The quantisation of the harmonic oscillator applied to the free Klein-Gordon field

In David Tong’s lecture notes on quantum field theory, at the bottom of page 23, we are applying the quantisation of the harmonic oscillator to the field to obtain expressions for the field operators in terms of ladder operators.

If we write the field $phi(vec x,t)$ as its Fourier transform: $$phi(vec x,t)=int frac{text d^3p}{(2pi)^3}e^{ivec pcdotvec x}phi(vec p,t),$$ and then expand the field in terms of the ladder operators we obtain: $$phi(vec x,t)=int frac{text d^3p}{(2pi)^3}frac{1}{sqrt{2omega_vec p}}left[a_vec pe^{ivec pcdotvec x}+a^dagger_vec pe^{-ivec pcdotvec x}right],$$ My problem is where the negative sign in the second exponential comes from, I can’t really even provide a guess, since I want to say that we’re just substituting in what is essentially the position operator from the regular quantum harmonic oscillator: $$hat x=frac{1}{sqrt{2omega}}(hat a+hat a^dagger).$$