# Why is the Pauli Exclusion Principle not a force?

• I
This is a question I've had for some time now.
Why is the exchange interaction not considered a force, like the other 4 fundamental forces? When reading solid-state physics texts, for example, I come across explanations of this kind: the atoms cannot get too close together because of the exchange interaction of the electrons. I guess an answer could be that there is no carrier particle supplying the force, such as the photon. But to me this is an unsatisfactory answer because why can't we simply expand our definition of force to interactions where no carriers are necessary? All we should care about is the net result on the particles experiencing the interaction, and in that sense I don't see why the exclusion principle cannot be considered a force. So, why is the exchange interaction not a force? What part of the interaction is different from among the other four forces that prevents it from being considered one?

A side-question: When you seat on a chair, my electrons are repeled by the electrons in the chair, so that I don't fall through. Is this electrical repulsion the only cause for me not falling through, or is it also the Pauli exclusion principle? Or is it ONLY the Pauli exclusion principle?
I find these things confusing, I hope someone can help me clear this up.

Related Quantum Physics News on Phys.org
Simon Bridge
Homework Helper
This is a question I've had for some time now.
Why is the exchange interaction not considered a force, like the other 4 fundamental forces? When reading solid-state physics texts, for example, I come across explanations of this kind: the atoms cannot get too close together because of the exchange interaction of the electrons. I guess an answer could be that there is no carrier particle supplying the force, such as the photon. But to me this is an unsatisfactory answer because why can't we simply expand our definition of force to interactions where no carriers are necessary?
We could but then why not define everything as a force? In order for our physics to make sense we need to be somewhat consistent with our definitions.

You realize that physics does not have to be satisfactory to you right?

All we should care about is the net result on the particles experiencing the interaction,
That's one point of view certainly - saves all that trouble looking for underlying laws of physics doesn't it?
and in that sense I don't see why the exclusion principle cannot be considered a force. So, why is the exchange interaction not a force? What part of the interaction is different from among the other four forces that prevents it from being considered one?
The Pauli principle is basically geometry - a statement of symmetry. There is no force making your mirror image look that way right?

A side-question: When you seat on a chair, my electrons are repeled by the electrons in the chair, so that I don't fall through. Is this electrical repulsion the only cause for me not falling through, or is it also the Pauli exclusion principle? Or is it ONLY the Pauli exclusion principle?
I find these things confusing, I hope someone can help me clear this up.
The chair thing is definitely coulomb force. PEP plays a role in determining the electro structure but that isn't what is holding you up.

We do sometimes talk about a Pauli pressure - it's what holds neutron-stars up.
... there are a bunch of useful links from there too.

We could but then why not define everything as a force? In order for our physics to make sense we need to be somewhat consistent with our definitions.
We could call a cat an animal, but then why not define everything as an animal?
You realize that physics does not have to be satisfactory to you right?
This is not what dsanz says. If somebody does't like coffee, he is not assuming coffee is made to please him
That's one point of view certainly - saves all that trouble looking for underlying laws of physics doesn't it?
Ridicule the argument. Again this is not what dsanz says.
The Pauli principle is basically geometry - a statement of symmetry. There is no force making your mirror image look that way right?
Just calling it a 'statement of symmetry' doesn't exclude it of being a force

I'm sure you mean well, but please use valid arguments. You might want to have a look at wiki.
So what exactly in the definition of a force does exclude the behavour of particles imposed by the PEP, like the Pauli pressure?

dsanz and Carrock
Simon Bridge
Homework Helper
Excuse me, perhaps I should not have relied so heavily on context to make my statements clear? I can certainly see how my comments could be taken the wrong way.

We could call a cat an animal, but then why not define everything as an animal?
Not a good comparison since cats are normally considered animals. However PEP is not normally considered a force and OP was asking why not extend the definition of force to include it anyway.

Using your analogy then, OPs question is like asking why not call a rock an animal... sure it is not normally considered an animal but maybe we can extend the definition to include rocks?

