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So there is this notion that all forces can be broken down into the four fundamental forces/interactions. However, I'm starting to wonder if that is really true.

The solidity of matter is explained using two things. The first thing is that the electromagnetic interactions keep atoms and molecules bounded together. This explains why something holds itself together, but it is not enough to explain why my hand can't push past my table. So the second relevant thing is that Pauli's exclusion principle prevents atoms and electrons from occupying the same volume, causing solid matter to be impenetrable. See this post.

But if it's Pauli's exclusion principle and fermionic statistics that prevent my hand from going through my table, then the normal force that prevents my hand from going through my table isn't really decomposable into the four fundamental interactions. So now I'm left wondering, is it really true that all forces break down into the four fundamental interactions?


Clarification

I've looked at various other posts and comments, and frankly I'm left with even more confusion than I started out with. John Rennie provided a helpful older post explaining how the Pauli exclusion principle and the electromagnetic field together play a non-trivial role in forming the stability of matter. A comment left by tobi_s further clarifies John Rennie's post,

John's argument is that the electrons need to be rearranged due to Pauli's exclusion principle, and this is made expensive by the electromagnetic forces which try to keep the electrons in place. So the force you feel is electromagnetic, but it is caused by the EM forces competing against Pauli's exclusion principle. I guess it comes down to semantics which of the two "leads to repulsion".

This makes sense, but nonetheless I feel as though my question isn't fully answered.

My main question is, besides the possible example of gravity (which may or may not have a corresponding graviton particle), is there a Newtonian force vector with no corresponding boson force carrier?

Notice the two different usages of the word "force." On one hand, I am talking about a Newtonian force vector. On the other hand, I am talking about boson force carriers in QFT. I am wondering if there is a correspondence between these.


Some More Comments

After reading over John Rennie's post, I had a lingering question (related to this post) that is now resolved. I thought I would write it here as part of my thought process.

Let's say I "turned the EM field off" completely. Electrons are now free particles and the EM field is not responsible for any force (because it's "turned off"). Nonetheless, if we imagine some cloud/gas of electrons, and we send this cloud directly to another cloud/gas of electrons, it seems as though there would be an "effective" collision between the two clouds due to the Pauli exclusion principle.

This post here touches exactly upon this type of thought experiment, and the answer there is that there isn't really any emergent or effective collision happening at all due to linearity of wavefunction evolution.

So it seems that we cannot say there is any kind of emergent Newtonian force at the macroscopic scale for non-interacting free fermions that comes solely from Pauli's exclusion principle.

It seems that the fact that electrons in solids are (generally) bounded to atoms (due to the electromagnetic force) plays a non-trivial role in explaining how a Newtonian normal force emerges on the macroscopic scale.

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    $\begingroup$ The Pauli exclusion force isn't a fundamental force. $\endgroup$ Commented Mar 9, 2022 at 16:45
  • $\begingroup$ "The Pauli exclusion principle is the quantum mechanical principle which states that two or more identical fermions (particles with half-integer spin) cannot occupy the same quantum state within a quantum system simultaneously." wikipedia. principle is an axiom for the quantum mechanical theories, imposed together with the postulates in order to pick up from the mathematical solutions those solutions that fit the data and are predictive of new observations. $\endgroup$
    – anna v
    Commented Mar 9, 2022 at 18:04
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    $\begingroup$ see physics.stackexchange.com/a/610128/50583 (I'd pretty much write that same answer here) $\endgroup$
    – ACuriousMind
    Commented Mar 9, 2022 at 18:59
  • $\begingroup$ @JohnRennie Ok, my question about clouds of non-interacting electrons was nonsense, because wavefunction evolution is linear and thus no "effective collision" would result. $\endgroup$ Commented Mar 17, 2022 at 15:44
  • $\begingroup$ Closely related: Is Pauli-repulsion a "force" that is completely separate from the 4 fundamental forces? $\endgroup$ Commented Jun 8, 2023 at 23:48

2 Answers 2

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Pauli's exclusion principle is not a force, but rather a constraint that has the effect of changing how other forces manifest.

The relevant force in this case is still just the electromagnetic force (what other ones could be involved, by process of elimination?). But because of how that quantum mechanics alters the informational attributes of mechanics, the quantity of "electromagnetic force" between two particles must, just like position, be described as a probability distribution, viz. a "force wave function", only this probability distribution is not over points in space but over possible values of the electric force felt by the electrons in the materials.

And what Pauli does is alter that probability distribution. Thus it does contribute to determining what the final force you feel and can exert with your hand is. But a nontrivial distribution for the force is only present to begin with by virtue of the fact that there is an electromagnetic force source available.

ADD: That said, though I haven't worked through the details, it might - so don't trust me on this - be that you could write that altered distribution as though it were generated by a "phantom force" operator acting on an otherwise non-Pauli "counterfactual" state (think combining electron states as though they were bosons) and, in this sense, you could think of it as a "force" in a manner analogous to how that centrifugal, Coriolis, etc. effects are understood in Newtonian mechanics. In any case, though, the counterfactual state cannot be correct because it will yield the wrong probabilities for other physical parameters (though maybe you can also distort their operators in a suitably complementary manner to even that out, so you are in effect "shifting a reference frame" on the operator space. Still, though, the main point holds that this is a mathematical trick.).

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One of the simplest example of attraction / repulsion force in QM is analized by Griffiths in the chapter 7.

It analizes the bound state of a H2 molecule with only one electron.

The starting point is the Hamiltonian for the electron, with a term for kinetic energy and another for (electrostatic) potential.

Initially, it is supposed a fixed distance $R$ between the protons, and the result is the energy of the electron. After adding the energy resulting form the repulsion between protons, (which is a function of $R$), he gets the total energy.

The total energy is plotted against $R$.

Taking the force as minus the gradient of that energy, it is attractive for distances greater than $2.4$ Bohr radius, and repulsive below that.

The gradient for smaller distances is very steep, what means huge repulsive force.

This strong repulsive force comes only from QM Hamiltonians with electrostatic potentials. There is no mention to Pauli exclusion principle.

I think that it is reasonable to extend the same explanation for complicated bound state in solids.

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  • $\begingroup$ I'm skeptical of your concluding remarks at the end, but I'll make sure to look over that part in Griffiths. Thank you. $\endgroup$ Commented Mar 17, 2022 at 2:37

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