SYMMETRIES in PHYSICS

Symmetry and Spontaneous Symmetry Breaking
Selected Groups and Symmetries
Spontaneous Symmetry Breaking
Identical Particles Must Be either Fermions or Bosons!
From definition of the permutation operator it follows that P 2
ij
= 1
while P ij Ψ = p ij Ψ → P 2
ij Ψ = p 2
ij Ψ ↔ p 2
ij = 1 → p ij = ±1
In the particle-number representation we may write for short
df .
Ψ 1,2, ... n = Ψ(x 1 , x 2 , . . . x n )
Conclusion: We thus Discover the Pauli Principle !
1 o P ij Φ n1 , ... n i , ...n j , ... n n
= +Φ n1 , ... n j , ...n i , ... n n
→ Bosons
2 o P ij Ψ n1 , ... n i , ...n j , ... n n
= −Ψ n1 , ... n j , ...n i , ... n n
→ Fermions
Jerzy DUDEK
SYMMETRIES in PHYSICS