Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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Nonzero Momentum Pplus Chart

Physical scope

Spatial-rotation charts and induced BRST/covariant-form transport for nonzero momenta in the 26-dimensional flat Minkowski target; it excludes p=0.

Statement

In $D = 26$ Minkowski target space, write $X^{+} = (X^{0} + X^{25}) / \sqrt{2}$ and $p^{+} = (p^{0} + p^{25}) / \sqrt{2}$ . Every nonzero momentum $p = (p^{0}, \vec{p}) \in R^{1, 25}$ can be carried by a spatial rotation $R \in S O (25)$ to a frame with $p^{+} = (p^{0} + (R \vec{p})^{25}) / \sqrt{2} \neq 0$ : if $p^{0} \neq 0$ , choose $R$ so $(R \vec{p})^{25} = 0$ ; if $p^{0} = 0$ , then $\vec{p} \neq 0$ and choose $R$ to align it with the $25$ direction. The nonzero-momentum chart argument does not cover $p = 0$ .

At $D = 26, a = 1$ , let $L_{n}^{m} = \frac{1}{2} \sum_{r \in Z} : α_{n - r} \cdot α_{r} :$ with $[α_{m}^{μ}, α_{n}^{ν}] = m η^{μ ν} δ_{m + n, 0}$ and ${b_{m}, c_{n}} = δ_{m + n, 0}$ , and define the single-chiral BRST operator on the algebraic matter--ghost Fock space by $Q_{B} = \sum_{n} c_{- n} (L_{n}^{m} - a δ_{n 0}) - \frac{1}{2} \sum_{m, n} (m - n) : c_{- m} c_{- n} b_{m + n} :$ . The ghost-number grading has $gh (c_{n}) = 1$ and $gh (b_{n}) = - 1$ , and $L_{0}^{t o t} = {Q_{B}, b_{0}}$ . Its relative spaces are $C_{r e l}^{q} (p) = {ψ : gh (ψ) = q, b_{0} ψ = 0, L_{0}^{t o t} ψ = 0}$ , with cohomology

H_{r e l}^{q} (p) = \ker (Q_{B} : C_{r e l}^{q} (p) \to C_{r e l}^{q + 1} (p)) / im (Q_{B} : C_{r e l}^{q - 1} (p) \to C_{r e l}^{q} (p)) .

The representative pairing at ghost number one is $⟨ ψ ∣ c_{0} ∣ ϕ ⟩_{K}$ . The covariant weighted level is the eigenvalue of $N_{l e v} = \sum_{n > 0} α_{- n} \cdot α_{n}$ ; in the covariant Fock space at level $N$ the physical subspace is $P_{N} (p) = {ψ : L_{n} ψ = 0 (n > 0), (L_{0} - a) ψ = 0}$ and its radical is $Rad (P_{N} (p)) = P_{N} (p) \cap P_{N} (p)^{⊥}$ , with orthogonality in the ambient indefinite form.

For $R \in S O (25)$ let $Λ_{R} = diag (1, R)$ . Define $U_{R}$ on the algebraic matter--ghost Fock basis by sending $∣ 0; p ⟩$ to $∣ 0; Λ_{R} p ⟩$ , each matter oscillator $α_{n}^{μ}$ to $(Λ_{R}^{- 1})^{μ}_{ν} α_{n}^{ν}$ (and each tilded oscillator by the same rule in a closed sector), and every $b_{n}, c_{n}$ mode to itself, and extend linearly to finite mode monomials. This map intertwines $Q_{B}$ between momentum fibers $p$ and $Λ_{R} p$ , sends each relative complex onto the corresponding relative complex, and is a Krein isometry. It preserves the representative pairing $⟨ ψ ∣ c_{0} ∣ ϕ ⟩_{K}$ and therefore every induced $c_{0}$ pairing on cohomology on which that pairing descends. It also maps each $P_{N} (p)$ isometrically onto $P_{N} (Λ_{R} p)$ , carries its radical to the corresponding radical, and induces an isometry of the radical quotients. Thus results proved in a $p^{+} \neq 0$ chart transport to every nonzero momentum.

Statement references

David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes, §2.2.1 (PDF file pp.40–42; printed pp.33–35); printed pp.33–35); printed pp.33–35); printed pp.33–35); printed pp.33–35); printed pp.33–35); printed pp.33–35); light-cone chart requires $p^{+}$ nonzero; the spatial-rotation proof is local.

Facts & Assumptions

[F1]

Flat Bosonic String Model fixes the 26-dimensional target metric $η = diag (-, +, \dots, +)$ .

