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Open Bosonic Lightcone Spectrum

Physical scope

The flat open tensile bosonic string with Neumann endpoints, p^+ nonzero, D=26, intercept a=1, and the standard transverse Fock representation.

Statement

Use the target $M = R^{1, 25}$ with $η = diag (- 1, + 1, \dots, + 1)$ , define $X^{\pm} = (X^{0} \pm X^{25}) / \sqrt{2}$ , $p^{\pm} = (p^{0} \pm p^{25}) / \sqrt{2}$ , and oscillator components $α_{n}^{\pm} = (α_{n}^{0} \pm α_{n}^{25}) / \sqrt{2}$ , and let $i = 1, \dots, 24$ denote transverse indices. The open worldsheet is the dimensionless Neumann strip $τ \in R$ , $σ \in [0, π]$ . The target light-cone chart is $p^{+} \neq 0$ with gauge condition $X^{+} (τ, σ) = x^{+} + 2 α^{'} p^{+} τ$ . In this chart the nonzero-mode Virasoro constraints determine the longitudinal modes $α_{n}^{-}$ from the transverse modes, while the gauge sets $α_{n}^{+} = 0$ for $n \neq 0$ . Define the weighted transverse level $N = \sum_{n > 0} \sum_{i = 1}^{24} α_{- n}^{i} α_{n}^{i}$ ; a monomial with $r_{n, i}$ quanta of mode $(n, i)$ has level $\sum_{n, i} n r_{n, i}$ . With mass convention $M^{2} = - p_{μ} p^{μ}$ , the zero-mode state constraint is $L_{0}^{m} - a = 0$ . At $D = 26$ and intercept $a = 1$ , the physical Fock space has 24 transverse oscillator families with positive oscillator norm and mass formula $α^{'} M^{2} = N - 1$ for $N \in Z_{\geq 0}$ .

Statement references

David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes, §§2.2–2.3.3 and 3.1 (PDF file pp.39–52 and 60–61; printed pp.32–45 and 53–54); printed pp.32–45 and 53–54); printed pp.32–45 and 53–54); printed pp.32–45 and 53–54); printed pp.32–45 and 53–54); printed pp.32–45 and 53–54); printed pp.32–45 and 53–54); light-cone gauge and open-string mass formula.

Facts & Assumptions

[F1]

Bosonic Matter Oscillator Modes fixes $p \in R^{1, D - 1}$ with dimension $L^{- 1}$ , dimensionless modes, $α_{0}^{μ} = \sqrt{2 α^{'}} p^{μ}$ , the oscillator algebra, normal ordering, and the weighted level $\sum_{n > 0, i} α_{- n}^{i} α_{n}^{i}$ in the transverse sector.

[F2]

Critical Dimension Flat Bosonic String states the conditional values $D = 26$ and $a = 1$ for the standard flat tensile model.

[F3]

The draft physics theorem Open Neumann Classical Mode Expansion locally proves the $C^{2}$ Neumann-strip mode expansion, its one-family normalization, and the classical $L_{n} = 0$ constraints used in the gauge reduction.

[F4]

Conformal Gauge and Residual Symmetry defines the local world-sheet conformal gauge, $σ^{\pm}$ coordinates, and residual world-sheet conformal maps. It does not define the target light-cone coordinates or target gauge; those are stated in this item's Statement.

[A1]

Work at fixed momentum with $p^{+} \neq 0$ and the standard transverse Fock vacuum; use the target chart and gauge condition stated above.

[A2]

In this light-cone quantization, impose the normal-ordered oscillator form of the nonzero-mode Virasoro constraints from [F3], solve them for the longitudinal oscillators, and impose the remaining zero-mode condition $(L_{0}^{m} - a) ∣ ψ ⟩ = 0$ . This is the adopted gauge-fixed quantization prescription.

Given: Assume the flat open tensile string with Neumann endpoints, the stated zero modes, and $D = 26$ , $a = 1$ .

Proof

1.1givenF1F3F4A1A2algebra

The gauge condition gives $α_{n}^{+} = 0$ for $n \neq 0$ and $α_{0}^{+} = \sqrt{2 α^{'}} p^{+} \neq 0$ . With $η_{+ -} = - 1$ , the nonzero-mode constraints from [F3] under [A2] become $0 = - α_{0}^{+} α_{n}^{-} + L_{n}^{⊥}$ , where $L_{n}^{⊥} = \frac{1}{2} \sum_{r \in Z} \sum_{i = 1}^{24} : α_{n - r}^{i} α_{r}^{i} :$ ; hence $α_{n}^{-} = (L_{n}^{⊥}) / (α_{0}^{+})$ . The zero-mode operator is $L_{0}^{m} = \frac{1}{2} α_{0}^{2} + N = α^{'} p^{2} + N$ , with $N = \sum_{n > 0, i} α_{- n}^{i} α_{n}^{i}$ the weighted level from [F1]. Thus the two longitudinal oscillator families are not independent. The transverse commutator is $[α_{m}^{i}, α_{n}^{j}] = m δ^{i j} δ_{m + n, 0}$ , so each normalized creation mode $α_{- n}^{i} / \sqrt{n}$ has positive norm and the transverse finite-particle Fock form is positive definite.

2.1givenF1F2A1A2step 1.1algebra

The zero-mode condition in [A2] and $a = 1$ from [F2] give $α^{'} p^{2} + N - 1 = 0$ . Since $M^{2} = - p^{2}$ , this is $α^{'} M^{2} = N - 1$ . The normalized oscillator number at mode $n$ is $α_{- n}^{i} α_{n}^{i} / n$ , so the weighted sum in [F1] has eigenvalue $\sum_{n, i} n r_{n, i} \in Z_{\geq 0}$ on occupation numbers $r_{n, i}$ .

3.1step 1.1step 2.1∎

Every state at level $N$ is a finite product of the 24 transverse creation modes, and step 1.1 shows these span the light-cone physical Fock space with positive oscillator norm; the result is restricted to the declared Neumann, nonzero- $p^{+}$ chart.

Depends on

Dependency tree · 2 levels

Sources

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