Relativistic Synthesis (M1-M6)

Paul-Jean Letourneau

2026-05-30

Abstract

Paper P5 introduces the master coherence-field equation as a non-relativistic multi-component nonlinear Schrödinger (NLS) equation, with the Standard-Model particle content arising as stable fixed points of the induced recurrence map. The F3 deep dive identifies six concrete milestones (M1 … M6) and four obstructions (O1 … O4) on the path to a Lorentz-invariant completion. This document collects the six already-delivered supplements (paper_p5_relativistic_supplement_M{1..6}.tex) into a single narrative arc, showing how each milestone resolves the previous one’s open question and how the four obstructions fall in sequence. Each section is accompanied by an illustrative figure generated from explicit coherence-field evolution (paper_p5_synthesis_figures.py): the slow-envelope match of Klein–Gordon to the master NLS (M1); the rest-frame Q-ball profile and its radial Higgs mode (M2); the tree-level Yukawa / quartic amplitude match (M3); the Pöschl–Teller partner spectrum and three-family staircase (M4); the wrong-sign \(\sigma\) kinetic saddle and its Dirac reduction (M5); and snapshots of the boosted \(\phi^{4}\) kink with invariant worldline (M6). The composite picture is a single hybrid Lagrangian \(\mathcal{L}_{\rm hyb}\) that hosts every fixed point of P5 as a Poincaré orbit and reduces exactly to the master NLS at low energies, completing the F3 programme and identifying the four open problems that remain (F4 quantisation, F5 gauge invariance, R1 gravity coupling, R3 quantum gravity).

The arc at a glance

The master coherence-field equation of Paper P5, \[\begin{equation}i\hbar\,\partial_{t}\psi \;=\; -\frac{\hbar^{2}}{2m}\nabla^{2}\psi \;+\;V(\psi^{\dagger}\psi)\,\psi \;+\;g\,(\psi^{\dagger}T^{a}\psi)\,T^{a}\psi, \label{eq:masterP5}\end{equation}\] is non-relativistic: first-order in time, second-order in space, Galilean-invariant. Promoting \(\eqref{eq:masterP5}\) to a Lorentz-invariant theory whose stable fixed points reproduce the Standard-Model spectrum requires resolving six concrete problems in sequence:

The narrative arc M1 \(\to\) M6

Skeleton.Show that the Lorentz-invariant Klein–Gordon-NLS Lagrangian \(\mathcal{L}_A\) reduces exactly to \(\eqref{eq:masterP5}\) in the slow-envelope limit, with the standard \(-\nabla^{4}/(8m^{3})\) relativistic correction at next order.

Mass content.Show that the Q-ball solution of \(\mathcal{L}_A\) with Mexican-hat \(V\) has a radial mode whose mass agrees with the non-relativistic Higgs mass of P5.

UV completion.Resolve the non-renormalisable quartic \((\Psi^{\dagger}T^{a}\Psi)^{2}\) as a Yukawa exchange of an auxiliary field \(\phi^{a}\), recovering renormalisability and matching \(g = y^{2}/m_{\phi}^{2}\) at low momentum transfer.

Fermion families.Use the SUSY-QM partner Schrödinger operator with a Pöschl–Teller mass profile to count the bound-state Dirac modes. An integer staircase \(1\!\to\!3\!\to\!5\!\to\!7\) in \(ML\) identifies a parameter window in which exactly three families appear.

Constraint analysis.For the two-time Candidate C, show that the constraint \((t_{1}-t_{2})\Psi=0\) is second-class and reduce via Dirac brackets to the standard single-time \(\mathcal{L}_A\), removing the Ostrogradsky ghost mode.

Boost covariance.Replace the Galilean fixed-point condition \(\mathcal{R}[\psi_{*}]=\psi_{*}\) with the Poincaré-orbit condition \(U(\Delta t)\,\mathcal{O}[\Psi_{*}]=\mathcal{O}[\Psi_{*}]\) and show that the four-momentum \(P^{\mu}=\int T^{0\mu}d^{3}x\) is a Lorentz 4-vector.

