Helium under the Refined Ansatz

Paul-Jean Letourneau

2026-05-30

Abstract

The M1–M6 milestones of the P5 relativistic deep dive deliver a single, self-consistent ansatz for the slow envelope of a coherence field: (M5) one physical time after the Dirac reduction of the two-time formalism, (M1) the leading \(1/m^{2}\) correction \(-(\nabla^{2})^{2}/(8m^{3})\) to the Schrödinger kinetic operator, (M3) a soft-Yukawa interpretation of contact interactions whose short-distance trace on the bare Coulomb kernel is the Darwin term, and (M6) rigid boost covariance for the centre-of-mass mode. Applied to the helium ground state, this reduces to: (i) an ordinary single-time variational \(1s^{2}\) wavefunction; (ii) a closed-form \(\mathcal{O}(\alpha^{2}Z^{4})\) correction consisting of mass-velocity (M1) and Darwin (M3) contributions; and (iii) a Z-scaling prediction along the He-isoelectronic series. We evaluate all pieces analytically on the screened-charge \(1s^{2}\) state \({Z^{\prime}}= Z - 5/16\), exhibit the \(Z^{2}\) scaling of the binding and the \(Z^{4}\) scaling of the relativistic shift, and identify the residual \(\sim 10^{-3}\,\mathrm{Ha}\) gap to experiment as electron correlation — a deficiency of the trial wavefunction, not of the M-series corrections. This supersedes the older two-time density-matrix treatment of paper_p3_relativistic_supplement.tex, which introduced spurious \(t_{1},t_{2}\) degrees of freedom that M5 has now shown to be pure Dirac-redundant gauge.

What the refined ansatz says, and what it kills

The two-time density operator \(\rho(t_{1},\mathbf{r}_{1};t_{2},\mathbf{r}_{2})\) of the original P3 helium supplement was introduced to keep each electron’s coordinate time manifestly Lorentz-covariant. In the M-series synthesis, M5 (two-time Dirac brackets) shows that the second time direction is a constraint surface populated by second-class constraints \(\phi_{1}=\sigma\Psi\), \(\phi_{2}=\sigma\pi\); the Dirac reduction kills the would-be ghost completely, leaving a single physical time \(\tau\) and an ordinary state \(\psi(\tau;\mathbf{r}_{1},\mathbf{r}_{2})\). Equivalently: \[\rho_{\mathrm{two\text{-}time}}(t_{1},\mathbf{r}_{1};t_{2},\mathbf{r}_{2}) \;\xrightarrow[\;\text{M5}\;]{\;\text{Dirac reduction}\;}\; \delta(t_{1}-t_{2})\,\psi(t;\mathbf{r}_{1},\mathbf{r}_{2})\psi^{*}(t;\mathbf{r}'_{1},\mathbf{r}'_{2}),\] not as a non-relativistic approximation but as a constraint identity at the Hamiltonian level.

The remaining content of relativity is then injected by:

• M1 (KG \(\to\) NLS reduction). Expanding \(\sqrt{p^{2}+m^{2}} - m = \tfrac{p^{2}}{2m} - \tfrac{p^{4}}{8m^{3}} + \mathcal{O}(p^{6}/m^{5})\) produces the mass-velocity correction \[H_{\mathrm{MV}} \;=\; -\,\frac{1}{8m^{3}c^{2}}\bigl(\nabla_{1}^{4}+\nabla_{2}^{4}\bigr).\] This is the only kinetic correction that survives at \(\mathcal{O}(v^{2}/c^{2})\) in the M5-reduced sector.

• M3 (quartic = soft-Yukawa contact). The four-fermion contact piece of the e–e interaction is the \(q^{2}\!\to\!0\) limit of a Yukawa exchange of mass \(m_{\phi}\). Its short-distance trace on the bare Coulomb propagator \(1/q^{2}\) is the Darwin contact term, generated separately for the nuclear attraction and the electron repulsion: \[H_{\mathrm{D}}^{(\mathrm{eN})} = \frac{\pi Z\alpha^{2}}{2}\bigl[\delta^{3}(\mathbf{r}_{1})+\delta^{3}(\mathbf{r}_{2})\bigr], \qquad H_{\mathrm{D}}^{(\mathrm{ee})} = -\pi\alpha^{2}\,\delta^{3}(\mathbf{r}_{12}).\]

• M6 (boost covariance). The full \(\psi^{4}\) kink obeys \(E^{2}-P^{2}=M^{2}\) exactly; in the bound atomic problem the centre-of-mass mode separates rigidly and the internal Hamiltonian above is the boost-invariant restatement.

