"QFT is simple harmonic motion taken to increasing levels of abstraction."

This is my memory of a quote from Sidney Coleman, which is probably in many textbooks.

What does it refer to, **if** he meant something specific?

If he did not, which is far most likely, then rather than asking a list question, if somebody can point to an example of how we move from SHM to, presumably some example of fields interacting, producing or destroying a particle.

I don't think I can go any further with asking a question, because of the restrictions when asking a specific recommendation, but if someone is able to say, "wait until you get to chapter X of Zee, Tong, P & S (please don't say Weinberg) and enlightenment will follow", that would be very much appreciated.

My apologies if I have mangled the quote, nobody has heard of it or I am trying to run way ahead of myself. No need to hold back on telling me that last part.

## Best Answer

The main idea is that you can take a complicated interacting or coupled system and write its solution as a sum of non interacting or free modes. Even in classical mechanics, if you have a linear chain of $N$ oscillators, one can show that the general solution is a sum of $N$ normal modes each of them being a simple harmonic oscillator. In Quantum Field Theory we can also write the fields as sums of modes, each mode behaving like a quantum harmonic oscillator so can accept energy in an integer number of lumps of size $\hbar\omega$. In the second quantization formalism, these lumps are created or annihilated by operators acting on vacuum and this is interpreted as particles being created or annihilated.

As a concrete example, consider a linear chain of (interacting) atoms whose Hamiltonian operator is $$H=\sum_i\left[\frac{p_i^2}{2m}+\frac 12k(x_{i+1}-x_i)^2\right]$$ After Fourier transforming both the positions and momenta, the Hamiltonian can be written in the reciprocal space as $$H=\sum_i\left[\frac{\tilde p_i\tilde p_{-i}}{2m}+\frac 12\omega_i^2\tilde x_i\tilde x_{-i}\right].$$ By defining the operators $$a_i=\sqrt{\frac{m\omega_i}{2\hbar}}(\tilde x_i+i\tilde p_i/m\omega_i),\quad a_i^\dagger=\sqrt{\frac{m\omega_i}{2\hbar}}(\tilde x_{-i}-i\tilde p_{-i}/m\omega_i),$$ the Hamiltonian can also be written as $$H=\sum_{i=1}^N\hbar\omega_i(\hat a^\dagger_i \hat a_i+1/2),$$ which is the sum of $N$ uncoupled quantum harmonic oscillators. Each $i$ in the above equation represent a mode or an oscillator and according to second quantization the $n$th excitation of this mode shall be interpreted as having $n$ particles, which in these example in particular we call phonons. The same approach can be carried out to a generic field. You Fourier transform the field and show that it can be written as a sum over modes. These modes decouples in the Hamiltonian and each of them gives a harmonic oscillator. The general solution is therefore a linear superposition of harmonic oscillators. If we quantize those oscillators we then quantize the field itself.