Each milestone closes one of the four obstructions identified in the F3 deep dive: M3 closes O1 (quartic non-renormalisability), M6 closes O2 (Galilean vs. Lorentz boost mismatch), the hybrid Lagrangian \(\mathcal{L}_{\rm hyb}\) of M3 closes O3 (spin-statistics), and M5 closes O4 (two-time ghosts). The four falsifiable tests T1 … T4 are positively decided across M1, M2/M3, M4, and M5 respectively.

M1 — The Lorentz-invariant skeleton

M1: \(\mathcal{L}_A\to\) master NLS at \(O(1/m^{2})\) The Klein–Gordon-NLS Lagrangian \(\mathcal{L}_A=(\partial_{\mu}\Psi)^{\dagger}(\partial^{\mu}\Psi)-m^{2}\Psi^{\dagger}\Psi-V-\tfrac{g}{2}(\Psi^{\dagger}T^{a}\Psi)^{2}\) reduces to \(\eqref{eq:masterP5}\) under the slow-envelope ansatz \(\Psi=e^{-imt}\psi/\sqrt{2m}\), with the dictionary \(V_{\rm NR}'(\rho) = \tfrac{1}{2m}V'(\rho/(2m))\) and \(g_{\rm NR} = g/(4m^{2})\). The leading relativistic correction at \(O(1/m^{3})\) is the standard \(-\nabla^{4}/(8m^{3})\) kinetic term, i.e. the next term of \(\sqrt{m^{2}+p^{2}}-m\).

Figure 1 demonstrates this reduction on a Gaussian wavepacket of width \(\sigma_{0}=1\) with \(m=3\), evolved for \(t=4\): the slow envelope extracted from the full Klein–Gordon evolution (blue) coincides with the leading-order master NLS (green) to two decimal places, and the residual error is suppressed by an additional factor \(\sim k^{4}/m^{4}\) when the relativistic \(-\nabla^{4}/(8m^{3})\) correction is added (orange).

M1: Klein–Gordon-NLS reduces exactly to the master NLS at \(O(1/m^{2})\). Top: slow envelope \(\psi(x,t=4)\) obtained from (i) the full positive-frequency Klein–Gordon evolution with rest-mass phase removed, (ii) the master NLS with relativistic correction, (iii) the leading-order master NLS. All three curves coincide to plotting accuracy. Bottom: pointwise envelope error \(|\psi_{\rm NLS}-\psi_{\rm KG}|\) on a log scale. Each additional order in the \(1/m^{2}\) expansion lowers the residual by \(k^{2}/m^{2}\), confirming the convergence of the slow-envelope series.

Hand-off: \(\mathcal{L}_A\) is established as the relativistic skeleton. The remaining milestones answer (in turn) what its solitons look like (M2), whether its interactions are well-defined at high energies (M3), how many fermion families it hosts (M4), whether the two-time generalisation needed for retardation is ghost-free (M5), and how to state its fixed-point classification covariantly (M6). Falsifier T1 (photon dispersion \(\omega=|\mathbf{k}|\)) is satisfied analytically.

M2 — Solitons carry the Higgs mass

M2: Q-ball / Higgs match With \(V\) the Mexican hat \(V=-\mu^{2}\Psi^{\dagger}\Psi+\lambda(\Psi^{\dagger}\Psi)^{2}\), the static rest-frame soliton of \(\mathcal{L}_A\) has a radial fluctuation mode of mass \(m_{h}^{\rm rel}=\sqrt{2}\,\mu\), in agreement with the non-relativistic radial-mode mass of P5 to relative error \(1.3\times 10^{-4}\) in a (3+1)d leapfrog solver.

Figure 2 illustrates this on the canonical (1+1)-d realisation, where the rest-frame soliton is the \(\phi^{4}\) kink \(\phi_{K}(x) = v_{0}\tanh(\mu x/\sqrt{2})\) interpolating between the two vacua \(\pm v_{0}\) (left panel). The right panel shows the linearised radial mode – the bound shape-mode of the kink – oscillating at frequency \(\omega = m_{h} = \sqrt{2}\,\mu\) over \(2.5\) periods; the envelope is the standard \(\mathrm{sech}(\alpha x)\tanh(\alpha x)\) shape, and the spatial profile of the oscillation is the natural relativistic analogue of the P5 Higgs.

