The numerical prefactor depends on the renormalisation convention and
finite-temperature corrections; only the order of magnitude is used
here. Quantitative definitions require finite-temperature QFT methods
\citep{Quiros1999} and are an open task (Section~\ref{sec:open}, item~7).
\paragraph{Properties.}
In the broken-phase minimum ($|H|^2=v^2/2$): $r_{\rm loc}=1$,
$\Tloc=1$, $\nabla\Tloc=0$ in the fixed-$\Theta_{\rm loc}$
approximation; the VEV, Higgs mass, and gauge boson masses are
unchanged relative to the Standard Model.~\checkmark
In the symmetric phase ($H\approx0$): $r_{\rm loc}<1$, $\Tloc<1$.
At the bubble wall: $\nabla\Tloc\neq0$.
% =============================================================================
\section{Implications for Quantum Field Theory}
\label{sec:implications}
% =============================================================================
\subsection{Gauge Fields}
\label{sec:gauge}
At the level of the classical fixed-$\Theta$ action the gauge
symmetries $U(1)_Y\times SU(2)_L\times SU(3)_c$ are intact. In the
GR sector all gauge field equations of motion reduce exactly to their
standard forms.
\subsection{Electroweak Phase Transition and the Higgs Mechanism}
\label{sec:higgs}
\subsubsection{Base model}
At the electroweak epoch $\Theta\approx1$ everywhere; the Higgs
mechanism is identical to the Standard Model and the phase transition
is a smooth crossover for $m_H=125$\,GeV.
\subsubsection{Scale-separation model}
\paragraph{Two-level description.}
Two conceptually distinct levels must be separated:
\textit{Level~1 (equations of motion in the fixed-$\Theta$ approximation):}
$\Theta_{\rm loc}$ is a prescribed background; equations of motion
for the Higgs field follow from variation of \eqref{eq:Sphi} with
$\Theta$ held fixed. This is the level at which \eqref{eq:KG} is
valid.
\textit{Level~2 (quasi-static diagnostic potential):}
After solving the self-consistent map that determines
$\Theta_{\rm loc}[\phi]$ from the field configuration, one may
substitute the result back to define an effective potential landscape:
\begin{equation}
V_{\rm eff}^{\rm ET}(\phi)
= \sqrt{\Tloc(\phi)}\cdot V(\phi),
\label{eq:Veff}
\end{equation}
where $\Tloc(\phi)$ is now evaluated on the field $\phi$ as a
quasi-static approximation, not as a fixed background. These two
levels are not equivalent; Level~2 is a diagnostic tool for
understanding the potential landscape and does not follow from the
fixed-$\Theta$ equations of motion. A complete treatment requires
the self-consistent formulation (Open Problem~1) and the
finite-temperature effective potential \citep{Quiros1999}.
\begin{remark}[On the cubic term]
The $C^\infty$ saturating function $S_\infty$ satisfies $S_\infty(r)
\sim e^{-1/r}$ as $r\to0^+$: it is smooth but not analytic at $r=0$,
so no polynomial expansion $\sqrt{\Tloc}\approx c_1|\phi|+\ldots$
holds. Whether $V_{\rm eff}^{\rm ET}$ develops a potential barrier
between the symmetric and broken-phase minima must be determined by
explicit numerical computation; no cubic term is claimed on analytical
grounds.
\end{remark}
In the broken-phase minimum $\Tloc=1$ by construction; VEV, Higgs
mass, and gauge boson masses are unchanged in the fixed-$\Theta_{\rm
loc}$ approximation.
\subsection{CP Violation and Baryogenesis}
\label{sec:cpbary}
\subsubsection{Base model}
The preferred frame $U_H^\mu$ breaks global Lorentz invariance.
The CPT theorem requires local Lorentz invariance, causality, and
unitarity \citep{Greenberg2002}; radiative corrections may generate
CPT- and Lorentz-violating effective operators of the Standard Model
Extension type \citep{Colladay1998}, but their magnitude is not
computed here.
\subsubsection{Scale-separation model}
At the electroweak bubble wall, $\Tloc$ transitions from its
symmetric-phase value to 1 over a distance $\sim m_H^{-1}$, giving
a spatially localised gradient. This couples to fermion currents:
\begin{equation}
\delta\mathcal{L}_{b}
= b_\mu\bar\psi\gamma^\mu\gamma_5\psi,
\qquad
b_\mu = \zeta\,\partial_\mu\Tloc,
\label{eq:bterm}
\end{equation}
with dimensionless coupling $\zeta$. \textit{Dimension check:}
$[\Tloc]=1$, $[\partial_\mu\Tloc]=[E]$, $[b_\mu]=[E]$,
$[b_\mu\bar\psi\gamma^\mu\gamma_5\psi]=[E^4]$.~\checkmark
No new dimensionful parameter is introduced. The coupling $\zeta$
is dimensionless but undetermined; it must be derived from the
constraint structure of the theory. The term \eqref{eq:bterm} is
CPT- and Lorentz-violating in the sense of the SME; whether it
provides CP violation in the form required for electroweak baryogenesis
is not established.
