Do not understand this question well.

The potential energy in the form of parabola or paraboloid is a scalar quantity.

What is it you want as a scalar?

If you want examples in nature, an atom close to another atom feels a potential that is locally a parabola. In first approximation there is the harmonic model for vibrations of atoms in molecules, ie, balls connected by springs.

Thanks Juan Weisz

I've rephrased the question and the description a bit. As you see, the background is that I look for a more fundamental way how a symmetry breaking potential well could be formed, but my main question concerns the existence of a scalar field harmonic oscillator of some form. It could be very similar to e.g. a quartic interaction, but then with specific (harmonic) wavelengths.

Beste Sydney

Ik heb vooralsnog geen slechte ervaringen met professionele wetenschappers hier op RG. Het spijt me als jij dat wel zo ervaart (absoluut niet negatief bedoeld). Toch dank voor je reactie.

A covalent bond is a good example of how you can find a double well potential felt by an electron, which can be on atom 1 or atom 2. Example molecular oxygen.

The quantum mechanical solution is that there are two eigenstates with two energy levels, the bonding and antibonding levels . You place two electrons in the bonding level and that gives the bond some certain stability.

You can symmetry break the simple harmonic potential with a linear term, that just give you a new displaced harmonic oscilator.

However this may not be your question, and you are looking for some other more theoretical question. Field theory like QED allows you for example of how you eventually get photons by starting from Maxwell equations, which reduces to a problem of harmonic oscillators. Or you get phonons from linear chain equations in Solid State Physics, within the Harmonic approximation. However you will find that

symmetry breaking in these cases is rather hard to do, save linear perturbation.

Perhaps you can have some more specific example of what you want to do.

You sound a bit like seeking the Higgs symmetry breaking process, that you can look up...but I am just gessing.

Dear Juan Weisz

We're inching closer. Perhaps by giving the standard QHO as an analogy, the question seemed more about potential wells among particles than I actually meant.

" You sound a bit like seeking the Higgs symmetry breaking process "

Absolutely. As you know, standard symmetry breaking using the Lagrangian with added quartic term (i.e. non-linear self-interaction) has certain issues at high energy scales. I used to attribute that specifically to the quartic term, but others such as Farhad Ghaboussi argue that it is caused by the wrong assumption of 4 independent degrees of freedom (dimensions) in the whole theory. Farhad thinks there are only 2 independent physical degrees of freedom, which in fact makes a lot of sense to me.

I have worked (and still am) on what you might call an alt-Higgs mechanism which originates from a synchronized self-interaction of the field itself. This is the "scalar harmonic oscillator" in my question. I.e. there is no added (quartic or etc..) term. The potential well associated with a massive interaction is parabolic, i.e. it does not need to be double to produce a useful mass term. Philosophically there is still symmetry breaking, but not like it is done now, which I think Farhad would appreciate.

In fact, as from your replies so far I already get the impression that a "scalar field harmonic oscillator" is not commonly known, you have already answered my question. If you're interested a bit further I'd gladly walk you through the proposed solution, your further feedback is much appreciated.

- regards

The property “harmonic” of an oscillator does not leave much play. Harmonic oscillators always follow a sine function pattern. If “non-trivial” means “not oscillating in a sine pattern” then the answer is simply “no”.

Examples of scalar fields are pressure, temperature, density … in their spatial distribution.

Sound waves as an example are nearly harmonic oscillations which involve pressure, temperature, and density.

However symmetry breaking effects indeed exist if we observe high energy sound waves in water. The phenomenon is named cavitation. If the pressure gradient becomes too high, the water vaporizes and suddenly alters its properties. This cavitation effect is a good example for bifurcation.

Generally physical media tend to bifurcation in dynamic situations close to a transition between different aggregate states.

Non-trivial here does not at all mean non-sinusoidal, on the contrary, but rather a modest departure, in special cases, from the (over) generalized quartic interaction resp. Lagrangian dynamics.

What does "non-trivial" mean here? What is a modest departure (deviation?) from quartic interaction? What is a modest deviation from Lagrangian dynamics? Is it non-Lagrangian? Is Lagrangian dynamics a synonym for generalized quartic interaction?

Instead of being interested in reasons for true bifurcation in physics, you here play the master of confusion.

Non-trivial means functional, e.g. like the standard theorem potential well. It would be only a modest deviation from the latter, in that it considers distinct harmonic interactions, rather than an interaction mode.

Sorry, what is a potential well? Do you mean a potential wall/barrier?

What is functional in respect to a potential wall?

To what is "It" and "the latter" pointing? To "Non trivial", "standard theorem", or "potential wall", or something else?

What are distinct harmonic interactions? Can we consider also indistinct harmonic interactions or anharmonic interactions?

What do you mean about interaction mode? Is it a certain type of interaction? Do you have an example?

This post is exceptionally unclear! I have rarely found anything comparable, which leads to so much interpretation possibilities within such a short text.

In one of my posts on RG, you will find in the work

On the sturm-Liouville pseudo Hermitian connection,

one of the strangest Harmonic oscilator posts ever.

It is actually a double peak potential, with approximate parabola in the middle,

that finally tapers off to zero on both sides.

It is solvable, not in Schrodinger sense, but in Sturm Liouville, but continuously approaches the harmonic oscilator as the peaks become very high and separated, and in this limit becomes a Schrodinger problem.

Strangely the ground state is always a bound state approaching the HO ground state, and the other levels also approach HO levels, as peaks become high, becoming bound states.

For electrons moving along a periodic potential like a cosine or sine function,

quantum mechanics predicts a transition with amplitude known as the Aubry

transition, between localized and extended states

It studied by those who have studied localization theory started by Anderson, mainly linked to Solid State Physics.

For weak potentials states are all extended. Localized states are similar to bound states.