This is not what dsanz says. If somebody does't like coffee, he is not assuming coffee is made to please him
Quite right - however, someone publicly proclaiming that the coffee is unsatisfactory may often be doing so under the impression that the coffee aught to have been made to please him. One cannot always tell and it is a common refrain in these forums so I asked ... just to be sure. You are reading too much into that statement.

Ridicule the argument.
That's not what I said either.

Just calling it a 'statement of symmetry' doesn't exclude it of being a force
But that's not all that I said.

I'm sure you mean well, but please use valid arguments.
I did not mean you to think I was attempting to produce arguments. I don't have to - others have done so before me. I was trying to explore OPs reasoning. The idea is that I get feedback from OP.

I had hoped that the reader, having got that far, would have also looked at the supplied link where this question has been addressed before. Perhaps I needed to make it more prominent?

Thanks for the heads-up. I'll have to take more care in future.

davidbenari and bhobba
Bill_K
Our mistake is in thinking of the particles in an N-particle state as being N independent objects. And so we ask ourselves why this particle can't be over there with the other one - there must be some force preventing it.

Whereas in fact, for each type of fermion there is only one independent object - the field. There is only one universal electron field, and electrons are excitations of this field. It's a property of the field that certain N-particle states exist while others do not.

A single electron can't have spin 3/2, but that just means there is no spin 3/2 state. You wouldn't claim it's the influence of a force that's preventing it. Likewise for two electrons the symmetric states simply don't exist.

PeterDonis, Kia Neves, Filip Larsen and 3 others
tom.stoer
Why is the exchange interaction not considered a force?
Using creation and annihilation operators free particles are described by a Hamiltonian with bilinear operators like $a_i^\dagger a_i$. Any interaction term in a Hamiltonian has four or more operators.

The Pauli exclusion principle says that no two fermions (I'll use b's instead of a's) can exist in the same quantum state i (where i is a multi-index for monentum, spin, etc.). The property of the fermionic operators encoding this principle reads

$$(b_i^\dagger)^2 = 0$$
which means that you cannot create two fermions for one single state i.

As you can see this has nothing to do with a specific index i or an interaction described by higher order terms. It is valid for any fermionic operator b and any multi-index i. Using fermionic operators to construct arbitrary Hamiltonians means that the Pauli principle is guarantueed by construct.

Therefore this principle is more fundamental than any interaction.

Kia Neves and Nugatory
Bill_K thank you so much! That was the kind of answer I was looking for.
Simon, I think you misunderstood me, but I was not trying to change the definitions of the physics language, but merely trying to understand what I was missing. I think you should listen to what ajw1 said and try to answer people's questions rather than trying to just ridiculize them.

Simon Bridge
Homework Helper
@dsanz: I figured as much ... but that was why I put that link at the bottom: so I could provide a range of answers while exploring the question ... just in case. Mostly what I was getting from your post was that you were confused and much of what I was reading was likely to be an artifact of this. On reflection, I could have made the strategy clearer.

Bill_K's sort of answer was what I was referring to as "geometry" and "symmetry" ... we don't see a force making all the angles of an (euclidean) equilateral triangle 60 degrees because the angles are not separate from the triangle... but if we saw a equi-quadrilateral with all 90 degree angles, we may wonder what force keeps it that way.

Tom.stoer's sort of answer would have been where I would have headed if it turned out you were more interested in taxonomy: why we label some things forces but exclude others despite their similarity? But it turns out that wasn't what you were getting at.

I had a choice of ways I could have responded to get a better picture of the question ... I could have just written out all the possible answers and had you pick the one you wanted. I had hoped to avoid a lot of writing: well that didn't work :)

There were several other ways of reading the first post too... To your point: I did not intend to "ridiculize" you or what you said exactly but to reflect the confusion in the reader (me) back at you... I expected that you'd reply tot tell me how I've got you totally wrong and set me right, which would have told me what sort of reply to give you.

The gambit usually works - in this case, it didn't.
I apologize for that and I hope the impression created hasn't put you off asking questions.

