One usually *starts* from the CCR for the creation/annihilation operators and derives from there the commutation rules for the fields.
However, one can start from either (see for example here about this).
Suppose we want then to start from the equal-time anticommutation rules for a Dirac field $\psi_\alpha(x)$:
$$ \tag{1} \{ \psi_\alpha(\textbf{x}), \psi_\beta^\dagger(\textbf{y}) \} = \delta_{\alpha \beta} \delta^3(\textbf{x}-\textbf{y}),$$
where $\psi_\alpha(x)$ has an expansion of the form
$$ \tag{2} \psi_\alpha(x) = \int \frac{d^3 p} {(2\pi)^3 2E_\textbf{p}} \sum_s\left\{ c_s(p) [u_s(p)]_\alpha e^{-ipx} + d_s^\dagger(p) [v_s(p)]_\alpha e^{ipx} \right\}$$
or more concisely
$$ \psi(x) = \int d\tilde{p} \left( c_p u_p e^{-ipx} + d_p^\dagger v_p e^{ipx} \right), $$

and we want to *derive* the CCR for the creation/annihilation operators:
$$ \tag{3} \{ a_s(p), a_{s'}^\dagger(q) \} = (2\pi)^3 (2 E_p) \delta_{s s'}\delta^3(\textbf{p}-\textbf{q}).$$
To do this, we want to express $a_s(p)$ in terms of $\psi(x)$. We have:
$$ \tag{4} a_s(\textbf{k}) = i \bar{u}_s(\textbf{k}) \int d^3 x \left[ e^{ikx} \partial_0 \psi(x) - \psi(x) \partial_0 e^{ikx} \right]\\
= i \bar{u}_s(\textbf{k}) \int d^3 x \,\, e^{ikx} \overset{\leftrightarrow}{\partial_0} \psi(x) $$
$$ \tag{5} a_s^\dagger (\textbf{k}) = -i \bar{u}_s(\textbf{k}) \int d^3 x \left[ e^{-ikx} \partial_0 \psi(x) - \psi(x) \partial_0 e^{-ikx} \right] \\
=-i \bar{u}_s(\textbf{k}) \int d^3 x \,\, e^{-ikx} \overset{\leftrightarrow}{\partial_0} \psi(x) $$
which you can verify by pulling the expansion **(2)** into **(4)** and **(5)**.
Note that these hold for any $x_0$ on the RHS.

Now you just have to insert in the anticommutator on the LHS of **(3)** these expressions and use **(1)** (I can expand a little on this calculation if you need it).

most sources simply 'pull the $u, u^\dagger$ out of the commutators' to get (anti)commutators of only the creation/annihilation operators. How is this justified?

There is a big difference between a polarization spinor $u$ and a creation/destruction operator $c,c^\dagger$.

For fixed polarization $s$ and momentum $\textbf{p}$, $u_s(\textbf{p})$ is a four-component spinor, meaning that $u_s(\textbf{p})_\alpha \in \mathbb{C}$ for each $\alpha=1,2,3,4$.
Conversely, for fixed polarization $s$ and momentum $\textbf{p}$, $c_s(\textbf{p})$ is an operator in the Fock space. *Not* just a number, which makes meaningful wondering about (anti)commutators.

This post imported from StackExchange Physics at 2014-12-06 00:45 (UTC), posted by SE-user glance