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Disclaimer: I know this is a vague question, so if this is not the appropriate thread, please direct me to the correct one.

On page 5 of Anthony Zee's Quantum Field Theory in a Nutshell he speaks of a "harmonic paradigm". He uses this term to describe physical theories which are based on studying the excitations of a field, like QFT and String theory do. He is surprised by the fact that even after 75 years we still rely on basic notions of oscillations and wave packets for our physical theories and have not yet found an alternative formulation.

My questions are:

  • What makes this "harmonic paradigm" so useful?

  • What, if any, alternative formulations have been proposed?

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  • $\begingroup$ That we teach the harmonic oscillator is simply because it is important for a large number of systems. And then there are the ones for which harmonic approximations do not work... those are the hard ones, but Zee's criticism is pretty shallow because even the most non-linear systems seem to be capable of nearly linear excitations and if we trust the phenomenology, we can almost assume that there is a general principle at work that forces almost all systems to have scales on which a linear approximation is at least halfway reasonable. $\endgroup$ – CuriousOne Jun 18 '16 at 0:33
  • $\begingroup$ Related question: physics.stackexchange.com/q/159021/2451 and links therein. $\endgroup$ – Qmechanic Jun 19 '16 at 12:54
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This might help you out a bit: In what sense is a quantum field an infinite set of harmonic oscillators?

From my understanding, most people think it provides a useful way to conceptualize uncoupled quantum fields physically. It doesn't, however, work for coupled quantum fields.

The main problem with this seems to be that infinite harmonic oscillators give you an infinite vacuum energy, something physicists work around through renormalization (basically looking only at energy differences instead of total energy). Some people are satisfied with this, some aren't.

Klauber, as mentioned in the question linked, published this paper in 2003, where he claims to have worked out a solution to this problem (see section 4 of the paper). I've looked at this paper and I didn't really understand it, so I can't verify if his claims are true, but section 4 seems to offer something of an alternative formulation (or at least a solution to the infinite energy problem).

Hopefully that helps!

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    $\begingroup$ I don't quite see how that paper solves the fundamental problems, especially since it doesn't address them. The crucial problem of quantum field theory is that it happens on a pre-geometry which is scale free, i.e. it sets an infinite amount of space at the bottom and at the top. We know that phenomenologically this can not be true (because of gravity) and the entire notion of pre-geometry is utterly false (but practical). To me it seems that Glauber's algebraic approach represents a standing wave approximation of qft, which, of course, is not physical. $\endgroup$ – CuriousOne Jun 18 '16 at 0:49
  • $\begingroup$ Very interesting, thank you! What exactly do you mean by pre-geometry? Is that the assumption that we're looking at an infinite volume (when we do the integrals, for example)? $\endgroup$ – Akorl Jun 19 '16 at 1:27
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    $\begingroup$ It's the assumption that a(n exactly) three dimensional space with a perfectly "ticking" (if local) time variable exists and that every event can be labeled by these space-time coordinates. General relativity tells us that the bending of spacetime has extremely deep consequences and so far we have not been able to successfully mary quantum theory with general relativity. It is fairly widely believed that one can't formulate a quantum field theory that resembles reality in fully self-consistent ways without replacing the spacetime parametrization with a new concept... but what? $\endgroup$ – CuriousOne Jun 19 '16 at 1:37
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Purely mathematically, a function $f(x)$ (that can be differentiated enough times and so on) can be Taylor expanded around a point $a$ as \begin{equation} f(x) = f(a) + (x-a) f'(a)+\frac{(x-a)^2}{2}f''(a)+.. \end{equation} Now if we're describing a physical system with $f$ and the point $a$ is an equilibrium of the system, $f'(a)=0$. Then we have \begin{equation} f(x) = f(a) + \frac{(x-a)^2}{2}f''(a)+.. \end{equation} We see that when describing the behaviour around an equilibrium, the simplest nontrivial approximation is of the form $x^2$, harmonic. Hence, the universal importance of the harmonic oscillator.

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