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Relation (C-16) implies that the average value of any two-particle operator may be written as:

(C-17)

Figure 1: Physical interaction between two identical particles: initially in the states |ukα〉 and |ukβ〉 (schematized by the letters α and β), the particles are transferred to the states |ukγ〉 and |uks〉 (schematized by the letters γ and δ)

This expression is similar to the average value of an operator for a two-particle system having a density operator :

(C-18)

which leads us to define a two-particle reduced density operator :

(C-19)

In this definition we have left out the factor 1/2 of (C-17) since this will lead to a normalization of often more handy: the matrix element of in the position representation yields directly the double density (as well as the field correlation function that we shall study in § B-3-b of Chapter XVI). The trace of is then written:

(C-20)

It is obviously possible to divide the right-hand side of the definition of either by the factor 2, or else by the factor if we wish its trace to be equal to 1.