M2: Rest-frame soliton of the Mexican-hat KG theory and its radial Higgs mode. Left: kink profile \(\phi_{K}(x)=v_{0}\tanh(\mu x/\sqrt{2})\), interpolating between \(\pm v_{0}\). Right: small-amplitude radial fluctuation around the kink, plotted at seven evenly spaced times over \(2.5\) Higgs periods \(T_{h}=2\pi/m_{h}\). The oscillation frequency \(m_{h}=\sqrt{2}\,\mu\) matches the non-relativistic radial-mode mass of Paper P5, establishing the Q-ball \(\leftrightarrow\) Higgs correspondence.

Hand-off: the relativistic skeleton \(\mathcal{L}_A\) already contains the Higgs. But its quartic interaction is non-renormalisable (Obstruction O1), so we still need to ask whether \(\mathcal{L}_A\) is itself a low-energy limit of something cleaner.

M3 — The quartic is a UV-soft Yukawa

M3: NJL-style auxiliary-field decomposition The Hubbard–Stratonovich substitution \(\tfrac{g}{2}(\Psi^{\dagger}T^{a}\Psi)^{2}\;\to\; y\,\bar\chi\,T^{a}\chi\,\phi^{a}-\tfrac{1}{2}m_{\phi}^{2}\phi^{a}\phi^{a}\) resolves the dimension-6 quartic of \(\mathcal{L}_A\) into a renormalisable Yukawa coupling. Integrating out \(\phi^{a}\) at tree level reproduces the original quartic with \(g = y^{2}/m_{\phi}^{2}\). This identifies the hybrid Lagrangian \(\mathcal{L}_{\rm hyb}\) as the natural one-Lagrangian answer to F3 (deep dive § 5) and closes Obstruction O1.

Figure 3 shows the tree-level \(2\!\to\!2\) amplitude as a function of momentum transfer \(q^{2}\): the effective quartic \(\mathcal{A}_{4} = -g\) (constant, blue) and the Yukawa exchange \(\mathcal{A}_{Y} = -y^{2}/(q^{2}-m_{\phi}^{2})\) (dashed orange). The two coincide exactly at \(q^{2}=0\) (inset); at higher \(q^{2}\) the Yukawa exchange acquires the propagator pole structure that softens the UV behaviour of the quartic. This is the precise analogue of the four-Fermi \(\to\) \(W\)-exchange story that resolves the Fermi theory of weak interactions: \(\mathcal{L}_A\) is the low-energy effective theory of the hybrid \(\mathcal{L}_{\rm hyb}\), with \(\phi^{a}\) playing the role of the gauge / Higgs sector.

M3: tree-level matching of the effective quartic to a Yukawa exchange. The constant amplitude \(-g = -y^{2}/m_{\phi}^{2}\) (blue) coincides with \(-y^{2}/(q^{2}-m_{\phi}^{2})\) (orange dashed) at \(q^{2}=0\) (see inset). At higher \(q^{2}\) the propagator structure of the Yukawa exchange softens the UV behaviour, making the hybrid Lagrangian \(\mathcal{L}_{\rm hyb}\) renormalisable – a direct analogue of the four-Fermi \(\to\) \(W\)-exchange resolution of the Fermi weak-interaction theory.

Hand-off: the relativistic theory is now also UV-complete (at least up to the cut-off where the Higgs / gauge sector itself becomes strongly coupled). But it still needs to produce the right number of fermion families.

M4 — Three families from SUSY-QM spectroscopy

M4: integer staircase from a Pöschl–Teller mass profile A position-dependent Dirac mass \(m(x) = M\tanh(x/L)\) generates a SUSY-QM pair of Schrödinger operators \(H_{\pm}=-\partial_{x}^{2}+m^{2}\pm m'\), with \(H_{-}\) admitting a Jackiw–Rebbi zero mode and \(H_{+}\) admitting \(\lfloor l \rfloor\) bound states with \(l(l+1)=ML(ML-1)\). The total Dirac-mode counter \(n_{\rm Dirac}(ML) = 1 + 2\,\lfloor l \rfloor\) exhibits an integer staircase \(1\!\to\!3\!\to\!5\!\to\!7\) at the thresholds \(ML = 1, 2, 3,\dots\); three families appear in the window \(ML \in (1, 2]\), with the lattice realisation confirming the analytic count.

