这是一份liverpool利物浦大学MATH423的成功案例

The operators $L_{m}$ satisfy the Virasora algebra:
$$
\left[L_{m}, L_{n}\right]=(m-n) L_{m+n}+\frac{c}{12}\left(m^{3}-m\right) \delta_{m+n, 0} .
$$
The Virasoro algebra is a central extension of the Witt algebra. On our Fock space $\mathcal{F}$ the central charge $c$ takes the value
$$
c=\eta^{\mu \nu} \eta_{\mu \nu}=D,
$$
i.e., each space-time dimension contributes one unit. Since the Poisson brackets of $L_{m}$ in the classical theory just give the Witt algebra, this dependence on the number of dimensions is a new property of the quantum theory. The extra central term occuring at the quantum level is related to a normal ordering ambiguity of commutators with $m+n=0$. This results in a new ‘anomalous’ term in the algebra. In the context of current algebras such terms are known as Schwinger terms.

MATH423 COURSE NOTES ：

As mentioned above the operator $L_{0}$ is the normal ordered version of with $m=0$. The original and the normal ordered expression formally differ by an infinite constant. Subtracting this constant introduces a finite ambiguity, which was parametrized by $a$. Unitarity then fixed $a=1$. The oscillator part of $L_{0}$ is
$$
N=\sum_{n=1}^{\infty} \alpha_{-n} \cdot \alpha_{n} .
$$
$N$ is called the number operator, because
$$
\left[N, \alpha_{-m}^{\mu}\right]=m \alpha_{-m}^{\mu}
$$
Since the total momentum is related to the mass of the string by $M^{2}+p^{2}=0$, the constraintsdetermine the mass of a physical states in terms of the eigenvalues of $N$ and of its right-moving analogue $\tilde{N}$. (We denote the operators and their eigenvalues by the same symbol.) We now use the above decomposition of $L_{0}$, take the sum and difference of the constraints and reintrodue the Regge slope $\alpha^{\prime}=\frac{1}{2}$ by dimensional analysis:
$$
\begin{aligned}
\alpha^{\prime} M^{2} &=2(N+\tilde{N}-2) \
N &=\tilde{N}
\end{aligned}
$$