Euclidean Coherence Lagrangian

derived

Overview

This derivation supplies a prerequisite for any thermal or tunneling calculation in the framework: what is the Euclidean continuation of the Coherence Lagrangian, and does it behave as standard Euclidean field theory on observer-projected spacetime?

The Lorentzian Coherence Lagrangian — scalar + gauge + gravity from Coherence Lagrangian Theorem 3.1, extended to the spinor sector via Spinor Coherence Lagrangian Theorem 2.7 — is the framework’s classical dynamics. Quantum effects in the semi-classical (WKB) limit, thermal field theory at Gibbons–Hawking temperature, and instanton/bounce amplitudes all require the Euclidean continuation. This derivation makes that continuation explicit.

The approach. Each observer projects its own Lorentzian static de Sitter patch $M_A$ (from Observer-Projected Spacetime Theorem 3.1). The patch has a distinguished static Killing vector — the generator of $A$‘s own rest-frame time translations — which fixes the Wick rotation $t \to -i\tau$ unambiguously. Absence of conical singularity at the horizon $r = L_A$ determines the Euclidean time periodicity, reproducing the Gibbons–Hawking inverse temperature $\beta_A = 2\pi L_A/c = \pi T_A$. The Euclidean Coherence Lagrangian is then the standard Wick-rotated form of the Lorentzian Lagrangian, with fermion fields treated as independent Grassmann variables and anti-periodic in Euclidean time.

The result. The Euclidean Coherence Lagrangian $\mathcal{L}_E$ is explicit on each observer’s Euclidean patch $M_A^E$, with:

• Bosonic action bounded below (modulo the Gibbons–Hawking–Perry conformal-mode treatment).
• Fermionic action producing the expected Grassmann determinants and Pfaffians.
• Lichnerowicz identity $(\gamma^\mu_E\nabla_\mu)^2 = \Box_E - R_E/4$ on the Euclidean de Sitter background.
• Cross-observer compatibility via the presheaf structure of observer-projected spacetime.

Why this matters. The Euclidean Coherence Lagrangian is the object used by Coherence Bounces to compute WKB tunneling amplitudes and by any future thermal field theory work in the framework (finite-temperature bootstrap analysis, early-universe crystallization, KMS state structure on observer-projected patches).

Statement

Theorem. Under the Wick rotation $t \to -i\tau$ on each observer’s projected static de Sitter patch $M_A$, the Lorentzian Coherence Lagrangian (scalar + gauge + gravity + spinor sectors) admits an unambiguous Euclidean continuation with the following properties:

1. The rotation is fixed by the static Killing vector $\xi_A = \partial_t$ of $M_A$ — a framework-intrinsic object.
2. Absence of conical singularity at the horizon $r = L_A$ forces $\tau$ to be periodic with period $\beta_A = 2\pi L_A/c = \pi T_A$, giving Gibbons–Hawking temperature $T_{\mathrm{GH},A} = \hbar c/(2\pi k_B L_A) = \hbar/(\pi k_B T_A)$.
3. The Euclidean bosonic action is positive-definite on the scalar and gauge sectors; the gravitational sector requires the standard Gibbons–Hawking–Perry conformal-mode treatment.
4. The Euclidean fermionic action takes the standard Grassmann form with $\bar\psi, \psi$ independent and $\tau$-anti-periodic; Majorana fermions contribute via Pfaffians, Dirac fermions via determinants.
5. Cross-observer consistency on shared Type III relational content is inherited from the observer-projected-spacetime presheaf structure — no additional postulate required.

Derivation

Step 1: Wick rotation on a single observer’s patch

Setup. By Observer-Projected Spacetime Theorem 3.1, observer $A$‘s projected continuous dual $M_A$ is the static patch of 4D de Sitter space with de Sitter radius $L_A = cT_A/2$:

$g_A = -\left(1 - \frac{r^2}{L_A^2}\right)c^2\,dt^2 + \left(1 - \frac{r^2}{L_A^2}\right)^{-1}dr^2 + r^2\,d\Omega^2, \qquad r \in [0, L_A).$

The static Killing vector is $\xi_A = \partial_t$; timelike in the interior and null on the horizon.

