Real Scalar Field Fourier Transform

[Xenosum]

Homework Statement

Silly question, but I can't seem to figure out why, in e.g. Peskin and Schroeder or Ryder's QFT, the Fourier transform of the (quantized) real scalar field $\phi(x)$ is written as

$$\phi (x) = \int \frac{d^3k}{(2\pi)^3 2k_0} \left( a(k)e^{-ik \cdot x} + a^{\dagger}(k)e^{ik \cdot x} \right) .$$

It's just weird because the Fourier transform usually only has one term, and one coefficient.

Thanks for any help.

Homework Equations

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The Attempt at a Solution

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[nrqed]

It just makes the field real. Note that the second term is the complex conjugate (or the hermitian conjugate once the coefficients have been promoted to operators) of the first term and in general, adding an expression to its complex conjugate gives something real.