From the group $U(1)$ in QED, via group representation we get the harmonic oscillators. However, I still have a hard time to imagine newtonian objects (computer, headphone, etc) as a combination of oscillators. I have take a look at classical limit but the visualization is still vague.

Let's say I have a glass breaking. How to visualize it in oscillations?

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  • $\begingroup$ It is a really, really long trip. The rugged path is dubbed "classical limit". $\endgroup$ – Cosmas Zachos Feb 14 at 20:30
  • $\begingroup$ @CosmasZachos can you elaborate? $\endgroup$ – Ooker Feb 15 at 10:06
  • 2
    $\begingroup$ You may, painfully, discuss the classical limit, in phase space, of one oscillator; repackage trillions of these to classical field theory, reversing second quantization. Now, you claim you have an adequate classical field theoretic description of your phenomenon. Shortcut to models. Good luck. $\endgroup$ – Cosmas Zachos Feb 15 at 12:00

Suppose the object comprises $n$ degrees of freedom, and we work in a Cartesian coordinate system for these degrees of freedom, so each coordinate $q_i$ is $0$ at a stable equilibrium. The kinetic energy is of the form $\sum_i\frac12m_i\dot{q}_i^2$; the potential energy is of the form $\sum_i\frac12k_iq_i^2$, plus higher-order terms if an exact description of the system is anharmonic. (Cross terms, e.g. a term proportional to $q_1q_2$, can be deleted by rotating the coordinate axes.) Close enough to the equilibrium, such terms can be neglected. Such a harmonic approximation is of $n$ uncoupled harmonic oscillators.

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