Last edited:
@dsanz: I think that's a very interesting question. I'm sorry I don't have an answer. but I have another question in the neighbourhood which might help. speaking very loosely, the reason you probably have to think that PEP should count as a force, is that you expect nature to oppose our attempts to violate it. and we know it does. now, how could this "opposition" manifest itself, if not by exerting (or seeming to exert) a force? However, PEP also has features which don't make it seem like it is linked to a force, and this brings me to my question. what is the magnitude of the "force" that seems to be associated with the PEP, and how does it change with distance? On the one side, I guess, it's magnitude is infinite (whatever this means) at 0 distance, as the probability associated with that state is 0; and it is 0 anywhere else. Am I wrong? This is very strange indeed for a force, and it makes it hard to understand how such a "force" might happen to cause anything. Or is there any gradual phenomenon which could be interpreted as nature's gradually beginning to "prevent" such violation, as distance approaches 0 (Pauli pressure?)?

bhobba
Mentor
Well it often manifests as a force eg Dyson showed it was basically responsible for solidity. But I think in physics we try to label things in the most elegant way. Its actually a direct consequence of a deep insight into QFT, the so called spin statistics theorem:
http://www.worldscientific.com/worldscibooks/10.1142/3457

It explains the behaviour of Fermions and Bosons and why we have this categorisation in the first place. At rock bottom it's not really a force which is why I think its not generally labelled that way.

Interesting question though.

Thanks
Bill

Thanks for your reply, Bill! I have a related question. Two electrons occupying the same orbital but with opposite spins ARE actually occupying the same (quantistic) location. There is no repulsive "force" acting between them in that case (in virtue of PEP). But what if all other degrees of freedom are saturated? I wonder if you could use a stable configuration of equilocated fermions to "push" another fermion, exerting a repulsive "force" on it, just in virtue of PEP.

In defense of Dsanz' question, gravity was called a force long before gravitons were postulated (notwithstanding that in general relativity it becomes a fictitious force). Since we still don't know if gravitons exist, why require a mediating particle to dignify a phenomenon with the title "force"? The Pauli exclusion principle undoubtedly causes repulsion. It requires work to compress fermions and some fermions must occupy higher energy levels to be in the same proximity. This energy is at the expense of the compressing mechanism. Recent texts I've read attribute the near incompressibility of liquids and solids to the Pauli exclusion principle, not Coulomb repulsion, which appears to make sense because neutral atoms have equal numbers of positive and negative charges, and from symmetry the mean positions of electrons (taken over the extent of their electron shells) are coincident with the nuclei. Until quantum gravity is established as a credible theory, we might as well limit the fundamental forces to three, electromagnetic, electro-weak, and strong, if we insist on a mediating particle.

Last edited:
Demystifier
Gold Member
Why is the exchange interaction not considered a force, like the other 4 fundamental forces?
The 4 fundamental forces are described in the language of field theory. Field theory is described by a Lagrangian density which contains terms quadratic in fields, as well as non-quadratic terms in which 3 or more fields are multiplied. In this language, by a "force" (or more precisely by interaction) one means all the non-quadratic terms in the Lagrangian density. Even gravity can be considered interaction in this language. On the other hand, the Pauli pressure is present even without the non-quadratic terms, provided that the fields are quantized according to the Fermi-Dirac statistics. Thus the Pauli pressure has a very different fundamental origin, even if a manifestation for many-particle systems looks similar to other forces.

Last edited:
PeterDonis, Elemental and DrClaude
Staff Emeritus
2019 Award
By the way, the number of fundamental forces is 5, not 4. One should not forget the force carried by the Higgs.
Is it five? Or is it three? (Strong, electroweak, gravity)

This urge to enumerate - we see the same thing with states of matter - often gets in the way of understanding rather than leading to understanding. Are there six or seven colors in a rainbow? Does that lead to any better understanding of refraction?

The truth is that nobody understands the PEP. It is indeed the PEP that prevents Dsanz from falling through his chair.
The result of PEP is therefore the force preventing this to happen. In fact, without PEP the universe would be very different.
I find it hard to accept that 'geometry' would have such an important physical effect.