The Sakharov conditions \citep{Sakharov1967} may find candidate
realisations:
\begin{enumerate}[label=(\roman*)]
\item Baryon number violation: electroweak sphaleron processes
for $T\gtrsim130$\,GeV.
\item C and CP violation: the operator \eqref{eq:bterm} is a
candidate source; quantitative sufficiency is not shown.
\item Departure from thermal equilibrium: a first-order phase
transition would provide the required non-equilibrium conditions,
if it occurs (see Remark above).
\end{enumerate}
The key quantities $v_c/T_c$, the baryon-to-entropy ratio,
the sphaleron rate, and the bubble-wall transport equations are not
computed (Section~\ref{sec:open}, items~8 and~9).
\subsection{Gluon Propagation}
\label{sec:gluon}
\begin{proposition}[Propagation requires $\Theta>0$]
\label{prop:prop}
In any spacetime region where $\Theta(x)=0$ the clock metric
$\hat{g}_{\mu\nu}$ degenerates; no physical field, including massless
gauge bosons, propagates on $\hat{g}_{\mu\nu}$.
\end{proposition}
This is a geometric statement about the degenerate $\Theta=0$ limiting
case. In the observable universe $\Theta\approx1$ everywhere, so
Proposition~\ref{prop:prop} does not apply to any accessible physical
region. It ties the propagation of gauge bosons ontologically to the
presence of temporal structure, which itself requires matter in the
causal past.
\begin{remark}[On confinement]
A connection between Proposition~\ref{prop:prop} and QCD colour
confinement has been considered but is not supported. In QCD the
confinement mechanism is associated with colour flux tubes whose
energy \emph{increases} as quark-antiquark separation grows
($E_{\rm tube}\sim\sigma r$, $\sigma\approx0.18$\,GeV/fm). A
na\"{\i}ve identification $\Tloc^{\rm QCD}\to0$ at large separation
would require the local causal energy density to \emph{decrease},
which is the opposite of the flux-tube picture. The relationship
between the causal time field and non-perturbative QCD is therefore
an open problem with no established direction.
\end{remark}
% =============================================================================
\section{Outlook: Planck-Scale Dispersion}
\label{sec:outlook}
% =============================================================================
The $C^\infty$ boundary kernel of \citet{Jaster2026b} introduces $\lP$
into the causal averaging, suggesting on dimensional grounds a possible
leading correction
\begin{equation}
E^2 = p^2c^2 + m^2c^4
+ \alpha\,\frac{E^3}{E_P}
+ \mathcal{O}\!\!\left(\frac{E^4}{E_P^2}\right),
\qquad E_P = \frac{\hbar c}{\lP}.
\label{eq:MDR}
\end{equation}
Equation~\eqref{eq:MDR} is a phenomenological parametrisation only.
The clock metric $\hat{g}_{\mu\nu}$ already modifies the null
condition; the relation between this and \eqref{eq:MDR} is not
established. Neither the existence of a cubic term, the sign, nor
the magnitude of $\alpha$ is derived; computing $\alpha$ requires
explicit kernel integrals and a preferred-frame analysis.
% =============================================================================
\section{Open Problems}
\label{sec:open}
% =============================================================================
\begin{enumerate}[label=(\arabic*)]
\item \textbf{Self-consistent variation: $\delta\Theta/\delta\phi$
(new).}
In the self-consistent theory, $\Theta$ is a functional of
$T_{\mu\nu}[\phi]$; its variation generates non-local additional
terms in all equations of motion. The two levels of
Section~\ref{sec:higgs} also require this formulation.
\item \textbf{Disformal vierbein and spin connection (new).}
Explicit construction of $\hat{e}^\mu{}_a$, the spin connection,
and their $(\Theta,U_H^\mu)$-dependent derivatives.
\item \textbf{State-dependence and backreaction of $\Theta$ (new).}
Since $\Theta$ is evaluated on $\langle\hat{T}_{\mu\nu}\rangle$,
it is quantum-state-dependent; the semiclassical backreaction
consistency must be analysed.