Proposition 1.1 (Wick rotation). The Euclidean continuation of $M_A$ is

$g_A^E = \left(1 - \frac{r^2}{L_A^2}\right)c^2\,d\tau^2 + \left(1 - \frac{r^2}{L_A^2}\right)^{-1}dr^2 + r^2\,d\Omega^2$

under $\tau := it$. This metric is positive-definite for $r \in [0, L_A)$ and is uniquely fixed by the static Killing vector $\xi_A$.

Proof. Positive-definiteness is direct: each diagonal entry of $g_A^E$ is positive on the stated domain. Uniqueness follows from static-patch rigidity: any other timelike Killing vector on $M_A$ coincides with $\xi_A$ up to a constant rescaling (standard result; see Hawking–Ellis, The Large Scale Structure of Space-Time, §6.3). $\square$

Proposition 1.2 (Euclidean periodicity). To avoid a conical singularity at the horizon $r = L_A$, the Euclidean time coordinate $\tau$ must be periodic with period

$\beta_A = \frac{2\pi L_A}{c} = \pi T_A.$

Equivalently, the Gibbons–Hawking temperature is $T_{\mathrm{GH},A} = 1/\beta_A = \hbar c/(2\pi k_B L_A) = \hbar/(\pi k_B T_A)$.

Proof. Near the horizon, set $r = L_A - \epsilon$ with small $\epsilon > 0$. Expanding: $1 - r^2/L_A^2 \approx 2\epsilon/L_A$. Change radial coordinate to $\rho := \sqrt{2L_A\epsilon}/\sqrt{c^2}$; then $d\rho^2 = (L_A/(2\epsilon))\,c^{-2}\,d\epsilon^2$ and the near-horizon metric reduces to

$g_A^E \approx d\rho^2 + \rho^2\bigl(c\,d\tau/L_A\bigr)^2 + L_A^2\,d\Omega^2.$

Setting $\theta := c\tau/L_A$, the $(\rho, \theta)$ plane is Euclidean flat with polar coordinates. Absence of conical defect requires $\theta$ to be $2\pi$-periodic, hence $\tau$ is $(2\pi L_A/c)$-periodic. $\square$

Remark 1.3 (boundary conditions for fields). Standard thermal-field-theory rules apply on the Euclidean patch: bosonic fields (scalars, gauge fields, metric perturbations) are $\tau$-periodic with period $\beta_A$; fermionic fields are $\tau$-anti-periodic with period $\beta_A$. The anti-periodicity is universal for KMS thermal states at finite temperature and carries through from standard flat-space thermal field theory unchanged.

Step 2: Euclidean bosonic Lagrangian

Proposition 2.1 (Euclidean continuation of the bosonic sectors). Under the Wick rotation of Proposition 1.1, the bosonic sectors of the Coherence Lagrangian transform as:

Scalar sector. The Lorentzian kinetic term $\tfrac12 \hbar G^{\mu\nu}\partial_\mu\phi^*\partial_\nu\phi$ (signature $(+, -, -, -)$) becomes, with $\mathcal{L}_E = -\mathcal{L}_L|_{\text{Wick}}$:

$\mathcal{L}_E^{\text{scalar}} = \tfrac12\hbar\, g_E^{\mu\nu}\partial_\mu\phi^*\partial_\nu\phi + V(\phi)$

with $g_E^{\mu\nu}$ positive-definite and $V(\phi) = m^2|\phi|^2 + \lambda|\phi|^4$ unchanged (potentials are Wick-invariant scalars).

Gauge sector. The Yang–Mills kinetic term $-\tfrac14 F_{\mu\nu}F^{\mu\nu}$ becomes $+\tfrac14 F^E_{\mu\nu}F^{E\,\mu\nu}$ under the Wick rotation, with indices raised by $g_E^{\mu\nu}$.

Gravity sector. The Einstein–Hilbert term $\tfrac{c^4}{16\pi G}\sqrt{-g}\,R$ becomes $\tfrac{c^4}{16\pi G}\sqrt{g_E}\,R_E$. On the Euclideanized static de Sitter patch, $R_E = 12/L_A^2$ (constant positive scalar curvature of $S^4$-type).

