Quantization is the procedure of going from a classical theory to a quantum theory. An important example is the canonical quantization procedure for going from classical mechanics to quantum mechanics. It amounts to replacing the classical dynamical variables and their Poisson brackets by quantum mechanical operators and their commutators. Considering a system of particles in an external electromagnetic field, the above procedure leads to the original formulation of quantum mechanics where the motion of the particles is quantized, while the applied fields are still treated classically. However, it turns out that the canonical quantization procedure can also be extended to field theory, such that the classical fields are replaced by quantum-mechanical creation and annihilation operators. Since the resulting formulation allows for the quantization of the fields that were in the original quantum theory still treated classically, it is commonly referred to as second quantization. Another way to understand this name is by introducing a Lagrangian density that generates the Schrödinger equation upon applying the Euler-Lagrange equations. This means that the fields of the Lagrangian density become the wavefunctions in the Schrödinger equation. Upon applying the canonical quantization procedure to these fields, the wavefunctions actually become quantized themselves, and the corresponding operators are creation and annihilation operators of particles. It is this last way of applying second quantization that corresponds to the formalism developed in this chapter.
The reason for introducing the language of second quantization is that it turns out to be extremely convenient in the formulation of a quantum theory for many interacting particles. The starting point of this chapter is the more familiar first-quantized N-body Schrödinger equation in the place representation, where the Hamiltonian of interest is motivated from the study of ultracold atomic quantum gases. However, the resulting Hamiltonian is actually seen to be much more general, such that it also applies to a large class of condensed-matter problems. For identical particles, the resulting many-body wavefunction needs to be fully symmetric for bosons, whereas it needs to be fully antisymmetric for fermions. Since a fully (anti)symmetrized wavefunction consists of about N! terms, we need to introduce a shorthand notation, because N is for our purposes typically a million or more. A convenient way to fully specify an (anti)symmetric wavefunction is in terms of the occupation numbers of the single-particle eigenstates from which the many-body state is constructed. This notation is appropriately called the occupation-number representation. Then, by relaxing the constraint of a fixed number of N particles, we introduce the Hilbert space of all (anti)symmetric many-body states, which is also known as Fock space. Since the number of particles in Fock space is not fixed, it is natural to define an annihilation operator, that can destroy a particle in a certain quantum state. From this definition, we immediately also obtain the creation operator, which is consequently used to construct all possible many-body states in Fock space. This procedure is then seen to incorporate automatically the statistics of the corresponding identical particles. By expressing also the many-body Hamiltonian in terms of the creation and annihilation operators, we arrive at our fully second-quantized many-body theory. Finally, we prove the complete equivalence between the old, more familiar formulation of the N-body Schr¨odinger equation in the place representation, also referred to as the first-quantized formalism, and the new formulation in terms of creation and annihilation operators, also referred to as the second-quantized formalism. This equivalence then ultimately validates all newly introduced definitions and expressions.
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