# QFT for dummies

Consider a chain of point masses connected by springs and attached to a wall. This is a set of *coupled* harmonic oscillators.


How can we characterize the set of all possible oscillations? We can do a Fourier decomposition. Plot the displacements ($x_1, x_2, ...$) on a graph. We will find that there exist so-called *normal modes*, which look like standing waves:


The set of all normal modes forms a basis for the solutions (i.e., every solution can be expressed as a weighted sum of such modes).

If you look at any individual mode in *frequency space*, it will look like a single simple harmonic oscillator. The set of all modes is therefore a set of *uncoupled* harmonic oscillators.

(Taken from https://physics.stackexchange.com/a/630627/47309.)

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If you solve Maxwell's equations in a box using the Coulomb gauge[^gauges], the solutions look like standing waves:

$${\displaystyle \mathbf {A} (\mathbf {r} ,t)=\sum _{\mathbf {k} }\sum _{\mu =\pm 1}\left(\mathbf {e} ^{(\mu )}(\mathbf {k} )a_{\mathbf {k} }^{(\mu )}(t)e^{i\mathbf {k} \cdot \mathbf {r} }+{\bar {\mathbf {e} }}^{(\mu )}(\mathbf {k} ){\bar {a}}_{\mathbf {k} }^{(\mu )}(t)e^{-i\mathbf {k} \cdot \mathbf {r} }\right)}$$
[(From Wikipedia)](https://en.wikipedia.org/wiki/Quantization_of_the_electromagnetic_field#Electromagnetic_field_and_vector_potential)

where each $k$ represents a single mode (frequency). This is a bit harder to read, but notice that each wave in the summation consists of three (k-dependent) parts:

* $\mathbf {e} ^{(\mu )}(\mathbf {k} )$: A fixed polarization vector.
* $a_{\mathbf {k} }^{(\mu )}(t)$: A purely time-varying component.
* $e^{i\mathbf {k} \cdot \mathbf {r} }$: A purely space-varying amplitude.

We will also find that $a_{\mathbf{k}}^{(\mu )}(t)=a_{\mathbf{k}}^{(\mu )}{{e}^{-iwt}}$.

[^gauges]: TODO: Understand gauge fixing better.