Nor can we forget what we owe to the physicists who introduced us to research, Alfred Kastler and Jean Brossel for two of us and Maurice Levy for the third. It was in the context of their laboratories that we discovered the beauty and power of quantum mechanics. Neither have we forgotten the importance to us of the modern physics taught at the C.E.A. by Albert Messiah, Claude Bloch and Anatole Abragam, at a time when graduate studies were not yet incorporated into French university programs.
We wish to express our gratitude to Ms. Aucher, Baudrit, Boy, Brodschi, Emo, Heywaerts, Lemirre, Touzeau for preparation of the mansucript.
Volume III:
We are very grateful to Nicole and Daniel Ostrowsky, who, as they translated this Volume from French into English, proposed numerous improvements and clarifications. More recently, Carsten Henkel also made many useful suggestions during his translation of the text into German; we are very grateful for the improvements of the text that resulted from this exchange. There are actually many colleagues and friends who greatly contributed, each in his own way, to finalizing this book. All their complementary remarks and suggestions have been very helpful and we are in particular thankful to:
Pierre-François Cohadon
Jean Dalibard
Sébastien Gleyzes
Markus Holzmann
Thibaut Jacqmin
Philippe Jacquier
Amaury Mouchet
Jean-Michel Raimond
Félix Werner
Some delicate aspects of Latex typography have been resolved thanks to Marco Picco, Pierre Cladé and Jean Hare. Roger Balian, Edouard Brézin and William Mullin have offered useful advice and suggestions. Finally, our sincere thanks go to Geneviève Tastevin, Pierre-François Cohadon and Samuel Deléglise for their help with a number of figures.
Chapter XV
Creation and annihilation operators for identical particles
1 A General formalism A-1 Fock states and Fock space A-2 Creation operators a† A-3 Annihilation operators a A-4 Occupation number operators (bosons and fermions) A-5 Commutation and anticommutation relations A-6 Change of basis
2 B One-particle symmetric operators B-1 Definition B-2 Expression in terms of the operators a and a† B-3 Examples B-4 Single particle density operator
3 C Two-particle operators C-1 Definition C-2 A simple case: factorization C-3 General case C-4 Two-particle reduced density operator C-5 Physical discussion; consequences of the exchange
Introduction
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.
A. General formalism
We denote N the state space of a system of N distinguishable particles, which is the tensor product of N individual state spaces
(A-1)
Two sub-spaces of N are particularly important for identical particles, as they contain all their accessible physical states: the space S(N) of the completely symmetric states for bosons, and the space A(N) of the completely antisymmetric states for fermions. The projectors onto these two sub-spaces are given by relations (B-49) and (B-50) of Chapter XIV:
(A-2)