\item \textbf{RG derivation of $\Theta=\Tcosm\cdot\Tloc$ (new).}
The factorisation is an approximation; its rigorous derivation
requires the RG flow of the causal time field.
\item \textbf{Smoothness of $d_q^{\rm sp}$ at caustics (new).}
The spatial geodesic distance $d_q^{\rm sp}(x,x')$ used in the
local bump kernel \eqref{eq:Bloc} may fail to be $C^\infty$ at
caustics or geodesic cut loci of the $q_{\mu\nu}$-foliation; the
local kernel is smooth away from these configurations.
\item \textbf{Covariant derivation of field-selective
$\mathcal{E}_{H,\rm loc}$ (new).}
The restriction to gauge-invariant field-sector quantities and its
derivation from a unified constraint action, including prevention
of double-counting with $\Tcosm$.
\item \textbf{Finite-temperature reference densities (new).}
The tree-level estimate \eqref{eq:OrefH} requires extension to
gauge-invariant, renormalisation-scale-dependent definitions at
finite temperature.
\item \textbf{Numerical computation of $V_{\rm eff}^{\rm ET}$ (new).}
Whether $\sqrt{\Tloc(\phi)}\cdot V_{\rm eff}^{\rm SM}(\phi,T)$
generates a potential barrier for a first-order EWPT at
$m_H=125$\,GeV must be determined numerically.
\item \textbf{Baryogenesis quantification (new).}
A complete calculation requires the coupling $\zeta$ from
\eqref{eq:bterm}, $v_c/T_c$, the sphaleron rate, and bubble-wall
transport equations.
\item \textbf{Planck-scale dispersion coefficient (new).}
Computing $\alpha$ in \eqref{eq:MDR} from the kernel geometry
and comparing with astrophysical bounds.
\item \textbf{Unitarity, stability, and microcausality (new).}
For $\Theta<1$, the modified metric may affect hyperbolicity,
Hamiltonian structure, and causal commutator relations.
\item \textbf{BRST consistency, ghosts, and anomalies (new).}
Gauge-fixing, ghost sector, and potential anomalies at the
quantum level.
\item \textbf{Constraint action $S_{\rm constr}$ and
$T^{\rm C}_{\mu\nu}$ (inherited from \citealt{Jaster2026a}).}
Dynamical closure of the gravitational field equations.
\item \textbf{Fixed-point existence (inherited from
\citealt{Jaster2026a}).}
Convergence of the self-consistent map
$T_{\mu\nu}\!\to\!\Pi_H^\mu\!\to\!U_H^\mu\!\to\!\Omega_H
\!\to\!\Theta\!\to\!\hat{g}_{\mu\nu}\!\to\!T_{\mu\nu}$.
\end{enumerate}
% =============================================================================
\section{Summary}
\label{sec:summary}
% =============================================================================
\begin{table}[h]
\centering
\caption{Comparison of base model and scale-separation model.}
\label{tab:comparison}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{>{\bfseries}p{4.7cm} p{3.4cm} p{4.6cm}}
\toprule
Aspect & Base model & Scale separation\\
\midrule
Sign convention & $-\tfrac{1}{2}\hat{g}^{\mu\nu}\partial\phi\partial\phi$
& Idem \\
$\sqrt{\Theta}$ in action
& From $\sqrt{-\hat{g}}$ (geometric)
& Idem \\
$\Theta$ treatment
& Fixed semiclassical background
& Idem \\
$\Theta$ in observable universe
& $\approx1$: SM reproduced$^{*}$
& $\Tcosm\approx1$; $\Tloc$ varies \\
Gauge principle (classical)
& Preserved
& Preserved \\
EWPT
& Crossover (as in SM)
& $V_{\rm eff}^{\rm ET}$ modified (Level~2);
barrier must be computed \\
Fixed-$\Theta$ EOM / diagnostic potential
& Not applicable
& Two-level structure (Sec.~\ref{sec:higgs}) \\
CP violation
& CKM; CPT from $U_H^\mu$ unquantified
& $+b_\mu\bar\psi\gamma^\mu\gamma_5\psi$;
$\zeta$ dimensionless, undetermined \\
Sakharov conditions
& Not provided
& Candidate ingredients; not sufficient \\
Gluon propagation
& $\Theta=0$ bound (not applicable today)
& Flux-tube connection not established \\
$\Theta$ quantisation
& Semiclassical; not independent
& Idem \\
UV suppression
& Superpolynomial (kernel)
& Idem \\
New dimensionful parameters
& None & None \\
New dimensionless parameters
& None & $\zeta$ (undetermined) \\
Additional structural assumption
& None
& Field-selective $\mathcal{E}_{H,\rm loc}$;
covariant derivation open \\
\midrule
\multicolumn{3}{l}{$^{*}$Metric predictions only; preferred-frame
effects from $U_H^\mu$ not estimated.}\\
\bottomrule
\end{tabular}
\end{table}
The causal-Machian emergent time theory provides a geometrically
grounded framework for embedding quantum field theory. The factor
$\sqrt{\Theta}$ in all field actions follows from the metric
determinant identity \eqref{eq:det}; the sign convention is
$-\frac{1}{2}\hat{g}^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$
throughout. $\Theta$ is a semiclassical constraint functional
requiring no independent quantisation within the present framework.
