Bosonic Covariant State Space

For each target momentum $p\in {\mathbb{R}}^{1,D-1}$, let $F_D(p)$ be the covariant oscillator Fock space generated by ${\alpha }_{-n}^{\mu }$, $n>0$, over a momentum ground state, with $[{\alpha }_m^{\mu },{\alpha }_n^{\nu }]=m{\eta }^{\mu \nu }{\delta }_{m+n,0}$ and the indefinite Hermitian form determined by $⟨0;p\mathrm{\mid }0;p⟩=1$ and $({\alpha }_n^{\mu }{)}^{†}={\alpha }_{-n}^{\mu }$. The weighted level operator is $N_{\mathrm{l}\mathrm{e}\mathrm{v}}={\sum }_{n>0}{\alpha }_{-n}\cdot {\alpha }_n$; a monomial ${\alpha }_{-n_1}^{{\mu }_1}\cdots {\alpha }_{-n_r}^{{\mu }_r}\mathrm{\mid }0;p⟩$ has eigenvalue $N=n_1+\cdots +n_r$. Let $a\in \mathbb{R}$ be the dimensionless normal-ordering intercept in $L_0-a$; its value is left unfixed here. The old-covariant physical subspace at level $N$ is $P_N(p)=\{\psi \in F_D(p):N_{\mathrm{l}\mathrm{e}\mathrm{v}}\psi =N\psi ,L_n\psi =0\text{ for }n>0\text{ and }(L_0-a)\psi =0\}$; its null physical subspace is $P_N(p)\cap P_N(p{)}^{\perp }$, and the candidate physical space is the quotient by that radical.

David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes, §§2.1–2.1.2 and 2.3–2.3.3 (PDF file pp.35–38 and 47–52; printed pp.28–31 and 40–45); printed pp.28–31 and 40–45); printed pp.28–31 and 40–45); printed pp.28–31 and 40–45); printed pp.28–31 and 40–45); printed pp.28–31 and 40–45); printed pp.28–32 and 40–45); source context for covariant Fock states and Virasoro conditions.

Bosonic Matter Oscillator Modes supplies the finite-particle oscillator Fock domain and its induced indefinite form. Polyakov Equations and Virasoro Constraints supplies the classical Virasoro constraints whose quantized conditions are imposed here. Here $a$ is the dimensionless normal-ordering intercept subtracted in $(L_0-a)\psi =0$; this definition leaves its value unfixed. The quotient is not identified with BRST cohomology, and no positivity claim is made.

The level $N$ of a finite-particle monomial ${\alpha }_{-n_1}^{{\mu }_1}\cdots {\alpha }_{-n_r}^{{\mu }_r}\mathrm{\mid }0;p⟩$ is $n_1+\cdots +n_r$. It is the nonnegative integer grading for which $[L_0,{\alpha }_{-n}^{\mu }]=n{\alpha }_{-n}^{\mu }$; in this oscillator realization it is the eigenvalue of the weighted operator ${\sum }_{n>0}{\alpha }_{-n}\cdot {\alpha }_n$. The operator is written using the Minkowski contraction, but its eigenvalue on every oscillator monomial is the positive sum of mode indices, not the indefinite norm of that state. The fixed-momentum Fock form is indefinite because the target time oscillator has negative metric. For a subspace $P_N$, its radical is $\mathrm{Rad}(P_N)=P_N\cap P_N^{\perp }$, where $P_N^{\perp }$ is taken in the ambient sesquilinear form. Each level is finite-dimensional, so this intersection is a linear subspace and the quotient is algebraically defined. This definition does not identify the quotient with BRST cohomology or assert positivity.