# Why isn't Pauli Exclusion Principle a force?

1. Apr 11, 2012

### azabak

In the same way we could create "principles" for the other forces which would not make them not forces. Is it a misunderstanding of the meaning of a force or principle? Could someone clarify this for me.

2. Apr 11, 2012

### Khashishi

The Pauli exclusion principle leads to degeneracy pressure, which acts sort of like a force. That's related to the Casimir effect, which also acts like a force. But these are both derived from the quantization of available states of the system, and not due to some field acting on the particles. I guess you are suggesting that we could "derive" other forces in a similar manner? Eric Verlinde published an article suggesting that gravity was due to entropy, and therefore also some result of quantization of available states. I don't know of any treatment of electroweak/strong in any such manner. But maybe you could try inventing it.

3. Apr 12, 2012

### tom.stoer

The Pauli principle is not a force b/c it is already present in the free theory w/o any specific interaction; it holds purely algebraically w/o specifying a dynamics i.e. w/o specifying a Hamiltonian.

Suppose you have fermionic creation and annihilation operators $b_i^\dagger$ and $b_i$ with the usual anti-commutators. Here 'i' is a general index containing all relevant numbers specifying a state like momentum, spin, isospin etc.

The relevant identity which follows from the anti-commutators is

$$\left(b_i^\dagger\right)^2 = 0$$

It says that you cannot create a two-particle state with two fermions having both the same state 'i'.

Of course you can construct arbitrary complex interaction terms

$$\sum_{ijk, \ldots pqr \ldots}h_{ijk, \ldots pqr \ldots} b_i^\dagger b_j^\dagger b_k^\dagger \ldots b_p b_q b_r \ldots$$

but in all those interactions every diagonal term with (e.g.) i=j vanishes

$$h_{iik, \ldots pqr \ldots} b_i^\dagger b_i^\dagger b_k^\dagger \ldots b_p b_q b_r \ldots = 0$$

So the Pauli principle eliminates all these terms from the theory w/o requiring a specific interaction; the number hiik...pqr... need not be zero; it's the anti-commutator itself that makes this term vanish.

4. Apr 12, 2012

### Zarqon

But saying that a physical phenomena is caused by humanly invented algebra is to me not very satisfying explanation. Shouldn't it be more accurate to say that the algebra describes the phenomena, but that the cause is a (a combination of) certain physical axiom(s)?

Could you say anything about which those more fundamental physical axioms would be? Or alternatively, could it be that the Pauli exclusion principle is a fundamental physical axiom in itself? (meaning that the creation and annihilation operators and their commutators was designed to with the need to fulfill the PEP in mind.)

5. Apr 12, 2012

### tom.stoer

The Pauli exclusion principle is due to the anti-commutators or Grassmann fields to be used for quantizing fermions; these anti-commutators or Grassmann fields are due to the spin-statistics theorem

Have a look at Wikipedia and references therein: http://en.wikipedia.org/wiki/Spin-statistics_theorem

I have to admit that I am not an expert in axiomatic quantum field theory and that I am not able to comment on formal aspects of these proofs. But up to my knowledge the spin-statistics-theorem is the most fundamental starting point

Last edited: Apr 12, 2012