A course into Quantum Field Theory (QFT) is passed through the passage of
Quantization. In QFT, we quantize our classical fields using
canonical quantization. However, one can also quantize the fields using the
path integrals. A canonical quantization, also known as second quantization,
is a series of steps. In order to quantize a field using canonical
quantization, we first find its Lagrangian. A Lagrangian is yet another
formalism to develop theories, and every Lagrangian gives an equation of
motion using the Euler Lagrange equation;
$$S({q})=\int_{a}^{b} \mathrm{d} t L(t, {q}(t),
\dot{{q}}(t)) $$
$$\frac{\partial \mathcal{L}}{\partial q_{i}}-\frac{\mathrm{d}}{\mathrm{d}
t}\left(\frac{\partial \mathcal{L}}{\partial q_{i}^{\prime}}\right)=0$$
In our following coordinates, we tend to use light-cone coordinates for a
target space (a space where our strings action are parameterized, typically
where our $\eta_{\mu \nu}$ is
$$X^{\pm} = \frac{1}{\sqrt{2}} (X^0 \pm X^{D-1})$$
$$\eta_{+-} = \eta_{-+} = -1,\ n_{ij}=\delta_{ij}$$
our inner product follows
$$X^2 = - 2X^+X^- + \dot{X}^i \dot{X}^i $$
For those who are familiar with the string theory notion, we
use coordinates as a function of $\tau, \sigma$. An expansion of $X^+(\tau,
\sigma)$ gives us
$$X^{+}(\tau, \sigma)=x^{+}+\alpha^{\prime} p^{+} \tau+i
\sqrt{\frac{\alpha^{\prime}}{2}} \sum_{n \in \mathbb{Z}, n \neq 0} \frac{1}{n}
\alpha_{n}^{+} e^{-i n \xi^{-}}+i \sqrt{\frac{\alpha^{\prime}}{2}} \sum_{n \in
\mathbb{Z}, n \neq 0} \frac{1}{n} \tilde{\alpha}_{n}^{+} e^{-i n \xi^{+}}$$
these equation contains residual infinite dimensional symmetry
(conforming killing vectors) which comes because of the choosen light cones
gauge. The equation has a lot of oscilaltor modes, which can be killed by
the residual infinite dimensional symmetry, hence we set the oscillator
modes to 0
$$X^{+}(\tau, \sigma)=x^{+}+\alpha^{\prime} p^{+} \tau$$
we impose Virasoro constaints
$$\partial_{\pm} X^{-}=\frac{1}{\alpha^{\prime} p^{+}}\left(\partial_{\pm}
X^{i}\right)^{2}$$
We can see that $X^-$ comes from the transverse oscillator $X^i$,
and the $X^i$ have independent degrees of freedom. And $X^i$ contains two independent oscillator modes in light cones gauge. And, clearly, it helps us to with two polarization of string, i.e. $X^+, X^-$.
Now the action, after turning to light cones, reads
$$\begin{aligned} S_{lc} &=\frac{1}{4 \pi \alpha^{\prime}} \int_{\Sigma} d
\tau d \sigma\left[\left(\partial_{\tau}
X^{i}\right)^{2}-\left(\partial_{\sigma}
X^{i}\right)^{2}+2\left(-\partial_{\tau} X^{+} \partial_{\tau}
X^{-}+\partial_{\sigma} X^{+} \partial_{\sigma} X^{-}\right)\right] \\
&=\frac{1}{4 \pi \alpha^{\prime}} \int_{\Sigma} d \tau d
\sigma\left[\left(\partial_{\tau} X^{i}\right)^{2}-\left(\partial_{\sigma}
X^{i}\right)^{2}\right]-\int d \tau p^{+} \partial_{\tau} q^{-} \\ &
\equiv \int d \tau L \end{aligned}$$
where
$$q^{-} \equiv \frac{1}{2 \pi} \int_{0}^{2 \pi} d \sigma X^{-} .$$
from the action, we can find out the canonical momenta
$$p_{-} \equiv \frac{\partial L}{\partial \dot{q}^{-}}=-p^{+}, \quad \Pi_{i}
\equiv \frac{\partial L}{\partial \dot{X}^{i}}=\frac{\dot{X}_{i}}{2 \pi
\alpha^{\prime}}$$
and the commutation relation, we can infer, is
$$\left[X^{\mu}(\tau, \sigma), \Pi^{\mu}\left(\tau,
\sigma^{\prime}\right)\right]=i \eta^{\mu \nu}
\delta\left(\sigma-\sigma^{\prime}\right)$$
The next step in the usual canonical process is tuning the oscillator modes to operators using construction and destruction operators.
We say, that $\alpha^i_{-n}$ are creation operators with $n>0$ and $\alpha^i_{n}$ is destruction operator that kills the vacuum with $n<0$.
They read
$$\alpha_{n}^{-}=\frac{1}{2 \sqrt{2 \alpha^{\prime}} p^{+}}
\sum_{m=-\infty}^{m=\infty} \alpha_{n-m}^{i} \alpha_{m}^{i}$$
After ordering operator
$$\alpha_{n}^{-}=\frac{1}{2 \sqrt{2 \alpha^{\prime}}
p^{+}}\left(\sum_{m=-\infty}^{m=\infty}: \alpha_{n-m}^{i} \alpha_{m}^{i}:-a
\delta_{n, 0}\right)$$
where we define
$$ \alpha_{m}^{i} \alpha_{n}^{i}:
\equiv\left\{\begin{array}{ll}\alpha_{m}^{i} \alpha_{n}^{i} & \text {
for } m \leq n \\ \alpha_{n}^{i} \alpha_{m}^{i} & \text { for }
n<m\end{array}\right.$$
This was the usual canonical quantization, that we read in
QFT, applied in string theory. There are other options to quantize strings,
but this introductory process is ideal for first-time string learners.