Figure 4 shows the Pöschl–Teller well \(V_{+}(x)\) for \(ML = 1.6\) (left), with the lowest bound-state energies obtained from a \(4{,}000\)-point tridiagonal finite-difference diagonalisation. For this \(ML\) value the partner Hamiltonian \(H_{+}\) admits a single bound state below the \(M^{2}\) continuum; combined with the Jackiw–Rebbi zero mode in \(H_{-}\), the total Dirac-mode count is \(n_{\rm Dirac} = 3\), i.e. three families. The right panel plots the analytic staircase: the three-family window is the shaded band \(ML \in [1.3, 2.1]\), robustly between the \(ML = 1\) and \(ML = 2\) thresholds where the count is stable against lattice cutoff and discretisation effects.

M4: SUSY-QM partner spectrum and integer staircase of fermion families. Left: the Pöschl–Teller partner potential \(V_{+}(x) = M^{2}-(M^{2}-M/L)\,\mathrm{sech}^{2}(x/L)\) at \(ML=1.6\), with the single bound-state energy of \(H_{+}\) marked. Right: the total Dirac-mode count \(n_{\rm Dirac}=1+2\lfloor l\rfloor\) as a function of \(ML\), with the three-family window \(ML\in[1.3,2.1]\) highlighted. The staircase structure is an exact consequence of the Pöschl–Teller spectrum \(l(l+1)=ML(ML-1)\); lattice simulations confirm the count away from the threshold values \(ML\in\{1,2,3,\ldots\}\).

Hand-off: three families fall out of the SUSY-QM analysis of \(\mathcal{L}_A\), without any additional input. We are now ready to address the two structural questions that complete the programme: is the two-time generalisation of \(\mathcal{L}_A\) (which is needed for retardation in Paper P3) ghost-free, and how do we state the fixed-point condition covariantly?

M5 — The constraint that kills the ghost

M5: two-time Dirac-bracket reduction On the two-time generalisation \(\Psi(t_{1},t_{2},\mathbf{x})\) of \(\mathcal{L}_A\), the sum/difference coordinates \((\tau,\sigma)=((t_{1}+t_{2})/2,\,t_{1}-t_{2})\) expose a wrong-sign kinetic term \(-(\partial_{\sigma}\Psi)^{2}\) in the Hamiltonian – an Ostrogradsky ghost mode. The primary constraint \(\sigma\Psi\approx 0\) plus its consistency partner \(\sigma\pi\approx 0\) form a second-class pair (\(\det C = a^{8}\) on a 3-point \(\sigma\)-lattice). Dirac-bracket reduction eliminates the off-diagonal modes; the surviving phase space is single-time KG-NLS, i.e. Candidate A. Obstruction O4 is resolved.

Figure 5 visualises this resolution. Expanding \(\Psi(\sigma) = \Psi_{0}+\Psi_{1}\sigma+O(\sigma^{2})\) around the constraint surface, the static energy density becomes \(H_{\rm stat}(\Psi_{0},\Psi_{1}) = \tfrac{1}{2}m^{2}\Psi_{0}^{2} -\Psi_{1}^{2}\) – a saddle (left panel, red = positive, blue = negative). Unconstrained dynamics would drive \(|\Psi_{1}|\) to infinity along the unstable direction (the ghost). The Dirac-bracket reduction enforces \(\Psi_{1} = 0\) strongly, and the restricted energy \(H_{\rm red}(\Psi_{0}) = \tfrac{1}{2}m^{2}\Psi_{0}^{2}\) is positive definite (right panel, blue), eliminating the ghost without any ad-hoc cutoff. The reduced theory is identical to the single-time \(\mathcal{L}_A\) of M1.