At the level of the classical fixed-$\Theta$ action the gauge
principle and Clifford algebra are preserved.
In the base model all Standard Model predictions that depend only on
the local metric are reproduced to the precision of
$1-\Theta\approx e^{-10^{60}}$. In the scale-separation model
$\Theta=\Tcosm\cdot\Tloc$ is an approximation scheme to
$\mathcal{O}(10^{-17})$ that separates a quasi-static effective
potential analysis (Level~2) from the fixed-$\Theta$ equations of
motion (Level~1). No cubic Higgs term is derived analytically;
its existence must be determined numerically. The CPT-violating
operator at the bubble wall has a consistent dimension with
undetermined coupling $\zeta$. A geometric connection between gluon
propagation and colour confinement is not established; the flux-tube
energy argument runs counter to the na\"{\i}ve expectation. The
central open task is the self-consistent formulation with
$\delta\Theta/\delta\phi\neq0$.
% =============================================================================
\bibliographystyle{plainnat}
\begin{thebibliography}{99}
\bibitem[Colladay and Kosteleck\'y(1998)]{Colladay1998}
Colladay, D. and Kosteleck\'y, V.~A. (1998).
\newblock Lorentz-violating extension of the standard model.
\newblock \textit{Physical Review D}, 58(11):116002.
\newblock \doi{10.1103/PhysRevD.58.116002}
\bibitem[Greenberg(2002)]{Greenberg2002}
Greenberg, O.~W. (2002).
\newblock {CPT} violation implies violation of Lorentz invariance.
\newblock \textit{Physical Review Letters}, 89(23):231602.
\newblock \doi{10.1103/PhysRevLett.89.231602}
\bibitem[Jaster(2026a)]{Jaster2026a}
Jaster, A. (2026a).
\newblock Emergent time: {A} causal-{M}achian time field coupled to matter.
\newblock Preprint, Zenodo.
\newblock \doi{10.5281/zenodo.20354175}
\bibitem[Jaster(2026b)]{Jaster2026b}
Jaster, A. (2026b).
\newblock Planck-scale smearing of the causal time field:
{B}oundary regularisation of the emergent time theory.
\newblock Preprint, Zenodo.
\newblock \doi{10.5281/zenodo.20135479}
\bibitem[Page and Wootters(1983)]{Page1983}
Page, D.~N. and Wootters, W.~K. (1983).
\newblock Evolution without evolution: Dynamics described by stationary
observables.
\newblock \textit{Physical Review D}, 27(12):2885--2892.
\newblock \doi{10.1103/PhysRevD.27.2885}
\bibitem[Parker and Toms(2009)]{ParkerToms2009}
Parker, L.~E. and Toms, D.~J. (2009).
\newblock \textit{Quantum Field Theory in Curved Spacetime:
Quantized Fields and Gravity}.
\newblock Cambridge University Press, Cambridge.
\newblock \doi{10.1017/CBO9780511813924}
\bibitem[Quiros(1999)]{Quiros1999}
Quiros, M. (1999).
\newblock Finite temperature field theory and phase transitions.
\newblock In \textit{Proceedings of the ICTP Summer School in High-Energy
Physics and Cosmology}, pages 187--259.
\newblock \href{https://arxiv.org/abs/hep-ph/9901312}{arXiv:hep-ph/9901312}
\bibitem[Sakharov(1967)]{Sakharov1967}
Sakharov, A.~D. (1967).
\newblock Violation of {CP} invariance, {C} asymmetry, and baryon asymmetry
of the universe.
\newblock \textit{JETP Letters}, 5:24--27.
\bibitem[Weinberg(1967)]{Weinberg1967}
Weinberg, S. (1967).
\newblock A model of leptons.
\newblock \textit{Physical Review Letters}, 19(21):1264--1266.
\newblock \doi{10.1103/PhysRevLett.19.1264}
\end{thebibliography}
\end{document}