Invariance under infinitesimal Lorentz transformations

Proved

PolyakovAction.lorentz_invariance_first_order

by Lucas · Oct 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

mathematical-physicsstring-theory

Let $g_{\mu\nu}$ be a constant symmetric target metric and $\omega^\mu{}_\nu$ a matrix with $\omega_{\mu\nu}=g_{\mu\lambda}\omega^\lambda{}_\nu=-\omega_{\nu\mu}$ . Let $X$ be differentiable and assume the Polyakov densities of $X$ and of $\omega X$ are integrable over the region $U$ . Then under $X^\alpha\to X^\alpha+\varepsilon\,\omega^\alpha{}_\beta X^\beta$ ,

S[X+\varepsilon\,\omega X]=S[X]+O(\varepsilon^2)\qquad(\varepsilon\to0).

This is transformation (ii) of the source's Poincaré symmetry: the action is invariant under infinitesimal Lorentz transformations, $\mathcal S'=\mathcal S+O(\omega^2)$ .

Formalization Note. "Infinitesimal" is encoded by scaling a fixed generator $\omega$ by a real parameter $\varepsilon$ and asserting a big- $O(\varepsilon^2)$ bound as $\varepsilon\to0$ . The integrability hypotheses express finiteness of the action, implicit in the source.

Preamble

import Mathlib
import Definitions.Def_PolyakovAction_Defs

open MeasureTheory Filter Asymptotics
open scoped Matrix Topology

Formal statement

namespace PolyakovAction

theorem lorentz_invariance_first_order {D : ℕ} (T : ℝ) (g : Matrix (Fin D) (Fin D) ℝ)
    (hg : gᵀ = g) (ω : Matrix (Fin D) (Fin D) ℝ) (hω : (g * ω)ᵀ = -(g * ω))
    (h : Worldsheet → Matrix (Fin 2) (Fin 2) ℝ) (X : Worldsheet → Spacetime D)
    (hX : Differentiable ℝ X) (U : Set Worldsheet)
    (hL : IntegrableOn (polyakovLagrangian (fun _ => g) h X) U)
    (hLω : IntegrableOn (polyakovLagrangian (fun _ => g) h (fun σ => ω *ᵥ X σ)) U) :
    (fun ε : ℝ => polyakovAction T (fun _ => g) h (fun σ => X σ + ε • (ω *ᵥ X σ)) U
        - polyakovAction T (fun _ => g) h X U) =O[𝓝 0] (fun ε : ℝ => ε ^ 2) := by sorry

end PolyakovAction

Source

Wikipedia, "Polyakov action", revision oldid=1292212311 (https://en.wikipedia.org/w/index.php?title=Polyakov_action&oldid=1292212311)

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic)

Non-blind read-back — not independent testimony. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statements of this proposal, at the account owner's explicit instruction. Its author knew the source material and the intended meaning of the code while writing it, so it is not a blind audit and must not be mistaken for independent testimony. Reviewers should compare it against the Lean code themselves.

Throughout, $\sigma=(\sigma^0,\sigma^1)\in\mathbb R^2$ is a worldsheet point (index $0$ is $\tau$ , index $1$ is $\sigma$ ), $X:\mathbb R^2\to\mathbb R^D$ is the embedding, $\partial_a X^\mu(\sigma)$ is the $\mu$ -th component of the Fréchet derivative of $X$ at $\sigma$ applied to the $a$ -th standard basis vector (it is $0$ wherever $X$ is not differentiable), $G_{ab}(\sigma)=\sum_{\mu,\nu} g_{\mu\nu}(X(\sigma))\,\partial_aX^\mu\,\partial_bX^\nu$ is the induced metric, $h^{ab}$ denotes the entries of the matrix inverse $(h_{ab})^{-1}$ (Mathlib's inverse, which is the zero matrix when $\det h=0$ ), and $\sqrt{\cdot}$ is the real square root, which returns $0$ on negative inputs. The Polyakov density is $\mathcal L_P=\sqrt{-\det h}\,\sum_{a,b}h^{ab}G_{ab}$ , the Polyakov action on a set $U$ is $S_P[U]=\frac T2\int_U\mathcal L_P\,d^2\sigma$ (Lebesgue measure; Bochner integral, equal to $0$ if the integrand is not integrable on $U$ ), and the Nambu–Goto action is $S_{NG}[U]=T\int_U\sqrt{-\det G}\,d^2\sigma$ . The stress–energy tensor is $T_{ab}=T\big(G_{ab}-\tfrac12 h_{ab}\sum_{c,d}h^{cd}G_{cd}\big)$ , and $\eta^{(n)}=\mathrm{diag}(1,-1,\dots,-1)$ is the $n\times n$ Minkowski matrix.

Statement. For every $D$ , real $T$ , matrix $g\in\mathbb R^{D\times D}$ with $g^{\mathsf T}=g$ , matrix $\omega\in\mathbb R^{D\times D}$ with $(g\omega)^{\mathsf T}=-(g\omega)$ , worldsheet metric $h$ (arbitrary), map $X:\mathbb R^2\to\mathbb R^D$ that is differentiable everywhere, and set $U\subseteq\mathbb R^2$ (measurability not assumed), assume

$\sigma\mapsto\mathcal L_P[g,h,X](\sigma)$ is integrable on $U$ ;
$\sigma\mapsto\mathcal L_P[g,h,\omega X](\sigma)$ is integrable on $U$ , where $(\omega X)(\sigma)=\omega\,X(\sigma)$ (matrix–vector product).

Here the target metric is the constant $g$ . Then the real function

\varepsilon\;\mapsto\;S_P\big(T,g,h,\;\sigma\mapsto X(\sigma)+\varepsilon\,\omega X(\sigma),\;U\big)-S_P(T,g,h,X,U)

is $O(\varepsilon^2)$ as $\varepsilon\to0$ (i.e. bounded by $C\varepsilon^2$ for some constant $C$ on a neighbourhood of $0$ ).

Human review

Endorsed by Shuze Chen · Oct 8, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 8, 2026
Confirmed by the mission captain (proposal self-audit).

View graph