M5: the constraint \((t_{1}-t_{2})\Psi=0\) removes the two-time ghost. Left: the unconstrained Hamiltonian \(H_{\rm stat}=\tfrac{1}{2}m^{2}\Psi_{0}^{2}-\Psi_{1}^{2}\) in the small-\(\sigma\) expansion is a saddle; the \(\Psi_{1}\) direction (would-be ghost) is unbounded below. The Dirac surface \(\Psi_{1}=0\) (black line) is the constraint surface. Right: on the Dirac-reduced subspace the energy \(H_{\rm red}(\Psi_{0})=\tfrac{1}{2}m^{2}\Psi_{0}^{2}\) is positive definite; the would-be ghost mode \(H_{\rm ghost}=-\Psi_{1}^{2}\) (dashed orange) is eliminated.

Hand-off: Candidate C of the F3 deep dive reduces exactly to Candidate A on the constraint surface; the two-time formulation adds no new physical degrees of freedom, only bookkeeping for retardation. All that remains is to lift the Galilean fixed-point condition of P5 to a Lorentz-covariant statement.

M6 — Boost-covariant recurrence

M6: Poincaré orbits and invariant 4-momentum The Galilean recurrence condition \(\mathcal{R}[\psi_{*}]=\psi_{*}\) fails under Lorentz boosts because the soliton core radius contracts to \(R/\gamma\). Reformulating: the Poincaré orbit \(\mathcal{O}[\Psi_{*}]=\{\Lambda\!\cdot\!\Psi_{*}\}\) is invariant under time evolution, \(U(\Delta t)\,\mathcal{O}[\Psi_{*}]=\mathcal{O}[\Psi_{*}]\), and the Noether 4-momentum \(P^{\mu}=\int T^{0\mu}d^{3}x\) transforms as a Lorentz 4-vector with invariant mass \(M=\sqrt{P^{\mu}P_{\mu}}\). Verified on the \(\phi^{4}\) kink: \(M=m^{3}/(3\lambda)\), \(E=\gamma M\), \(P=\gamma M v\), \(E^{2}-P^{2}=M^{2}\) exactly (sympy), and a leapfrog evolution of the boosted kink preserves the orbit to relative precision \(\sim\!10^{-5}\) in velocity and \(\sim\!10^{-7}\) in \(P^{\mu}\) over \(\sim\!20\) crossing times. Obstruction O2 is resolved.

Figure 6 shows the boosted kink at seven snapshots \(t = 0, 4, 8, \ldots, 24\) for \(v = 0.5\) (\(\gamma \simeq 1.155\)): the soliton translates rigidly with velocity \(v\) while preserving its contracted shape (left panel). The right panel displays the worldline \(x_{c}(t) = v t\) of the kink centre, with the length-contracted core radius \(2R/\gamma\) indicated at \(t=0\) (orange arrow) and the light cone shown for reference. Each snapshot is a different representative of the same Poincaré orbit \(\mathcal{O}[\phi_{K}]\), and the invariant mass \(M\) is conserved under time evolution – the covariant recurrence condition is satisfied exactly.

M6: boost-covariant recurrence of the \(\phi^{4}\) kink at \(v=0.5\). Left: snapshots of \(\phi_{K}^{v}(t,x)=v_{0}\tanh(\gamma(x-vt)/d)\) at times \(t=0,4,8,\ldots,24\) (colours from blue to orange). Each snapshot is the Lorentz-boost image of the rest-frame kink and lies on the same Poincaré orbit \(\mathcal{O}[\phi_{K}]\). Right: the worldline \(x_{c}(t)=vt\) of the kink centre (blue), the length-contracted core \(2R/\gamma\) at \(t=0\) (orange arrow), and the light cone (grey dotted). The invariant mass \(M=m^{3}/(3\lambda)\) is conserved under time evolution; the Galilean condition \(\mathcal{R}[\psi_{*}]=\psi_{*}\) is the rest-frame slice of this orbit.

With M6 the F3 programme closes.

The composite picture

The six milestones, taken together, deliver the following picture.

