Second quantization describes identical-particle quantum states in Fock space using creation operators and annihilation operators. Relative to an orthonormal basis of one-particle states, a one-body operator becomes . The canonical commutation relation implements bosonic statistics, while the canonical anticommutation relations implement fermionic statistics. Expanding the quantized field in a complete basis gives a delta-function equal-time bracket; restricting the expansion to finitely many modes replaces that delta function by the corresponding projection operator kernel.
In a one-dimensional identical-particle model, a contact interaction has two-body potential , using the Dirac delta distribution. For bosons, its second quantization is the displayed normally ordered Hamiltonian operator. Projection onto orthonormal modes gives , where . The overlap integrals determine which mode-changing processes survive. A finite-mode model describes that projection; discarded modes can still matter to the full physical system.
For distinct box modes and , the displayed contact interaction overlaps produce self terms , a density term , and pair conversion . Pair conversion preserves total particle number but prevents diagonalization by individual occupation numbers. The additional overlaps are and , following from . Third-harmonic mode pairs therefore also permit single-particle transfer assisted by the other particles. For the first two box modes these additional terms vanish.
Two distinct normalized plane waves on a periodic one-dimensional ring of length give the displayed projected contact interaction. Momentum conservation excludes pair conversion between these two modes. Four mixed operator orderings give the coefficient of . At fixed particle number , the energy is . For repulsive coupling, its minima are the two endpoint Fock states with all particles in one mode; a one-particle energy difference selects the lower mode. This ground-state statement concerns the two-mode projection.
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Second quantization is a formalism used in quantum mechanics and quantum field theory to describe and manipulate systems with varying particle numbers. It is particularly useful for dealing with many-body systems, where traditional first quantization methods become cumbersome. In the first quantization approach, particles are described by wave functions, and the focus is on the states of individual particles. However, this approach struggles to accommodate phenomena like particle creation and annihilation, which are crucial in fields like quantum field theory.
Second quantization also appears to be useful not only for relativistic quantum mechanics, but also for condensed matter physics. The reason is that the basis idea is to use the number occupation basis. This basis is:
- convenient for quantum field theory because of particle creation and annihilation changes the number of particles all the time
- convenient for condensed matter physics because there you have a gazillion particles occupying entire energy bands
Bibliography:
- www.youtube.com/watch?v=MVqOfEYzwFY "How to Visualize Quantum Field Theory" by ZAP Physics (2020). Has 1D simulations on a circle. Starts towards the right direction, but is a bit lacking unfortunately, could go deeper.