void_dynamics_theory.md

Note (2025‑08‑20): Canonical model set to reaction-diffusion (RD); the second‑order EFT is quarantined to EFT docs. Mass numerics are parameter‑dependent (m_eff=√(α-β) in EFT). The “promote to second order” gap is closed via a discrete action derivation with wave speed c^2=2 J a^2 (per‑site convention), see derivation/kinetic_term_derivation.md.

This document presents a comparative analysis with Bordag (Universe 2024, “Tachyon Condensation in a Chromomagnetic Center Vortex Background”) and enumerates required corrections.

Agreement with prior literature

• Tachyon → condensation story. The continuum limit yields a tachyonic origin (negative curvature at ϕ=0) with a non‑zero vacuum $v = 1-\beta/\alpha = 0.6$ and positive mass about the minimum $m_\text{eff}^2=\alpha-\beta$【turn3file11】. Bordag likewise starts with tachyonic modes $(m_l^2=-\kappa_l^2)$, expands around constant condensates $v_l$, and obtains positive masses for fluctuations plus massless phase modes (Goldstones) after symmetry breaking【turn4file10】. The potential $V(\phi)=\tfrac{\alpha}{3}\phi^3-\tfrac{\alpha-\beta}{2}\phi^2$ and the corresponding vacuum analysis are explicit【turn3file11】; Bordag’s tree‑level effective potential and minimization procedure are spelled out via the $L^\wedge_0,L^\wedge_1,L^\wedge_2$ expansion and mass matrix $m^2_{ll'}$【turn4file10】【turn3file16】.

• EFT mindset. The EFT note lays out the appropriate checklist: derive $V(\phi)$, establish $Z(\phi)$, and bound higher‑derivative operators【turn3file0】. The paper’s workflow-write an effective 2D Lagrangian, parameterize fields $\psi_l=\tfrac1{\sqrt2}\phi_l e^{i\Theta_l}$, expand about constant backgrounds, read off masses-mirrors that approach【turn3file19】.

Differences and implications

• Degrees of freedom + symmetry. The framework employs a single real scalar. In Bordag, unstable modes are complex and carry a phase; after condensation, the phase modes are Goldstone modes【turn4file10】. A real scalar does not exhibit Goldstone or phase dynamics; the symmetry analysis correctly identifies no nontrivial internal symmetry for the logistic on‑site law【turn3file1】【turn3file12】. The IR theory is therefore a real scalar EFT unless a U(1) extension is introduced.

• Dimensionality + provenance of derivatives. Earlier drafts promoted a first‑order update to a second‑order PDE and obtained a reaction-diffusion term before moving toward $\Box\phi$【turn4file7】. In Bordag, the $-\partial_\alpha^2$ kinetic form arises directly from the quadratic part of the action after mode reduction to two longitudinal coordinates $x_\alpha$【turn3file17】. The discrete model should be recast into a discrete action and taken to the continuum via a variational limit so that the $\partial_t^2$ term appears from first principles rather than assumption.

• Kinetic normalization. The temporal term $\tfrac12(\partial_t\phi)^2$ follows from the discrete kinetic energy with target $Z(\phi)=\tfrac12$【turn3file4】, while the spatial prefactor should be extracted explicitly from $\sum J(W_j-W_i)^2$ (compute the exact coefficient of $(\nabla\phi)^2$, not merely proportionality)【turn4file13】. In Bordag, the canonical normalization is fixed at the Lagrangian level and phase modes are manifestly massless【turn4file10】.

• Stability structure. The cubic-quadratic $V(\phi)$ is tachyonic at the origin and stabilized by the cubic; adding a $\lambda\phi^4$ term is natural【turn3file2】【turn3file3】. In Bordag, stabilization arises from quartic interactions and selecting a condensate minimum (mass matrix positive)【turn4file10】. A publishable baseline requires either (i) an explicit $\phi^4$ term (bounded below) or (ii) a clearly stated domain of validity for the cubic potential.

• Target theory mismatch. The foundational paper claims a free KG Lagrangian with $m=1$ and a conformal metric $g_{\mu\nu}=\phi^2\eta_{\mu\nu}$ leading to EFE【turn4file1】【turn4file3】. These elements are absent in Bordag, which treats non‑Abelian YM in a center‑vortex background with a 2D effective theory for tachyon modes【turn4file9】. Conclusion: Bordag should be used for methodology (condensation workflow), not for importing claims.

Required corrections

1. Derive the spatial kinetic prefactor exactly. Start from the discrete interaction energy $\tfrac12\sum_{j\in N(i)}J(W_j-W_i)^2$. Do the Taylor expansion on a cubic lattice and keep the full constant: show

   $$ \sum_{j}(W_j-W_i)^2 \to c_\text{lat},a^2(\nabla\phi)^2+\mathcal{O}(a^4) $$

   then match $\tfrac12(\partial_t\phi)^2-\tfrac12 c_\text{lat}J a^2(\nabla\phi)^2$ so Lorentz invariance fixes $c_\text{lat}J a^2=1$ in the chosen units【turn4file13】【turn4file4】. Write the steps and the value of $c_\text{lat}$ for 3D cubic.