[F2]

The draft physics theorem Critical Dimension Flat Bosonic String locally proves $D = 26$ and $a = 1$ for the standard flat tensile model under its stated Weyl and finite-part conventions.

Given: Let $p = (p^{0}, \vec{p}) \in R^{1, 25}$ be nonzero. Define $X^{+} = (X^{0} + X^{25}) / \sqrt{2}$ and $p^{+} = (p^{0} + p^{25}) / \sqrt{2}$ .

[F3]

The draft definition Bosonic BRST Complex fixes $Q_{B}$ , the matter--ghost Krein form, the relative constraints, and the representative pairing with $c_{0}$ .

[F4]

The draft definition Bosonic Covariant State Space defines $P_{N} (p)$ , its ambient form, and its radical quotient.

[F5]

Bosonic BRST Nilpotence proves $Q_{B}^{2} = 0$ at $D = 26, a = 1$ for the charge and ordering in [F3]. The same identity applies to a single closed chiral copy because it uses only the common zero-mode parameter $k$ , not its conversion to target momentum.

Proof

1.1givenF1F2choosealgebra

The criticality input [F2] sets $D = 26$ , so the spatial rotation group is $S O (25)$ . If $p^{0} \neq 0$ , choose a unit spatial vector orthogonal to $\vec{p}$ when $\vec{p} \neq 0$ and an orientation-preserving rotation $R \in S O (25)$ sending that vector to the 25-axis; then $(R \vec{p})^{25} = 0$ . If $\vec{p} = 0$ , take $R = I$ . In either case the rotated momentum has $p^{+} = p^{0} / \sqrt{2} \neq 0$ .

1.2givenF1F2choosealgebra

If $p^{0} = 0$ , nonzero $p$ implies $\vec{p} \neq 0$ . Choose $R \in S O (25)$ carrying $\vec{p} / ∣ \vec{p} ∣$ to the 25-axis; then $(R \vec{p})^{25} = ∣ \vec{p} ∣$ and $p^{+} = ∣ \vec{p} ∣ / \sqrt{2} \neq 0$ .

2.1F1F3F4F5step 1.1step 1.2algebra∎

By [F5], $Q_{B}^{2} = 0$ at the stated critical parameters; [F3] then gives actual relative subcomplexes and cohomologies. Fix any $R \in S O (25)$ and put $Λ_{R} = diag (1, R)$ . On each finite-particle state define $U_{R}$ by the rule in the Statement. Since $Λ_{R}^{T} η Λ_{R} = η$ , the oscillator commutators, adjoints, and the Minkowski contractions in every $L_{n}^{m}$ are preserved, and the same calculation holds for the commuting tilded copy. For the zero mode this convention gives $(Λ_{R}^{- 1})^{μ}_{ν} k^{ν} (Λ_{R} p) = k^{μ} (p)$ ; for every nonzero mode, contraction with $k (Λ_{R} p)$ gives the original contraction with $k (p)$ . Thus the zero-mode terms in every $L_{n}^{m}$ intertwine as well as the quadratic nonzero-mode terms. The scalar momentum-fibre vacuum pairing is preserved because both vacua have norm one and the oscillator contraction metric is invariant; the ghost pairing is unchanged because ghost modes are fixed. Hence $U_{R}$ is a Krein isometry. The BRST formula is built from the invariant matter $L_{n}^{m}$ , the scalar intercept, and unchanged ghost modes, so $Q_{B} (Λ_{R} p) U_{R} = U_{R} Q_{B} (p)$ ; likewise $b_{0}$ and $L_{0}^{t o t}$ intertwine. Thus $U_{R}$ maps the relative complexes and their cycles and boundaries bijectively. Since it fixes $c_{0}$ , $⟨ U_{R} ψ ∣ c_{0} ∣ U_{R} ϕ ⟩_{K} = ⟨ ψ ∣ c_{0} ∣ ϕ ⟩_{K}$ . Finally, the same invariance of $L_{n}$ and of the ambient matter form maps $P_{N} (p)$ onto $P_{N} (Λ_{R} p)$ , maps $P_{N} (p) \cap P_{N} (p)^{⊥}$ onto the corresponding radical, and preserves the quotient form. Applying this to the rotations in steps 1.1--1.2 proves all stated transports. No such rotation makes $p = 0$ have nonzero $p^{+}$ , so zero momentum is excluded.

Depends on

Dependency tree · 2 levels

Sources

David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes