Mode expansion of a free field

A free field is expanded in positive- and negative-frequency plane waves multiplied by annihilation and creation operators.

For a free field in the annihilation-operator vacuum, the coincident two-point function is its mode-power integral. Unsmeared vacuum products generally have an ultraviolet divergence. A finite band or spatial smearing defines a regulated variance without altering the meaning of a mode-by-mode dimensionless power spectrum.

For a free field, its positive-frequency part consists of the annihilation terms multiplying $e^{-i\omega t}$ with $\omega >0$; its negative-frequency part consists of the creation terms. The sign refers to positive energy, even though the exponential has a minus sign. Imposing the Lorenz gauge only through $(\partial \cdot A{)}^{(+)}|\mathrm{phys}⟩=0$ is the subsidiary condition in Gupta-Bleuler quantization, not an operator identity for the whole divergence.

The canonical commutation relation fixes the antisymmetric product of a mode function $f$ and its canonical-momentum mode $g$. In the convention $[a,a^{†}]=1$, one commonly has $f{g}^{\ast }-f^{\ast }g=i$.