where the position vector r i of particle i is a vector in R 3 and σ i takes on 2 s +1 values, where s is the half-integral intrinsic spin of the fermion. For electrons s = 1/2 and σ can have two values ("spin-up": 1/2 and "spin-down": −1/2). It is assumed that the positions of the coordinates in the notation for Ψ have a well-defined meaning. For instance, the 2-fermion function Ψ(1,2) will in general be not the same as Ψ(2,1). This implies that in general Ψ Ψ ( 1 , 2 ) − − Ψ Ψ ( 2 , 1 ) ≠ ≠ 0 and therefore we can define meaningfully a transposition operator P ^ ^ i j that interchanges the coordinates of particle i and j. In general this operator will not be equal to the identity operator (although in special cases it may be).