I have written a lot about energy, but I’ve put off introducing the most important type of energy again and again. It’s the most important, because it is this type of energy that is responsible for all the structure in the universe, from galaxy clusters down to protons and everything in between. It is the most challenging to write about because it is not particularly intuitive. All the types of energy we intuitively understand, such as the energy of motion, are positive, but this type of energy, crucially, can be **negative**. On this website I’ll call it “interaction energy” (not the technical term, but my own, chosen to avoid misconceptions that might otherwise arise) because it is associated with the interactions among fields — including their little ripples that we call “particles”. *If you’ve taken physics you’ve heard of “potential” energy; what you learned within that concept is a subset of what is included under interaction energy.*

I’ve been wanting to address this for a while, because many of you have asked penetrating and central questions about the basic structure of matter, such as:

- Why is the neutron stable inside of atomic nuclei, given that on its own it is unstable?
- Why is the proton arguably heavier than the quarks and gluons that make it up?

And there are other equally important questions that no readers have yet stumbled upon but that I ought to address. Before I can answer any of those questions, however, I have to first describe interaction energy and the role that it plays in structure.

So — without further ado, here’s the article. This was an especially hard article to write and it may well be confusing in places — so I very much welcome your feedback, in order that I can try to make it clearer, if necessary, in later versions.

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Using an accounting scheme for total energy that allows for any purely negative contributions, even in the interaction energy, seems a little problematic, for the following reason: According to what you have said so far, there’s nothing to rule out the possibility that the total interaction energy in a bound state could be negative in sign and greater in magnitude than the sum of the mass and motion contributions of the components, giving an overall negative mass for the composite particle, which would then be repelled from black holes, spontaneously arise in stable particle / anti-particle pairs from the vacuum, and generally promulgate a panoply of paradoxes. (I realize that there are some formal mechanisms which would view this as no paradox, at least for fermions, by the accounting trick of assuming that all the negative-energy states must already be filled, so that removing a negative-energy electron is isomorphic to the creation of a positive-energy particle called a positron, but I don’t see how this would work for bosons or Majorana fermions, and the required cancelling of infinities is unsettling to me as well. I realize also that there is a formal sense in which fixed energy levels of a Hamiltonian are precisely as fictitious as the total phase of the quantum system, and it is only honest to talk about energy level differences—but the convention of zero energy for our comfortable (probably false) vacuum seems a useful one, and precisely the energy level it is catastrophic ever to stably dip below.) Is the following an accurate explanation for why this does not occur?:

What we call the mass of an individual particle—say the blue ripple—already includes not only the energy contribution of the disturbance in the blue field, but also the sum total of all energy contributions from other fields, such as the green field, that are unavoidably disturbed by the presence of the blue fluctuation. Each of these contributions is positive, and the standard calibration for absolute mass is for an absolutely isolated ripple. It is convenient to lump all of these energies into a single mass, just as it might be convenient, from an accounting perspective, to lump the shipping costs into the purchase price of an item ordered online. But according to that accounting, the possibility of combining shipping costs on two items ordered from the same place must be calculated as an “interaction cost” which is negative! This only appears to contradict the axiom that nothing is truly free or, more so, negative cost; a more meticulous accounting shows a contribution of three separate, positive costs: the true price of the first item, the true price of the second item, and the positive combined shipping cost, which is less than the sum of the individual shipping costs. In the same way, the positive green-field contribution to the overall mass of an isolated blue ripple plus the positive green-field contribution to the overall mass of an isolated yellow ripple is more than the total green-field contribution to the overall mass of the composite state that includes both the blue ripple and the yellow ripple in proximity, interacting via the green field. If we choose the simple accounting scheme that lumps all interactions with all fields into a single mass for isolated particles, we are required to treat this interaction energy as a negative contribution to the balance sheet, but it is important to note that such negative contributions cannot overwhelm the positive contributions to create an overall negative mass for a bound state, as can be seen by a more careful accounting that only acknowledges a more complicated set of exclusively positive contributions.

You are correct that nothing I have said would preclude the interaction energy overwhelming the other energies in the system.

So far, I have only talked about **weakly-bound systems** (the earth-moon system and atoms, and soon, nuclei) for which the binding energy, and the interaction energy which contributes to it, is much less than the masses of the particles being bound.

Things will get much more subtle when we try to understand hadrons (such as protons and pions) which are **strongly-bound systems** for which the interaction energy plays a hugely important role in the overall mass of the state.

The reason there are no negative-mass states is subtle. It simply violates various basic rules of quantum field theory; and so any physical system built out of quantum fields that respects those rules will never generate such a state.

But it is possible for interaction energies to *exactly* cancel the masses of the particles that make up a composite object, leaving it massless. There’s typically a very good reason [a symmetry or principle] that explains why this occurs.

And it is, in fact, possible for boson mass-squareds (but not masses!) to go negative, leading to a cascade effect known as a condensate. (This is also known as a tachyon; people mistakenly think there will be an associated particle that travels faster than light when mass-squared is negative, but in fact what happens instead in field theory is that a cascade occurs: the field develops a non-zero value, and the vacuum rearranges itself until there is no tachyon any longer. In fact this is just what happens with the Higgs field! and one ends up with a Higgs particle of positive mass-squared [and positive mass].)

But these are extremely advanced concepts in quantum field theory. We must walk before we run, and crawl before we walk.

Thank you for the careful and patient (as always) reply, and for being willing to venture briefly outside the scope of the article’s pupose. Any example of a negative interaction energy that exactly cancels out positive contributions would seem to preclude an explanation as simple as my combined-shipping-costs analogy for why negative total energies are forbidden. It’s interesting that there exists a (famous) mechanism in the bosonic case as well for “healing” any negative contributions. I suppose in some sense (that I don’t have the background to make precise) the vacuum is by definition what remains once you have eliminated the negative.

That’s kind of right, it’s just a bit more subtle than that. But yes, if you try to drive the world to do something that would cause an instability, the vacuum tends to rearrange itself until the instability is eliminated. So it is no surprise we have no obvious instabilities in the world we live in; they would have been eliminated by such a rearrangement long ago.

Condensate? You mean like the gluon and quark QCD condensates?

My heart froze…

Yes, or the Cooper pair condensate in a superconductor. There are many examples. Why did your heart freeze over this? Take an aspirin…

Because thats SO cool. I’ve never imagined a relationship between tachyons and condensates. Actually, I didnt know they were possible.

In a few GR books, tachyons are taken to be FTL particles. So, always that a particle “would be” a FTL one, the vaccum rearranges itself so that this tachyon becomes a condensate; or there are actually two types of them?

The field whose ripples [i.e. particles] would be tachyonic (if you were naive about it) instead becomes unstable, and the field develops a condensate. I don’t have time to write about this now though.