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Suppose we want to construct a wave function for 2 free (relativistic) fermions. As we are dealing with fermions the total wave function has to be antisymmetric under interchange of the coordinates,Suppose we want to construct a wave function for 2 free (relativistic) fermions. As we are dealing with fermions the total wave function has to be antisymmetric under interchange of the coordinates,
Ψ ( x 1 , x 2 ) = Ψ ( x 2 , x 1 )
If we assume that we can factorize the wave function in terms of single particle wave functions we can write
Ψ ( x 1 , x 2 ) = ψ 1 ( x 1 ) ψ 2 ( x 2 ) ψ 1 ( x 1 ) ψ 2 ( x 2 )
which fulfills the anti-symmetry requirement. The plane wave single particle states are given by,
ψ k , m s ( x ) = u k , m s ( s ) ϕ ( k r )
So expect the total wavefunction to be
Ψ ( x 1 , x 2 ) = u k 1 , m s 1 ( s 1 ) ϕ ( k 1 r 1 ) u k 2 , m s 2 ( s 2 ) ϕ ( k 2 r 2 ) u k 1 , m s 1 ( s 2 ) ϕ ( k 1 r 2 ) u k 2 , m s 2 ( s 1 ) ϕ ( k 2 r 1 ) = u k 1 , m s 1 ( s 1 ) u k 2 , m s 2 ( s 2 ) ϕ ( k 1 r 1 ) ϕ ( k 2 r 2 ) u k 1 , m s 1 ( s 2 ) u k 2 , m s 2 ( s 1 ) ϕ ( k 1 r 2 ) ϕ ( k 2 r 1 )
However
u ( k 1 , m s 1 ) u ( k 2 , m s 2 ) ϕ ( k 1 r 1 ) ϕ ( k 2 r 2 ) u ( k 2 , m s 2 ) u ( k 1 , m s 1 ) ϕ ( k 1 r 2 ) ϕ ( k 2 r 1 )
If I'm not mistaking one cannot freely change the order of the Dirac spinors u ( k 1 , m s 1 ) u ( k 2 , m s 2 ) u ( k 2 , m s 2 ) u ( k 1 , m s 1 ) so these expressions seem to be uncompatible. What would the correct expression look like?

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