The free field Klein-Gordon equation $$(\Box+m^{2})\phi(t,\mathbf{x})=0$$ may be solved to give $$\phi(t,\mathbf{x})=\int d\omega d\mathbf{k}\widetilde{\phi}(\omega,\mathbf{k})\delta(\omega^{2}-\mathbf{k}^{2}-m^{2})e^{i(\omega t-\mathbf{k}\cdot\mathbf{x})}$$ which can be expressed as the superposition of harmonic oscillators $$\phi(t,\mathbf{x})=\int\frac{d\mathbf{k}}{2\omega(\mathbf{k})}\left(a(\mathbf{k})e^{i(\omega(\mathbf{k})t-\mathbf{k}\cdot\mathbf{x})}+a^{\ast}(\mathbf{k})e^{i(-\omega(\mathbf{k})t-\mathbf{k}\cdot\mathbf{x})}\right)$$ where $\omega(\mathbf{k})=\sqrt{\mathbf{k}^{2}+m^{2}}$ (see for example here: http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf)
My question is about getting from the second to the third equation, and what the exact form of the coefficients is. To do this we can use the identity $$\delta(\omega^{2}-\mathbf{k}^{2}-m^{2})=\frac{1}{2\omega(\mathbf{k})}[\delta (\omega-\omega(\mathbf{k}))+\delta (\omega+\omega(\mathbf{k}))]$$ and carry out the integral over $\omega$ which implies that \begin{align}a(\mathbf{k})&=\widetilde{\phi}(\omega(\mathbf{k}),\mathbf{k}) \\ a^{\ast}(\mathbf{k})&=\widetilde{\phi}(-\omega(\mathbf{k}),\mathbf{k})\end{align} This implies that \begin{equation}\tag{1}\widetilde{\phi}(\omega(\mathbf{k}),\mathbf{k})=\widetilde{\phi}^{\ast}(-\omega(\mathbf{k}),\mathbf{k})\end{equation}
We know that the scalar field $\phi$ is real. In one dimension, $$f^{\ast}(x)=f(x)$$ implies that $$\widetilde{f}(k)=\widetilde{f}^{\ast}(-k)$$ So in four dimensions, I would expect that \begin{equation}\tag{2}\psi(\omega(\mathbf{k}),\mathbf{k})=\psi^{\ast}(-\omega (\mathbf{k}),-\mathbf{k})\end{equation} should hold. Note that $\widetilde{\phi}$ is not quite the Fourier transform of $\phi$, which is instead $$\psi(\omega,\mathbf{k})=\widetilde{\phi}(\omega,\mathbf{k})\delta(\omega^{2}-\mathbf{k}^{2}-m^{2})$$ However, the delta function doesn't really seem to affect anything (since it is real and depends only on squared quantities) and this distinction doesn't seem that relevant to my problem. So my question is how do I reconcile Equations 1 and 2, which appear to disagree?
