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I've reading an introduction lecture on QFT, but I didn't started very well. I have troubles to understand some concepts of the very first example (But I think that they apply with the other fields): The scalar field, I will try to explain me the best I can.

As my lectures says, first we have the Equation of motion of the field: $$(\partial_\mu\partial^\mu + m^2) \phi = 0 \tag{1}$$

The author takes this Fourier transform: $$\phi(\vec{x},t) = \int \frac{\mathrm{d}^3p}{(2\pi)^3} \mathrm{e}^{\mathrm{i}\vec{p} \cdot \vec{x}} \phi(\vec{p},t) \tag{2}$$

Obtaining this equation of motion $(*)$: $$\frac{\partial^2}{\partial t^2} \phi( \vec{p}, t) + [\vec{p}^2 + m^2]\phi(\vec{p},t) = 0 \tag{3}$$

Where the solutions are a colleccion of harmonic oscillators labeled with $\vec{p}$ and with angular frecuency $ \omega^2_\vec{p} = \vec{p}^2 + m^2$

So my main quiestion is this: How do I interpret this?

The text says: "We learn that the most general solution to the KG equation is a linear superposition of simple harmonic oscillators, each vibrating at a different frequency with a different amplitude." But I can't see it since $\phi(\vec{p},t)$ is the Fourier Transform of $\phi(\vec{x},t)$, what I see is a linear superposition of $\mathrm{e}^{\mathrm{i}\vec{p} \cdot \vec{x}}$ with "weight" $\phi(\vec{p},t)$ which also varies in time. So I'm really confused of how the harmonic oscillators "make" the field.

Later the author continues with saying that we need to quantize that oscilators to quantize the field so this equation appears: $$\phi(\vec{x}) = \int \frac{\mathrm{d}^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_p}}(a_\vec{p} \mathrm{e}^{\mathrm{i}\vec{p} \cdot \vec{x}} + a^\dagger_\vec{p} \mathrm{e}^{\mathrm{i}\vec{p} \cdot \vec{x}})$$

I know $a_\vec{p}$ and $a^\dagger_\vec{p}$ are the annhilation/creation operator for the $\vec{p}$-th (for say it in a way) oscillator, but why the expression has this form? And how it relates with the superposition of oscillators?

$(*)$ Note: When I calculate by myself the Fourier transform of the equation I get a minus sign in the time derivative term since it have different sign from the $\nabla^2$ term in the KG-Equation. This doesn't give me the Harmonic oscillator solution so I asumme I'm wrong, but what is my mistake?

Sorry for my bad English.

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  • $\begingroup$ Its a bit more clearly seen if you try to solve the above KG equation assuming the spatial directions are periodic. Lets say in 1+1 dimensions with the space being compact $x = x+2\pi$. In that case you'll literally see an infinite set of oscillators with frequencies $\omega_n^2 = m^2 + n^2$. $\endgroup$ – childofsaturn Jun 6 '18 at 3:18

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