Total Euclidean bosonic action:

$S_E^{\text{boson}}[\phi, A_\mu, g_E] = \int_{M_A^E}d^4x_E\,\sqrt{g_E}\,\Bigl[\tfrac12\hbar|\partial\phi|^2 + V(\phi) + \tfrac14 F^E_{\mu\nu}F^{E\,\mu\nu} - \tfrac{c^4}{16\pi G}(R_E - 2\Lambda)\Bigr].$

The scalar and gauge contributions are manifestly non-negative; the gravity sector requires the Gibbons–Hawking–Perry conformal-mode treatment (Gibbons–Hawking–Perry 1978).

Proof. Each term is the direct Wick rotation of the corresponding Lorentzian term via the standard rules: $t \to -i\tau$ (hence $\partial_t \to i\partial_\tau$), $d^4x \to -i\,d^4x_E$, $\eta^{\mu\nu} \to \delta^{\mu\nu}$ (flat-space reduction; on curved $M_A^E$ the metric $g_E^{\mu\nu}$ replaces $\delta^{\mu\nu}$). The sign convention $\mathcal{L}_E = -\mathcal{L}_L|_{\text{Wick}}$ renders the scalar kinetic term positive; the gauge kinetic term changes sign relative to Lorentzian because $F_{\mu\nu}F^{\mu\nu} = -2|E|^2 + 2|B|^2$ in Minkowski and the electric-field term flips under Wick rotation. Full details in Weinberg QFT Vol. II §23.2. $\square$

Step 3: Euclidean fermionic Lagrangian

Setup. From Spinor Coherence Lagrangian Theorem 2.7, the Lorentzian Dirac action on $M_A$ is

$S_L^{\text{spinor}} = \int_{M_A}d^4x\,\sqrt{-g_A}\,\Bigl[i\hbar\,\bar\psi\gamma^\mu\nabla_\mu\psi - mc^2\,\bar\psi\psi\Bigr]$

with $\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}\mathbf{1}$ in signature $(+,-,-,-)$.

Proposition 3.1 (Euclidean Dirac action). Under the Wick rotation of Proposition 1.1 and standard Euclidean conventions, the Dirac action on $M_A^E$ is

$S_E^{\text{spinor}}[\bar\psi, \psi] = \int_{M_A^E}d^4x_E\,\sqrt{g_E}\,\bar\psi\bigl(\gamma^\mu_E\nabla_\mu + m\bigr)\psi$

where:

• The Euclidean gamma matrices $\gamma^\mu_E$ satisfy $\{\gamma^\mu_E, \gamma^\nu_E\} = 2\delta^{\mu\nu}\mathbf{1}$ (positive-definite signature), with $\gamma^0_E := i\gamma^0$ and $\gamma^i_E := \gamma^i$.
• $\bar\psi$ and $\psi$ are treated as independent Grassmann fields — not hermitian conjugates (standard Euclidean Dirac convention; see Zinn-Justin, Quantum Field Theory and Critical Phenomena, §5.1).
• $\nabla_\mu$ is the $M_A^E$-covariant Dirac operator with spin connection from the $M_A^E$ tetrad.
• $\psi$ is $\tau$-anti-periodic with period $\beta_A$ (Remark 1.3).

Proof. Direct Wick rotation of the Dirac action (Zinn-Justin §5.1; Peskin–Schroeder §9.6). The convention $\gamma^0_E = i\gamma^0$ gives $\{\gamma^\mu_E, \gamma^\nu_E\} = 2\delta^{\mu\nu}$. The $i\bar\psi\gamma^0\partial_t\psi$ Lorentzian kinetic piece becomes $\bar\psi\gamma^0_E\partial_\tau\psi$ (real under Euclidean conjugation, as required). The mass term’s sign follows from the $\mathcal{L}_E = -\mathcal{L}_L|_{\text{Wick}}$ convention. $\square$

Proposition 3.2 (Euclidean Majorana action). For a Majorana field $\nu$ satisfying the Majorana condition $\nu^c = \nu$ (from Spinor Coherence Lagrangian Theorem 5.3 or Neutrino Masses Theorem 1.3), the Euclidean action on $M_A^E$ is

$S_E^{\text{Majorana}}[\nu] = \tfrac12\int_{M_A^E}d^4x_E\,\sqrt{g_E}\,\nu^T C\bigl(\gamma^\mu_E\nabla_\mu + M\bigr)\nu$

with $C$ the charge-conjugation matrix. The factor of $1/2$ reflects the Majorana condition; Grassmann Gaussian integration yields a Pfaffian rather than a determinant.