• M2 (soliton mass). The single-electron rest mass is the trivial soliton mass \(m_{e}\); no Q-ball internal structure is excited in the ground state.

• M4 (three families). Irrelevant to a single isotope of helium; would govern any putative second-family analog \(\mu^{-}\mu^{-}\) muonic helium.

Variational baseline: screened \(1s^{2}\)

We take the simplest ansatz that the M5 reduction permits, \[\psi(\mathbf{r}_{1},\mathbf{r}_{2}) \;=\; \phi_{{Z^{\prime}}}(\mathbf{r}_{1})\,\phi_{{Z^{\prime}}}(\mathbf{r}_{2})\,\chi_{\mathrm{singlet}}, \qquad \phi_{{Z^{\prime}}}(\mathbf{r}) = \sqrt{{Z^{\prime}}^{3}/\pi}\,e^{-{Z^{\prime}}r},\] with \({Z^{\prime}}\) a single variational screened-charge parameter. The non-relativistic expectation value of \(H_{0} = T_{\mathrm{NR}} + V_{\mathrm{eN}} + V_{\mathrm{ee}}\) is the textbook closed form \[E_{0}({Z^{\prime}}) \;=\; {Z^{\prime}}^{2}\;-\;2Z{Z^{\prime}}\;+\;\tfrac{5}{8}{Z^{\prime}}, \qquad {Z^{\prime}}_{*} = Z-\tfrac{5}{16},\quad E_{0,*} = -\bigl(Z-\tfrac{5}{16}\bigr)^{2}.\] For \(Z=2\) this gives \({Z^{\prime}}_{*} = 27/16 = 1.6875\) and \(E_{0,*} = -2.847656\,\mathrm{Ha}\), which the script paper_p5_helium_refined.py reproduces to six digits.

Refined-ansatz corrections, closed-form on \({Z^{\prime}}_{*} = Z - 5/16\)

Substituting the matrix elements above: \[\begin{aligned} \Delta E_{\mathrm{MV}}^{(\mathrm{M1})} &= -\,\frac{1}{8c^{2}}\cdot 2\cdot\langle p^{4}\rangle_{1s,{Z^{\prime}}} \;=\; -\,\frac{5\,{Z^{\prime}}^{4}}{4c^{2}},\\[2pt] \Delta E_{\mathrm{D,eN}}^{(\mathrm{M3})} &= \frac{\pi Z\alpha^{2}}{2}\cdot 2|\phi_{{Z^{\prime}}}(0)|^{2} \;=\; Z\,\alpha^{2}\,{Z^{\prime}}^{3},\\[2pt] \Delta E_{\mathrm{D,ee}}^{(\mathrm{M3})} &= -\pi\alpha^{2}\cdot\frac{{Z^{\prime}}^{3}}{8\pi} \;=\; -\,\frac{\alpha^{2}{Z^{\prime}}^{3}}{8}. \end{aligned}\] For \(Z=2\), \({Z^{\prime}}=27/16\), these evaluate to \[\Delta E_{\mathrm{MV}} = -5.398\times 10^{-4}\,\mathrm{Ha},\quad \Delta E_{\mathrm{D,eN}} = +5.118\times 10^{-4}\,\mathrm{Ha},\quad \Delta E_{\mathrm{D,ee}} = -3.199\times 10^{-5}\,\mathrm{Ha},\] summing to \(\Delta E_{\mathrm{rel}}^{\mathrm{M1+M3}} = -6.00\times 10^{-5}\,\mathrm{Ha}\). The near-cancellation between M1 (negative) and M3-\(eN\) (positive) is the leading structural feature of the refined ansatz for hydrogen-like \(ns\) states; it survives all the way down the isoelectronic series.

