Let E=E(1)ÄE(2) be the state space of two non-identical particles with the same internal degrees of freedom (i.e. the same spin).  Each space E(1) and E(2) is spanned by the same orthonormal basis {|u_j>}.  E is spanned by the basis vectors {|1:u_i>Ä|2:u_j>=|1:u_i;2:u_j>=|u_i,u_j>}.  Note the different notations.

|1:u_i;2:u_j>¹|1:u_j;2:u_i> unless u_i=u_j.

The permutation operator P₂₁ is defined through its action on the basis vectors.

P₂₁|u_i,u_j>=|u_j,u_i>.

P₂₁²=1, P₂₁^-1=P₂₁.

The matrix elements of P₂₁^T and P₂₁ between different basis vectors are identical.

Therefore P₂₁= P₂₁^T=P₂₁^-1, P₂₁P₂₁^T=P₂₁^TP₂₁=I.

P₂₁ is Hermitian and unitary.  Its eigenvalues are real.  Since P₂₁²=1, its eigenvalues are ±1.

P₂₁|y_S>=|y_S>: symmetric eigenvector

P₂₁|y_A>=-|y_A>: anti-symmetric eigenvector

The operators S=(1/2)(1+P₂₁) and A=(1/2)(1-P₂₁) are projectors onto orthogonal subspaces, the spaces of symmetric and anti-symmetric kets.

S²=S, S^T=S, P₂₁S|y>=S|y>. S|y> is symmetric.

A²=A, A^T=A, P₂₁A|y>=-A|y>. A|y> is anti-symmetric.

SA=AS=0, S+A=I.

S is called the symmetrizer and A is called the anti-symmetrizer.  Operators that commute with P₂₁ are called symmetric observables.

For a system of three particles {|1:u_i>Ä|2:u_j>Ä|3:u_k>=|u_i;u_j;u_k>} is a basis for the state space E.  Six permutation operators exist, P₁₂₃, P₃₂₁, P₂₃₁, P₁₃₂, P₃₂₁, and P₂₁₃.

P_npq(|1:u_i>Ä|2:u_j>Ä|3:u_k>)=(|n:u_i>Ä|p:u_j>Ä|q:u_k>).

P_npq assigns the quantum numbers of particle 1 to particle n, of particle 2 to particle p, and of particle 3 to particle q.

Example:

• P₃₁₂|u_i;u_j;u_k>= |u_j;u_k;u_i>. (Note: P₁₂₃=I.)

N! permutation operators are associated with a system of N particles.  One of them is the identity operator.  The set of permutation operators forms a group (identity, product, inverse).  The elements of this group, in general, do not commute.

A transposition is a permutation, which exchanges two particles.  Any permutation can be written as a product of transpositions.  This decomposition is not unique.  However, it always takes an even or it always takes an odd number of transpositions to write a particular permutation.  If it takes an even number of transpositions, the permutation is said to have even parity, and if it takes an odd number of transpositions, the permutation is said to have odd parity.  This defines the parity of the permutation.

Example:

• P₁₂₃, P₂₃₁, P₃₁₂ are even permutations.
• P₁₃₂, P₃₂₁, P₂₁₃ are odd permutations.

Permutation operators are products of unitary operators and are therefore unitary.  Let P_a denote an arbitrary permutation.  A completely symmetric ket satisfies

And a completely anti-symmetric ket satisfies

P_a|y_A>=e_a|y_A>,
where e_a=1 if P_a=even and e_a=-1 if P_a=odd.

In the space of N particles the symmetrizer and anti-symmetrizer are defined by

.

A²=A, A^T=A, P_aA=AP_a=e_aA..

SA=AS=0, but S+A¹I if N>2.

S and A are projectors onto orthogonal subspaces, but E¹E_AÅE_S if N>2.