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:
$$
\begin{aligned}
\alpha^{\prime} M^{2} &=2(N+\tilde{N}-2) \
N &=\tilde{N}
\end{aligned}
$$

这是一份liverpool利物浦大学MATH423的成功案例

$$
\partial^{2} X^{\mu}=\partial^{\alpha} \partial_{\alpha} X^{\mu}=0 .
$$
Note that when imposing the conformal gauge on the Polyakov action the equation of motion for $h_{\alpha \beta}$, i.e., $T_{\alpha \beta}=0$, becomes a constraint, which has to be imposed on the solutions of (11).

The general solution is a superposition of left- and right-moving waves,
$$
X^{\mu}(\sigma)=X_{L}^{\mu}\left(\sigma^{+}\right)+X_{R}^{\mu}\left(\sigma^{-}\right) .
$$
However, we also have to specify boundary conditions at the ends of the string. One possible choice are periodic boundary conditions,
$$
X^{\mu}\left(\sigma^{0}, \sigma^{1}+\pi\right)=X^{\mu}\left(\sigma^{0}, \sigma^{1}\right) .
$$

MATH423 COURSE NOTES ：

$$
X^{\mu}(\sigma)=x^{\mu}+\left(2 \alpha^{\prime}\right) p^{\mu} \sigma^{0}+\mathrm{i} \sqrt{2 \alpha^{\prime}} \sum_{n \neq 0} \frac{\alpha_{n}^{\mu}}{n} e^{-\mathrm{i} n \sigma^{0}} \cos \left(n \sigma^{1}\right)
$$
There is, however, a second possible choice of boundary conditions for open strings, namely Dirichlet boundary conditions. Here the ends of the string are kept fixed:
$$
\left.X^{\mu}\right|{\sigma^{1}=0}=x{(1)}^{\mu},\left.\quad X^{\mu}\right|{\sigma^{1}=\pi}=x{(2)}^{\mu}
$$
With these boundary conditions the solution takes the form
$$
X^{\mu}(\sigma)=x_{(1)}^{\mu}+\left(x_{(2)}^{\mu}-x_{(1)}^{\mu}\right) \frac{\sigma^{1}}{\pi}+\mathrm{i} \sqrt{2 \alpha^{\prime}} \sum_{n \neq 0} \frac{\alpha_{n}^{\mu}}{n} e^{-\mathrm{in} \sigma^{0}} \sin \left(n \sigma^{1}\right)
$$