# Pauli's exclusion principle

1. Oct 30, 2013

### Ravi Mohan

How do we prove Pauli's exclusion principle? My professor makes a Slater determinant and then merrily shows how it disappears when two columns or rows are same.
That is not Pauli's principle, is it? It is based on an assumption that certain particles are described by certain states.
So my question translates to why fermions (half integral spin particles) are described by antisymmetric states while bosons (integral spins) are described by symmetric states?

2. Oct 30, 2013

### vanhees71

This is a pretty deep question. The first step to understand the answer is the proof that in both relativistic an non-relativistic quantum theory for particles in spaces with dimension $d \geq 3$ identical particles must be described either as bosons or fermions, i.e., with states in the Fock space spanned by totally symmetrized (bosons) or antisymmetrized (fermions) product bases. This is shown in the paper

Laidlaw, M. G. G., DeWitt, Cécile Morette: Feynman Functional Integrals for Systems of Indistinguishable Particles, Phys. Rev. D 3, 1375 (1970)

The second step is to understand the connection between spin and statistics in relativistic quantum field theory. The spin-statistics theorem tells us that for any local Poincare symmetric qft with a stable ground state (spectrum of the Hamiltonian bounded from below) particles with integer spin are bosons and those with half-integer spin fermions. The best book about QFT is

Weinberg, Quantum Theory of Fields, Cambridge University Press

For the very clear and careful treatment of the representation theory of the Poincare group, see vol. 1.

3. Oct 30, 2013

### Ravi Mohan

Thank you Vanhees. I will read the literature you mentioned.