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For a system composed of identical particles, the particle numbering used in Chapter XIV, the last chapter of Volume II [2], does not really have much physical significance. Furthermore, when the particle number gets larger than a few units, applying the symmetrization postulate to numbered particles often leads to complex calculations. For example, computing the average value of a symmetric operator requires the symmetrization of the bra, the ket, and finally the operator, which introduces a large number of terms¹. They seem different, a priori, but at the end of the computation many are found to be equal, or sometimes cancel each other. Fortunately, these lengthy calculations may be avoided using an equivalent method based on creation and annihilation operators in a “Fock space”. The simple commutation (or anticommutation) rules satisfied by these operators are the expression of the symmetrization (or antisymmetrization) postulate. The non-physical particle numbering is replaced by assigning “occupation numbers” to individual states, which is more natural for treating identical particles.

The method described in this chapter and the following is sometimes called “second quantization”². It deals with operators that no longer conserve the particle number, hence acting in a state space larger than those we have previously considered; this new space is called the “Fock space” (§ A). These operators which change the particle number appear mainly in the course of calculations, and often regroup at the end, keeping the total particle number constant. Examples will be given (§ B) for one-particle symmetric operators, such as the total linear momentum or angular momentum of a system of identical particles. We shall then study two-particle symmetric operators (§ C), such as the energy of a system of interacting identical particles, their spatial correlation function, etc. In quantum statistical mechanics, the Fock space is well adapted to computations performed in the “grand canonical” ensemble, where the total number of particles may fluctuate since the system is in contact with an external reservoir. Furthermore, as we shall see in the following chapters, the Fock space is very useful for describing physical processes where the particle number changes, as in photon absorption or emission.