Virasoro Algebra

To implement the classical constraints in canonical quantisation, we must promote the Fourier modes of the energy-momentum tensor to quantum operators. Classically,

$L_m:=\frac{1}{2}{\sum }_{n=-\infty }^{\infty }{\alpha }_{m-n}.{\alpha }_n$ where ${\alpha }_0:=l{p}^{\mu }$ for the open string and ${\alpha }_0:=l{p}^{\mu }/2$ for the closed string.

Since the classical expressions are quadratic in the raising and lowering operators, promoting them to quantum operators is ambiguous due to non-trivial commutation relations. Since ${\alpha }_{m-n}$ and ${\alpha }_n$ commute unless $m=0$, the only ambiguity arises in $L_0$: we would choose ${\alpha }_{-n}.{\alpha }_n$ or ${\alpha }_n.{\alpha }_{-n}$ or a linear combination of both for each $n\ne 0$.

Let us define quantum operators $L_m$ by normal ordering and then parametrise by a constant.

$L_0:=\frac{1}{2}{\sum }_{n=-\infty }^{\infty }:{\alpha }_{-n}.{\alpha }_n:=\frac{{\alpha }_0^2}{2}+{\sum }_{n=1}^{\infty }{\alpha }_{-n}.{\alpha }_n=\frac{{\alpha }_0^2}{2}+N$.

We can parametrise this by allowing a shift of $L_0\to L_0-a$, where $a$ is an arbitrary constant parameter. Then, from the properties of raising and lowering operators, $L_m^{†}=L_{-m}$.

Now, $[L_m,L_n]=(m-n)L_{m+n}+\frac{D}{12}m(m^2-1){\delta }_{m+n}$, known as Virasoro algebra, is the result because manipulations used in the computation of Poisson brackets will no longer be valid due to infinite ordering ambiguities.