secondquantization

Second quantization is a formalism used in quantum mechanics and quantum field theory in which the fundamental dynamical variables are field operators that create or annihilate particles in specific quantum states. Unlike first quantization, where particle coordinates are quantized with a fixed particle number, second quantization allows particle number to change and treats excitations as quanta of underlying fields.

One introduces mode creation and annihilation operators. For bosons, these are a_k^\dagger and a_k, and for fermions, c_k^\dagger and c_k, acting on Fock space whose basis states are occupation-number states |n_1, n_2, ...>. The vacuum |0> is defined by a_k|0> = 0 (and similarly c_k|0> = 0). The operators satisfy the bosonic commutation relations [a_k, a_{k'}^\dagger] = δ_{k k'}, [a_k, a_{k'}] = 0, and the fermionic anticommutation relations {c_k, c_{k'}^\dagger} = δ_{k k'}, {c_k, c_{k'}} = 0.

Observables and dynamics are expressed in terms of field operators ψ(x) and ψ^\dagger(x), which create or destroy a particle at position x. A common Hamiltonian in second quantization is H = ∫ dx ψ^\dagger(x) [ - (ħ^2/2m) ∇^2 + V(x) ] ψ(x) + (1/2) ∫ dx dy ψ^\dagger(x) ψ^\dagger(y) V(x,y) ψ(y) ψ(x). In momentum space, H is written using a_k^\dagger a_k and interaction terms involving products of creation and annihilation operators.

Key features include the ability to handle variable particle number, the use of normal ordering to manage vacuum contributions, and transformations that mix operators (such as Bogoliubov transformations) to describe phenomena like superfluidity and superconductivity.

Second quantization is central to quantum many-body physics, quantum optics, and quantum field theory, providing a concise framework for describing Bose-Einstein condensates, electronic systems, lattice models, and a wide range of interacting quantum systems.