Canonical Quantisation II

We will now canonically quantise the theory in the light-cone gauge. . The Poisson brackets and therefore commutators of position, momentum and transverse oscillators are the same as before

$[x^{-},p^{+}]=-i[x^i,p^j]=i{\delta }^{ij}[{\alpha }_m^i,{\alpha }_n^j]=m{\delta }_{m+n}{\delta }^{i,j}$

with all other commutators vanishing.

The composite operator ${\alpha }_0^{-}$ involves an ordering ambiguity since $[{\alpha }_{-n}^i,{\alpha }_n^i]\ne 0$. We deal with this by using normal ordering

${\alpha }_m^{-}=\frac{1}{\ell p^{+}}\left(\frac{1}{2}{\sum }_{n=-\infty }^{\infty }:{\alpha }_{m-n}^i{\alpha }_n^i:-a{\delta }_{n,0}\right)$

and parametrising our ignorance by a real parameter $a$. In particular,

${\alpha }_0^{-}=\frac{1}{\ell p^{+}}\left(\frac{1}{2}{\alpha }_0^i{\alpha }_0^i+{\sum }_{n=1}^{\infty }{\alpha }_{-n}^i{\alpha }_n^i-a\right)=\frac{1}{\ell p^{+}}\left(\frac{1}{2}{\ell }^2{p}^i{p}^i+N-a\right)$

and the mass-shell condition ${\alpha }^{-}=\ell p^{-}$ becomes

${\alpha }^{\prime }{M}^2=N-a$.

An analogous formula holds for the closed string.

Under what circumstances does the light-cone quantisation produce a Poincaré invariant theory?

The gauge fixing $\tau =X^{+}/p^{+}+\text{constant}$ explicitly breaks the symmetry group $SO(1,D-1)$ of Lorentz transformations $X^{\mu }\to {\Lambda }_{\nu }^{\mu }{X}^{\nu }$ to the subgroup $SO(D-2)$ of rotations $X^i\to {\Lambda }_j^i{X}^j$. The excitations generated by ${\alpha }_n^i$ will transform in a representation of $SO(D-2)$ but in a D-dimensional Lorentz invariant theory, they must lift to representations of $SO(1,D-1)$.

The states ${\alpha }_{-1}^i|0;k⟩$ transform in the vector representation of $SO(D-2)$ with ${\alpha }^{\prime }M=1-a$. Now, recall that a transversely polarised vector particle will acquire a longitudinal component after a general Lorentz transformation, unless it is massless. Thus, in a D-dimensional Poincaré invariant theory, the spin of a massive particle is a representation of $SO(D-1)$ while that of a massless particle is $SO(D-2)$. Therefore, light-cone gauge cannot give a Poincaré invariant theory unless $a=1$.

Now, $\frac{1}{2}{\sum }_{n=-\infty }^{\infty }{\alpha }_{-n}^i{\alpha }_n^i=\frac{1}{2}{\sum }_{n=-\infty }^{\infty }:{\alpha }_{-n}^i{\alpha }_n^i:-a$.

This gives $a=-\frac{D-2}{2}{\sum }_{n=1}^{\infty }n$ from the relation $[{\alpha }_n^i,{\alpha }_{-n}^i]=n$ for each $i=2,\dots ,D-2$ and $n=1,\dots ,\infty$.

Since the sum is divergent, we need to regularise it. We will use the zeta-function regularisation, which is common in QFT. This starts from the more general infinite sum

${\sum }_{n=1}^{\infty }{n}^{-s}$

which converges to the Riemann zeta function $\zeta (s)$ for $\text{Re}(s)>1$. This has an analytic continuation in the complex $s$-plane with $\zeta (-1)=-\frac{1}{12}$. Then, the regularised normal ordering constant is

$a=\frac{D-2}{24}$.

If $a\stackrel{!}{=}1$, then $D\stackrel{!}{=}26$. This is the critical dimension of the bosonic theory.