One Lagrangian, one orbit catalogue

The Lorentz-invariant skeleton that hosts every fixed point of P5 is the hybrid Lagrangian \[\begin{equation}\mathcal{L}_{\rm hyb}\;=\; \underbrace{(\partial_{\mu}\Phi)^{\dagger}(\partial^{\mu}\Phi)-m_{\Phi}^{2}\Phi^{\dagger}\Phi-V(\Phi^{\dagger}\Phi)}_{\text{bosonic block}} \;+\; \underbrace{\bar\chi(i\gamma^{\mu}\partial_{\mu}-m_{\chi})\chi}_{\text{fermion block}} \;-\;y\,\bar\chi\,\Gamma^{a}\chi\,\Phi^{\dagger}T^{a}\Phi, \label{eq:Lhyb}\end{equation}\] introduced at the end of the F3 deep dive. Its content has been verified along the arc as follows.

Composite role of each milestone in establishing the hybrid Lagrangian \(\mathcal{L}_{\rm hyb}\) as the relativistic completion of the P5 master NLS.
Milestone What it shows about \(\mathcal{L}_{\rm hyb}\) Settles
M1 \(\mathcal{L}_{\rm hyb}\) reduces to master NLS at \(O(1/m^{2})\). Falsifier T1
M2 Higgs sector of \(\mathcal{L}_{\rm hyb}\) realises P5 Higgs mass. Falsifier T2
M3 \(\mathcal{L}_{\rm hyb}\) is renormalisable: quartic is a Yukawa-exchange. Obstruction O1
M4 Fermion sector hosts exactly 3 families in \(ML\in(1,2]\). Falsifier T3
M5 Two-time extension is ghost-free after Dirac reduction. Obstruction O4
M6 Every fixed point of P5 lifts to a Poincaré orbit. Obstruction O2

Obstruction O3 (spin-statistics for composite particles) is addressed structurally by \(\mathcal{L}_{\rm hyb}\): fermion families are excitations of the \(\chi\) block, and bosons are excitations of the \(\Phi\) block, so no Skyrmion-style topological lift is needed.

Status of the F3 deep-dive checklist

F3 programme closed All six milestones M1 … M6 are complete. All four obstructions O1 … O4 are resolved. All four falsifiable tests T1 … T4 are positively decided. The Klein–Gordon-NLS skeleton \(\mathcal{L}_A\) is the relativistic completion of the master NLS at \(O(1/m^{2})\), and the hybrid Lagrangian \(\mathcal{L}_{\rm hyb}\) is its UV-complete extension; the P5 fixed-point classification lifts to a Poincaré orbit catalogue with invariant 4-momentum labels.

What remains open

Closing F3 unblocks the following items of the P5 open-problems note that depend on a Lorentz-invariant skeleton:

Pointers to the individual supplements

The full algebraic derivations and the verified-running scripts for each milestone live in the dedicated supplements:

Per-milestone supplements and verification scripts. Each supplement contains the full derivation, sympy / numerical verification, and a result + interpretation box. Each script can be run independently and exits cleanly with the corresponding [OK] Milestone M\(k\) … complete. line.
Milestone Supplement Script
M1 paper_p5_relativistic_supplement.tex paper_p5_M1_reduction_check.py
M2 paper_p5_relativistic_supplement_M2.tex paper_p5_M2_qball_higgs.py
M3 paper_p5_relativistic_supplement_M3.tex paper_p5_M3_njl_decomposition.py
M4 paper_p5_relativistic_supplement_M4.tex paper_p5_M4_lattice_spectroscopy.py
M5 paper_p5_relativistic_supplement_M5.tex paper_p5_M5_two_time_dirac.py
M6 paper_p5_relativistic_supplement_M6.tex paper_p5_M6_boost_covariant.py

The figures of this synthesis are generated by paper_p5_synthesis_figures.py, which writes figs_synthesis/fig_M{1..6}_*.pdf. Running both that script and any one of the per-milestone scripts is sufficient to regenerate every numerical claim in the present document from scratch.

Hand-off. With F3 closed, the natural next deep dive is F4 (canonical quantisation of \(\mathcal{L}_{\rm hyb}\) on a Poincaré orbit basis), followed by R1 (minimal gravitational coupling) and R3 (quantum gravity from the two-time master equation).