Canonical Quantisation I

This refers to promoting Poisson brackets of functions of position and momentum to commutators of operators on Hilbert space

$\{.,.\}\to -i[.,.]$.

Such a map typically does not exist. But, we can focus on a distinguished class of elementary functions of position and momenta.

In the canonical formalism of the bosonic string, the position and momentum are $X^{\mu }(\sigma )$ and ${\Pi }^{\mu }(\sigma )$ with their canonical Poisson bracket. These can be promoted to Hermitian operators with commutators

$[X^{\mu }(\sigma ),{\Pi }^{\nu }({\sigma }^{\prime })]=i{\eta }^{\mu \nu }\delta (\sigma -{\sigma }^{\prime })$,

where we set $\hslash =1$.

We also promote the coefficients to operators with commutation relations

$[x^{\mu },p^{\nu }]=i{\eta }^{\mu \nu }$

$[{\alpha }_m^{\mu },{\alpha }_n^{\nu }]=m{\delta }_{m+n}{\eta }^{\mu \nu }$

$[{\tilde{\alpha }}_m^{\mu },{\tilde{\alpha }}_n^{\nu }]=m{\delta }_{m+n}{\eta }^{\mu \nu }$

with all other commutators vanishing. In order for $X^{\mu }(\sigma )$ and ${\Pi }^{\mu }(\sigma )$ to be Hermitian, we require $x^{\mu }$ and $p^{\mu }$ to be Hermitian while $({\alpha }_n^{\mu }{)}^{†}={\alpha }_{-n}^{\mu }$ and $({\tilde{\alpha }}_n^{\mu }{)}^{†}={\tilde{\alpha }}_{-n}^{\mu }$.

Let us now focus on the open string and write formulae for the modes ${\alpha }_n$ with the understanding that it holds good for modes ${\tilde{\alpha }}_n$ in the closed string.

We define

$a_m^{\mu }:=\frac{1}{\sqrt{m}}{\hat{\alpha }}_m^{\mu }m>0$

$(a_m^{\mu }{)}^{†}:=\frac{1}{\sqrt{m}}{\hat{\alpha }}_{-m}^{\mu }m>0$

such that

$[a_m^{\mu },(a_n^{\nu }{)}^{†}]={\eta }^{\mu \nu }{\delta }_{m,n}$

with all other commutators vanishing.

We also define a number or level operator

$N:={\sum }_{m=1}^{\infty }{\alpha }_{-m}.{\alpha }_m={\sum }_{m=1}^{\infty }m{a}_m^{†}.a_m$ such that

$[N,a_m]=-m{a}_m[N,a_m^{†}]=+m{a}_m^{†}$

and commutes with $x^{\mu },p^{\mu }$. The modes therefore take the form of an infinite sequence of harmonic oscillators indexed by $m>0$ and $\mu =0,1,...,D-1$.