2. Replace “promote to second order” with a discrete action derivation. Postulate a lattice Lagrangian density per node $\mathcal{L}_i=\tfrac12(\Delta_t W_i)^2-\tfrac12\sum_j J(W_j-W_i)^2 - V(W_i)$ and apply discrete Euler-Lagrange ⇒ a second‑order time difference naturally. Then take the continuum limit (no hand‑waving). This will close the main rigor gap noted in the own write‑up【turn4file15】.

3. Stabilize the potential (publishable baseline). Add $\lambda\phi^4/4$ (small $\lambda$) and redo: vacua, $m_\text{eff}^2=V''(v)$, and parameter ranges where the minimum is global【turn3file2】【turn3file3】. Report $(v,m_\text{eff})$ as functions of $(\alpha,\beta,\lambda)$. This mirrors the paper’s “choose a condensate, expand, read masses” procedure【turn4file10】.

4. Optional U(1) extension (if you want Goldstones like the paper). Promote $\phi \rightarrow \tfrac{1}{\sqrt2}\rho e^{i\theta}$ and check whether the microscopic rule is invariant under a global phase at leading order. If yes, derive the broken‑phase spectrum: $m_\theta=0$, $m_\rho^2=V''(\rho)|_{\rho=v}$ (cf. Bordag’s $\Theta_l$ masslessness)【turn4file10】. If not, keep the real‑scalar story and don’t overclaim.

5. Document the EFT truncation clearly. Finish the explicit computation of $Z(\phi)$ (show it’s constant) and bound the first nonzero higher‑derivative operator coefficients $c_1,c_2$ by scale separation from the lattice spacing $a$【turn3file0】.

6. Symmetry/Noether story. the logistic on‑site law has time‑translation invariance; a constant of motion $Q$ for the 1‑DOF ODE has been derived【turn3file12】. In the continuum field theory, focus on spacetime translations ⇒ stress‑energy conservation; if a complex field is adopted, also show the U(1) current and its fate in the broken phase (again aligning with the paper’s Goldstone structure).

Mapping summary

• Kinetic term

  • FUM derivation: aiming for $\tfrac12(\partial\phi)^2$, temporal part shown; spatial constant still to fix【turn3file4】.
  • Paper: canonical $-\partial_\alpha^2$ for modes; phases massless after SSB【turn4file10】.

• Potential / masses

  • FUM derivation: $V(\phi)=\frac{\alpha}{3}\phi^3-\frac{\alpha-\beta}{2}\phi^2$; $v=0.6$; $m_\text{eff}^2=\alpha-\beta=0.15$【turn3file7】【turn3file10】.
  • Paper: tachyonic $m_l^2=-\kappa_l^2$, quartic couplings; expand about $v_l$ ⇒ mass matrix $m^2_{ll'}$ positive at minimum【turn4file10】【turn3file16】.

• Method

  • Earlier draft: reaction-diffusion obtained first, with subsequent encouragement toward $\Box\phi$【turn4file4】【turn4file5】.
  • Paper: derive an effective action, then expand around constants【turn3file17】【turn4file9】.

Formal derivation implementing steps (1)-(2)

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Note: If the comparison target differs, update the reference accordingly. Two branches are available: kinetic+action and a U(1) extension with Goldstones.

The following provides a formal derivation of steps (1)-(2) with consistent normalization.

Discrete action → second‑order dynamics (no hand‑waving)

Lattice + notation.

• Spatial lattice: cubic, spacing $a$, dimension $d$ (take $d=3$ in practice).
• Time step: $\Delta t$.
• Site field: $W_i^n \equiv W(\mathbf{x}_i, t_n)$, $t_n=n\Delta t$.
• Neighbor directions: $\mu\in{1,\dots,d}$, unit vectors $\hat e_\mu$.
• On‑site potential: $V(W)$ (keep general here; plug the $V(\phi)$ later).

Discrete Lagrangian (per time step).

$$ L^n ;=; a^d \sum_i\Bigg[ \frac{1}{2}\Big(\frac{W_i^{,n+1}-W_i^{,n}}{\Delta t}\Big)^2 ;-; \frac{\kappa}{2}\sum_{\mu=1}^d\big(W_{i+\mu}^{,n}-W_i^{,n}\big)^2 ;-; V!\big(W_i^{,n}\big) \Bigg] $$

• $\kappa$ is the per‑edge coupling (undirected edges counted once). If you prefer the per‑site convention $\frac{1}{2}\sum_{j\in N(i)}J(W_j-W_i)^2$ that sums both $\pm\mu$, then $\kappa = 2J$. This keeps the algebra consistent with the write‑up.

Euler-Lagrange on the lattice (central in time). Varying $W_i^n$ gives

$$ \frac{W_i^{,n+1}-2W_i^{,n}+W_i^{,n-1}}{(\Delta t)^2} ;-;\kappa,\sum_{\mu=1}^d \big(W_{i+\mu}^{,n}+W_{i-\mu}^{,n}-2W_i^{,n}\big) ;+;V'!\big(W_i^{,n}\big)=0. $$

That’s the second‑order discrete equation (no “promotion” needed). This replaces the first‑order heuristic in the earlier continuum note.

