Bosonic Matter Oscillator Modes

Use units $\mathrm{\hslash }=c=1$. For target dimension $D\ge 3$, a fixed target momentum is a contravariant vector $p=(p^{\mu })\in {\mathbb{R}}^{1,D-1}$, with $p_{\mu }={\eta }_{\mu \nu }{p}^{\nu }$ and dimension $L^{-1}$. The parameter ${\alpha }^{\mathrm{\prime }}>0$ has dimension $L^2$. Thus the zero modes and oscillator modes below are dimensionless.

For fixed $p$, take the algebraic open-string matter Fock space generated by ${\alpha }_{-n}^{\mu }$, $n>0$, from $\mathrm{\mid }0;p⟩$, with $[{\alpha }_m^{\mu },{\alpha }_n^{\nu }]=m{\eta }^{\mu \nu }{\delta }_{m+n,0}$, adjoint $({\alpha }_n^{\mu }{)}^{†}={\alpha }_{-n}^{\mu }$, and normal ordering relative to ${\alpha }_n\mathrm{\mid }0;p⟩=0$ for $n>0$. Set ${\alpha }_0^{\mu }=\sqrt{2{\alpha }^{\mathrm{\prime }}}{p}^{\mu }$ for open strings. For closed strings use two independent commuting left/right oscillator families: $[{\alpha }_m^{\mu },{\alpha }_n^{\nu }]=m{\eta }^{\mu \nu }{\delta }_{m+n,0}$, $[{\tilde{\alpha }}_m^{\mu },{\tilde{\alpha }}_n^{\nu }]=m{\eta }^{\mu \nu }{\delta }_{m+n,0}$, $[{\alpha }_m^{\mu },{\tilde{\alpha }}_n^{\nu }]=0$, $({\tilde{\alpha }}_n^{\mu }{)}^{†}={\tilde{\alpha }}_{-n}^{\mu }$, and ${\alpha }_0^{\mu }={\tilde{\alpha }}_0^{\mu }=\sqrt{{\alpha }^{\mathrm{\prime }}\mathrm{/}2}{p}^{\mu }$. On the finite-particle domain define $L_n^m=\frac{1}{2}{\sum }_k:{\alpha }_{n-k}\cdot {\alpha }_k:$ and the independent tilde copy for closed strings; the dot is contraction by $\eta$, and each sum acts finitely on each state.

Define the weighted matter levels on the algebraic Fock domains by $N_{\mathrm{l}\mathrm{e}\mathrm{v}}={\sum }_{n>0}{\alpha }_{-n}\cdot {\alpha }_n$ and, for closed strings, ${\tilde{N}}_{\mathrm{l}\mathrm{e}\mathrm{v}}={\sum }_{n>0}{\tilde{\alpha }}_{-n}\cdot {\tilde{\alpha }}_n$. These sums act finitely on every finite-particle vector. The oscillator commutators give $[N_{\mathrm{l}\mathrm{e}\mathrm{v}},{\alpha }_{-n}^{\mu }]=n{\alpha }_{-n}^{\mu }$ and $N_{\mathrm{l}\mathrm{e}\mathrm{v}}\mathrm{\mid }0;p⟩=0$, so a monomial ${\alpha }_{-n_1}^{{\mu }_1}\cdots {\alpha }_{-n_r}^{{\mu }_r}\mathrm{\mid }0;p⟩$ has level $n_1+\cdots +n_r$; the right-moving closed level is defined in the same way. The per-mode occupation number is $n^{-1}{\alpha }_{-n}\cdot {\alpha }_n$ (and its tilde copy); its unweighted sum counts quanta and is not the level. In a transverse sector, the dot in these level formulas is the Euclidean contraction over transverse indices.

David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes, §§1.4–1.4.1 and 2.1–2.3.3 (PDF file pp.32–34 and 35–52; printed pp.25–27 and 28–45); printed pp.25–27 and 28–45); printed pp.25–27 and 28–45); printed pp.25–27 and 28–45); printed pp.25–27 and 28–45); printed pp.25–27 and 28–45); printed pp.25–27 and 28–45); open/closed oscillator conventions.

Flat Bosonic String Model fixes the target as flat Minkowski space with signature $(-,+,\dots ,+)$ and excludes background fields and interactions. String Tension Alpha Prime fixes ${\alpha }^{\mathrm{\prime }}>0$, $[{\alpha }^{\mathrm{\prime }}]=L^2$, and $T=(2\pi {\alpha }^{\mathrm{\prime }}{)}^{-1}$. Canonical Phase Space Constraints gives only the classical constraint normalization used as formulation context. The oscillator commutators, Fock representation, adjoints, and normal ordering above are adopted algebraic quantization data; they are not consequences of that classical theorem. Open and closed zero-mode factors are kept distinct throughout.