Demystifier
Gold Member
Is it five? Or is it three? (Strong, electroweak, gravity)
Or is it one? (string theory)

You are right that there are different ways of counting, but I think the best answer is still five. Before the symmetry breaking, the "electroweak" interaction consists of three logically and mathematically independent sectors. Two of them are described by two different gauge groups (U(1) and SU(2)), and the third (Higgs) is not described by a gauge theory at all. Of course, after the symmetry breaking these three sectors get mixed, but the picture before symmetry breaking is more fundamental.

Dale
Mentor
Recent texts I've read attribute the near incompressibility of liquids and solids to the Pauli exclusion principle, not Coulomb repulsion
Do you have a reference for this?

Staff Emeritus
2019 Award
The truth is that nobody understands the PEP.
That you don't understand it does not mean nobody understands it.

Staff Emeritus
2019 Award
here is no repulsive "force" acting between them in that case
Actually, "force" - as measured in Newtons - works better here than it does in the case of the weak nuclear interaction, where getting to Newtons is as best contrived.

Consider a particle of mass m in a box of size x. It has energy

$$E = \frac{n^2 h^2}{8m x^2}$$

If I have three non-interacting bosons in the ground state, the total energy is

$$E = \frac{3 h^2}{8m x^2}$$

and if I make an adiabatic change in size dx, I get

$$F = - \frac{dE}{dx} = \frac{3 h^2}{4m x^3}$$

Essentially, there is a pressure on the walls of the box from the particles inside - i.e. a force - with the value as calculated above.

Now, let's make them three non-interacting fermions (with spin-1/2). Instead of having all three in the n=1 state, one has to be in the n=2 state. That changes the force to:

$$F = - \frac{dE}{dx} = \frac{3 h^2}{2m x^3}$$

In other words, doubling it. So there is a very real increase in force and pressure due to the fermionic nature of the particles in the box. You can measure this force in Newtons and everything.

CalcNerd, PeterDonis and Nugatory
DaleSpam,

Do you have a reference for this?
I think the current view traces back to Dyson as referenced here: https://en.wikipedia.org/wiki/Electron_degeneracy_pressure. See Introductory Solid State Physics, 2nd Ed., by H.P. Myers, pg. 19; http://www.physik.uni-augsburg.de/theo3/Talks/Surprising_effects_Inauguration_Pauli-Centre_DESY-Hamburg_2013-04-17_homepage.pdf [Broken], and http://motls.blogspot.com/2011/09/ten-new-things-that-science-has-learned.html .

Interatomic forces in large systems are complicated and hard to model so I'm probably oversimplifying it.

Last edited by a moderator:
vanhees71
Gold Member
2019 Award
The truth is that nobody understands the PEP. It is indeed the PEP that prevents Dsanz from falling through his chair.
The result of PEP is therefore the force preventing this to happen. In fact, without PEP the universe would be very different.
I find it hard to accept that 'geometry' would have such an important physical effect.
Everybody successfully taking quantum mechanics I understands the Pauli exclusion principle. It's valid for fermions, which by definition are particles, for which the ##N##-body Hilbert space is spanned by the totally antisymmetrized Kronecker products of single-particle Hilbert spaces. This is due to the indistinguishability of particles.

It is not a "force", which is an interaction term in the Lagrangian, but a constraint on the possible states of the many-body system and on the many-body operators which must be such that they are symmetric under exchange of indistinguishable particles.

This has, of course, also consequences for the dynamics, because it influences the Hamiltonian, leading to the socalled "exchange terms", which can play an important role. An example is Heisenberg's theory of ferromagnetism.

bhobba
bhobba
Mentor
That you don't understand it does not mean nobody understands it.
I was scratching my head about that one as well. We have the famous spin statistics theorem I gave a link to previously.

Thanks
Bill

bhobba
Mentor
I think the current view traces back to Dyson as referenced here.
I think Dales comment was because what Dyson showed was a little different to the incompressibility of liquids etc. He showed electron degeneracy pressure was responsible for the imperviousness of matter ie is the origin of solidity.

Thanks
Bill

Last edited by a moderator:
"From the point of view of logic, my report on « Exclusion principle and quantum mechanics » has no conclusion. "
Pauli Nobel lecture.