Proof. Standard Euclidean Majorana formalism (Wetterich 1990; Zinn-Justin §5.1). $\square$

Proposition 3.3 (Euclidean Lichnerowicz identity). The square of the Euclidean Dirac operator on $M_A^E$ is

$\bigl(\gamma^\mu_E\nabla_\mu\bigr)^2 = g_E^{\mu\nu}\nabla_\mu\nabla_\nu - \frac{R_E}{4} = \Box_E - \frac{3}{L_A^2}$

where the last equality uses $R_E = 12/L_A^2$ for the Euclidean de Sitter patch. The Laplacian $\Box_E = g_E^{\mu\nu}\nabla_\mu\nabla_\nu$ is negative-semi-definite in Euclidean signature.

Proof. Direct application of the Lichnerowicz identity (Lawson–Michelsohn, Spin Geometry, §II.8) in Euclidean signature using $R_E = 12/L_A^2$ for the 4-sphere-like curvature at radius $L_A$. $\square$

Remark 3.4 (Grassmann path integration). The Euclidean Dirac action is bilinear in Grassmann fields $\bar\psi, \psi$. The path integral

$Z = \int\mathcal{D}\bar\psi\,\mathcal{D}\psi\,e^{-S_E^{\text{spinor}}[\bar\psi,\psi]}$

is Grassmann Gaussian and evaluates to $\det(\gamma^\mu_E\nabla_\mu + m)$ — a determinant in the numerator, not the denominator as for bosonic Gaussian integrals. The Majorana case gives a Pfaffian $\mathrm{Pf}(C\cdot(\gamma^\mu_E\nabla_\mu + M)) = \sqrt{\det(\cdots)}$. These determinantal factors are the fermion one-loop prefactors multiplying any bosonic-sector bounce saddle and are the source of, e.g., the top-quark contribution to the Standard Model Higgs effective potential.

Step 4: Cross-observer consistency

Setup. The Wick rotation of Proposition 1.1 is defined on each observer’s patch $M_A$ individually, using that observer’s own static Killing vector. For observers $A, B$ sharing a Type III relation $I_{AB}$ with restriction morphism $\rho_{AB}: M_A|_{I_{AB}} \to M_B|_{I_{AB}}$ (from Observer-Projected Spacetime Definition 1.2), the induced Wick-rotated restriction $\rho_{AB}^E$ between the Euclidean patches must be consistently defined.

Proposition 4.1 (Cross-observer compatibility). The induced Wick-rotated restriction morphism

$\rho_{AB}^E: M_A^E|_{I_{AB}} \to M_B^E|_{I_{AB}}$

is well-defined. Both sides identify the same level-$(n+1)$ relational content at their common slice. No additional postulate is required; the compatibility is inherited from the observer-projected-spacetime presheaf structure.

Proof. The Type III relation $I_{AB}$ carries level-$(n+1)$ relational coherence content (Coherence Conservation Proposition 5.7). On $M_A$, this content lives on a distinguished timelike worldline $\gamma_B^{(A)}$ with periodicity $T_B$; on $M_B$, on $\gamma_A^{(B)}$ with periodicity $T_A$. Each patch’s Wick rotation acts on its own static Killing vector ($\partial_{t_A}$ for $M_A$; $\partial_{t_B}$ for $M_B$) and Wick-rotates its own partner line with respect to its own ambient time structure. The restriction morphism $\rho_{AB}^E$ identifies the two resulting Euclidean worldlines as the same relational invariant content embedded into the two patches’ Euclidean sections; the relational coherence (a scalar quantity) is invariant under the embedding, so the identification is consistent at the level of Euclidean action integrals over the shared content. $\square$

Remark 4.2. This is a weaker claim than “the two Euclidean patches glue into a single Euclidean manifold” — the framework’s observer-projected structure does not admit a single global Euclidean spacetime any more than it admits a single global Lorentzian one. What Proposition 4.1 establishes is that the relational content — which is all that enters bounce-action integrals and other observables of physical interest — is invariantly defined across observers.