Continuum limit and the exact spatial prefactor

Set $W_i^n\approx \phi(\mathbf{x}_i,t_n)$. Use standard Taylor expansions:

• Time: central difference $\to$ $\partial_t^2\phi + O((\Delta t)^2)$.
• Space: for each $\mu$,

$$ W_{i+\mu}+W_{i-\mu}-2W_i ;=; a^2,\partial_\mu^2\phi ;+; O(a^4). $$

Summing over $\mu$ yields $a^2\nabla^2\phi + O(a^4)$.

Taking $\Delta t\to 0,; a\to 0$, the discrete EOM becomes:

$$ \boxed{;\partial_t^2\phi ;-; \kappa,a^2,\nabla^2\phi ;+; V'(\phi);=;0;} $$

So the small‑fluctuation wave speed is

$$ \boxed{,c^2 = \kappa,a^2,}\quad\text{(or }c^2=2J,a^2\text{ in the per‑site convention).} $$

Drop‑in continuum Lagrangian density.

$$ \boxed{;\mathcal{L} ;=; \frac{1}{2}(\partial_t\phi)^2 ;-; \frac{\kappa a^2}{2}(\nabla\phi)^2 ;-; V(\phi);} $$

• If you keep the per‑site $J$ (both $\pm\mu$ counted in $N(i)$), it’s equivalent to $\mathcal{L}=\tfrac12(\partial_t\phi)^2 - J a^2(\nabla\phi)^2 - V(\phi)$ and the EOM carries $c^2=2Ja^2$. Both conventions are fine; just be consistent about whether $\kappa$ is per edge or per oriented difference. the current draft used this per‑site convention and landed on $J a^2(\nabla\phi)^2$; the only fix is to not force $J a^2=\tfrac12$-just read off $c^2=2J a^2$.

Consistency check (Dirichlet energy mapping). From the interaction energy $\frac12\sum_{j\in N(i)}J(W_j-W_i)^2$ on a cubic lattice, the per‑site continuum limit is

$$ \frac12,J\sum_{j\in N(i)}(W_j-W_i)^2 ;\longrightarrow; J a^2 (\nabla\phi)^2, $$

precisely what you wrote; the “factor of 2” is the $\pm\mu$ neighbor pair. Choosing the per‑edge $\kappa$ makes the canonical $\tfrac12(\nabla\phi)^2$ structure explicit and avoids having to pin $J a^2$ to a number.

Plugging in the potential (and optional $\lambda\phi^4$)

• With the $V(\phi)=\frac{\alpha}{3}\phi^3-\frac{\alpha-\beta}{2}\phi^2$, the linearized mass about a vacuum $v$ is $m^2=V''(v)$.
• If you include the stabilization you sketched, $V\to V+\frac{\lambda}{4}\phi^4$, all formulas remain the same; only $V'(\phi)$ and $m^2=V''(v)$ update.

Changes relative to earlier drafts

• Replaced “promote to second order” with a variational derivation from a discrete action → central‑difference EOM.
• Made the spatial prefactor exact: $c^2=\kappa a^2$ (or $2Ja^2$ in the notation). No need to impose $J a^2=\tfrac12$.
• Keeps the earlier gradient‑from‑neighbors derivation intact, but clarifies the edge‑counting convention so factors are unambiguous.

Proposition: Continuum limit of the FUM lattice action

Proposition (Continuum limit of the FUM lattice action). Consider the lattice action

$$ S=\sum_n \Delta t, a^d \sum_i\Big[\tfrac12\big(\tfrac{W_i^{,n+1}-W_i^{,n}}{\Delta t}\big)^2-\tfrac{\kappa}{2}\sum_{\mu}(W_{i+\mu}^{,n}-W_i^{,n})^2 - V(W_i^{,n})\Big]. $$

The discrete Euler-Lagrange equation is

$$ \frac{W_i^{,n+1}-2W_i^{,n}+W_i^{,n-1}}{(\Delta t)^2} -\kappa\sum_{\mu}\big(W_{i+\mu}^{,n}+W_{i-\mu}^{,n}-2W_i^{,n}\big)+V'(W_i^{,n})=0. $$

Setting $W_i^n\approx \phi(\mathbf{x}_i,t_n)$ and taking $\Delta t\to 0,,a\to 0$ yields

$$ \partial_t^2\phi - \kappa a^2\nabla^2\phi + V'(\phi)=0, $$

which follows from the continuum Lagrangian

$$ \mathcal{L}=\tfrac12(\partial_t\phi)^2 - \tfrac{\kappa a^2}{2}(\nabla\phi)^2 - V(\phi). $$

Hence the propagation speed is $c^2=\kappa a^2$. (In the per‑site convention $\frac12\sum_{j\in N(i)}J(W_j-W_i)^2$, set $\kappa=2J$, so $c^2=2Ja^2$.)

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This normalization aligns with derivation/kinetic_term_derivation.md; the action‑based derivation supersedes the earlier heuristic step and makes any fixed choice of $J a^2$ unnecessary.