The point of the budget is not the absolute number; the screened \(1s^{2}\) trial state is too crude to compete with Hylleraas. The point is that the M1+M3 corrections are evaluated in closed form on the trial state, they sum to \(\sim 10^{-4}\,\mathrm{Ha}\) as expected from \((\alpha Z)^{2}E_{0}\), and they realise the predicted sign pattern (mass-velocity lowers, nuclear-Darwin raises, electron-Darwin lowers slightly).

Isoelectronic scaling: \(Z^{2}\) for binding, \(Z^{4}\) for relativity

The closed-form expressions above are functions only of \(Z\) and \({Z^{\prime}}= Z - 5/16\). Evaluating them across the He-isoelectronic series \(\{H^{-},\,He,\,Li^{+},\,Be^{2+},\,B^{3+},\,C^{4+},\,N^{5+},\,O^{6+},\,F^{7+},\,Ne^{8+}\}\) exposes the structural \(Z\)-scaling of every refined-ansatz piece (Fig. 1).

Cumulative budget and where the residual lives

Fig. 2 stacks the four pieces of the refined-ansatz budget against the experimental ground-state energy. On the resolution of the binding energy (\(\sim 0.1\,\mathrm{Ha}\)) the relativistic corrections are invisible; on the milli-Hartree resolution of the right panel the M1/M3 cancellation pattern is clear.

Bridge to F4: where Breit and Lamb enter

The refined ansatz does not pretend to deliver the \(\sim 3\times 10^{-4}\,\mathrm{Ha}\) QED-level accuracy of modern multi-thousand-term Hylleraas calculations. What it does is identify, in closed form, exactly which \(\mathcal{O}(\alpha^{2})\) operators are mandated by the M5-reduced single-time Hamiltonian (M1 + M3-Darwin) and which require additional input. The remaining \(\mathcal{O}(\alpha^{2})\) pieces and their M-series provenance are:

The Breit interaction is the leading retardation correction to \(V_{ee}\) and lives outside the M3 soft-Yukawa reduction (M3 captures the contact piece; Breit is the magnetic piece of the same expansion to one order higher). The Lamb shift requires F4-level quantisation of the slow envelope and is by construction beyond the F3 classical-field programme that the M-series closes.

Comparison to the older two-time treatment

The older P3 supplement paper_p3_relativistic_supplement.tex introduced a two-time density operator \(\rho(t_{1},\mathbf{r}_{1};t_{2},\mathbf{r}_{2})\) together with a synchronisation penalty \(\gamma(t_{1}-t_{2})^{2}\), recovering single-time quantum mechanics only in the limit \(\gamma\!\to\!\infty\). The M5 analysis now shows this synchronisation limit is not a limit at all: the constraints \(\phi_{1}=\sigma\Psi\), \(\phi_{2}=\sigma\pi\) are second-class with non-degenerate Dirac matrix \(\det C = a^{8}\), so the \(\sigma\)-direction is pure Dirac-redundant gauge and \(\delta(t_{1}-t_{2})\) is enforced as an identity at the Hamiltonian level. Consequently:

• The auxiliary parameter \(\gamma\) disappears from every observable; it is a Lagrange multiplier for an identically-vanishing constraint.

• The retarded-Coulomb interaction kernel \(V_{12}(t_{1},t_{2},\mathbf{r}_{1},\mathbf{r}_{2})\) collapses to the instantaneous Coulomb \(1/r_{12}\) in the M5-reduced sector, with the leading retardation correction being precisely the Breit term that lives in F4.

• The two-time Monte-Carlo apparatus of helium_relativistic_cft.py is superseded by the closed-form algebra above on the M5-reduced single-time state; the new script paper_p5_helium_refined.py is \(\sim 250\) lines and runs in seconds.

The improvement in compactness reflects the structural lesson of M5: relativistic covariance for a bound atomic system does not require giving each particle its own time coordinate. Boost covariance lives on the centre-of-mass mode (M6); the internal dynamics live in the single-time M5-reduced sector and are governed by \(T_{\mathrm{NR}} + V + H_{\mathrm{MV}} + H_{\mathrm{D}}\).

Summary

Pointers