# The simple harmonic oscillator model relating particles and fields in QFT

In all of the introductory Quantum Field Theory texts I gave read so far, (such as Zee, Srednicki, Luke), there is an introduction to the concept of fields as operators, following the simple harmonic oscillator analogy.

After an illustration of how creation and annihilation operators methods are applied, the texts then go on to discuss the various (and large) range of other topics related to QFT.

My question is: does the level of discussion specifically around the process of creation of a particle from a field in these texts reflect as much as we can say about the subject, or do more advanced texts go further into this mechanism?

My problem here is that I don't know enough about QFT when searching online to recognize if there are more advanced discussions in standard texts, for example Weinbergs' Fields, which is why I am asking here.

At this stage, I simply want to know is the topic taken further later in more advanced texts, or does the simple harmonic model pretty much sum up our current model? (Probably not).

The reason I ask is simple curiosity, I certainly am not ready to tackle anything more than is contained in the excellent texts listed above.

I have read questions such as Excitations in a Field but that doesn't really address my question.

• Although the formulation using creation/annihilation ops is very useful for perturbative calculations, the "annihilation" operators used in perturbation theory don't annihilate the true vacuum state (not even approximately), and the "creation" operators don't create true single-particle states. As an analogy, consider the QM of a single quartic oscillator: $\ddot \phi + \omega^2\phi + \lambda\phi^4=0$. Cre/ann op's have limited utility here. Nonperturbative QFT studies, like numerical studies of the meson/baryon spectrum in QCD, don't usually try to express things using cre/ann ops. – Chiral Anomaly Dec 14 '18 at 3:33

One cannot overstate (underestimate, or undervalue) the value of the SHO. To fully understand it consider transforming the wave equation (and all QFT equations are a form of the wave equations) into momentum space. There, one has a representation of the field amplitudes that looks an awful lot like SHOs, possible coupled. In fact many models of classical fields can be built up from coupled oscillators bringing the analogy around full circle. However that approach also give rise to the interpretation of creating and annihilating particle states which is useful. This paradigm is also more useful, imo, when coupling between fields is weak. Some QFT books start with step up and step down operators then build a field state in space-time representation while others start in the space time representation then introduce these operators later. It really doesn't matter imo which way you go but I prefer to look at things in space-time field representation. In the very old days I believe the Schwinger-Dyson-Tomonaga approach looked like this. I have not heard of the authors you cite except for Zee. You can look at Ramond, itzykson and zuber. There are a lot of great old texts on QFT and I am sure that they go into these interpretations in detail.

• Thank you, it's just that Zee makes a reference (I think) to wanting another perspective than SHM, but I don't know enough to judge why. But I do fully take your point. Your first line should say overstate, I think....:) – user214814 Dec 14 '18 at 0:41

The standard model of particle physics is based on quantum field theory,the Feynman diagrams of the interactions being the output of this usage.

AFAIK, as an experimentalist, the field is an operator field distinct for each particle in the table of particles ( the anti also) of the standard model.

It is posited that all of space is covered by electron fields, neutrino fields etc , whose vacuum expectation value is zero, (except for the Higgs field) and mathematically, they are the solutions of the corresponding quantum mechanical equation for free particles, i.e. plane waves. They are solutions of the Dirac equation for fermions, Klein-Gordon for bosons, and quantized Maxwell for photons.

Creation and annihilation operators are used in constructing the Feynman diagrams for the calculation of interactions.

Thus the basis of the fields is not the simple harmonic oscillator solutions in the standard model of particle physics, but the mathematical logic is the same.