The Time Development of Field Operators

The creation and annihilation operators are related to the time dependent coefficients in our Fourier expansion of the radiation field.

$$\begin{eqnarray*} c_{k,\alpha}(t)&=&\sqrt{\hbar c^2\over 2\omega}a_{k,\alpha} \\... ...ha}^*(t)&=&\sqrt{\hbar c^2\over 2\omega}a_{k,\alpha}^\dagger \\ \end{eqnarray*}$$

This means that the creation, annihilation, and other operators are time dependent operators as we have studied the Heisenberg representation. In particular, we derived the canonical equation for the time dependence of an operator.

$$\begin{eqnarray*} {d\over dt}B(t)&=&{i\over\hbar}[H,B(t)] \\ \dot{a}_{k,\alpha}... ...}[H,a_{k,\alpha}^\dagger(t)]=i\omega a_{k,\alpha}^\dagger(t) \\ \end{eqnarray*}$$

So the operators have the same time dependence as did the coefficients in the Fourier expansion.

$$\begin{eqnarray*} a_{k,\alpha}(t)&=&a_{k,\alpha}(0)e^{-i\omega t} \\ a_{k,\alpha}^\dagger(t)&=&a_{k,\alpha}^\dagger(0)e^{i\omega t} \\ \end{eqnarray*}$$

We can now write the quantized radiation field in terms of the operators at $t=0$.

$$A_\mu ={1\over\sqrt{V}}\sum\limits_{k\alpha}\... ...0) e^{ik_\rho x_\rho}+a_{k,\alpha}^\dagger(0)e^{-ik_\rho x_\rho}\right)$$

Again, the 4-vector $x_\rho$ is a parameter of this field, not the location of a photon. The field operator is Hermitian and the field itself is real.