Step 5: Boundedness

Proposition 5.1 (Boundedness of the bosonic action). For scalar fields $\phi$ satisfying $V(\phi) \geq 0$ (which holds by Coherence Lagrangian Theorem 2.2(a)), gauge fields with finite norm, and metric configurations near the classical $M_A^E$ background, the bosonic Euclidean action is bounded below, with the gravitational sector handled by the standard Gibbons–Hawking–Perry prescription.

Proof. Each of the scalar kinetic, scalar potential, and gauge kinetic terms is manifestly $\geq 0$ under Euclidean positive-definite contractions. The gravitational action’s conformal-mode problem is resolved by contour deformation along the imaginary axis for the conformal factor (Gibbons–Hawking–Perry 1978), giving a bounded path integral. At the level of bounce calculations for sub-Planckian fields, gravitational fluctuations are suppressed by $M_{\mathrm{Pl}}^{-2}$ and the conformal-mode issue does not affect leading-order results. $\square$

Remark 5.2 (fermion sector). The Euclidean fermion action is bilinear in Grassmann variables and has no conventional “positivity” — the relevant criterion is the determinant of $(\gamma^\mu_E\nabla_\mu + m)$ (Dirac) or the Pfaffian (Majorana). These are non-zero for generic mass $m \neq 0$ on $M_A^E$ by the Lichnerowicz identity (Proposition 3.3): the operator’s spectrum avoids zero when $m^2 \neq R_E/4 = 3/L_A^2$ (the cosmological-scale fermion mass, negligible for fundamental particles).

Physical Interpretation

Consistency Model

Theorem 6.1. On the minimal observer’s projected patch $M_A$ (de Sitter radius $L_A = cT_A/2$), the Euclidean Coherence Lagrangian reproduces standard thermal de Sitter field theory with Gibbons–Hawking temperature $T_{\mathrm{GH},A} = \hbar/(\pi k_B T_A)$.

Verification:

• Wick rotation (Proposition 1.1) gives the standard Euclidean de Sitter metric on the static patch.
• Periodicity (Proposition 1.2) matches the standard Gibbons–Hawking (1977) result.
• Boson/fermion boundary conditions (Remark 1.3) match standard KMS thermal field theory.
• Lichnerowicz identity (Proposition 3.3) with $R_E = 12/L_A^2$ matches standard $S^4$ curvature result.

*All structural elements reduce to standard thermal de Sitter QFT in the appropriate limit. $\square$

Rigor Assessment

Fully rigorous:

• Propositions 1.1, 1.2: Wick rotation and Gibbons–Hawking period (standard de Sitter thermal field theory)
• Propositions 2.1, 3.1, 3.2: Euclidean bosonic and fermionic actions (standard QFT Wick-rotation rules)
• Proposition 3.3: Lichnerowicz identity Lichnerowicz, 1963
• Proposition 4.1: Cross-observer compatibility (inherits from Observer-Projected Spacetime presheaf structure)
• Proposition 5.1: Bosonic boundedness Gibbons–Hawking–Perry, 1978
• Theorem 6.1: Consistency with standard thermal de Sitter QFT

Assessment: Derived. All structural content is standard QFT Wick-rotation applied to the framework’s already-derived Lorentzian Lagrangian. The framework-specific content — that the rotation lives on observer-projected spacetime rather than a shared background — inherits consistency from Observer-Projected Spacetime Definition 1.2. No structural postulates introduced.

Open Gaps

1. Euclidean–Lorentzian correspondence at cosmological scales. For observables with characteristic scale comparable to $L_A$ (cosmological scale), the simple Wick rotation may receive corrections from boundary conditions at the de Sitter horizon. Addressing these requires Coleman–De Luccia-type analysis of bubble nucleation on curved backgrounds. Not relevant for sub-cosmological bounces (fermion-mass scales) targeted by Coherence Bounces, but an open question for cosmological applications.

2. Infinite-dimensional rigor. The derivation assumes finite-dimensional observer Hilbert spaces. For quantum field theory (infinite-dimensional state spaces), the Wick rotation inherits the standard functional-analytic issues shared with all continuum QFT; no framework-specific new issues arise at this layer.