| second quantization | |
|---|---|
| Name | Second quantization |
| Field | Quantum mechanics |
| Introduced | 1920s–1930s |
| By | Paul Dirac; development by Pascual Jordan and others |
| Related | Fock space, Quantum field theory, Many-body theory |
second quantization
Second quantization is a formalism in Quantum mechanics and Quantum field theory in which fields and many-particle states are quantized using operator-valued creation and annihilation operators, rather than quantizing single-particle wavefunctions. It provides a convenient and manifestly symmetric description of systems with variable particle number and identical particles, underpinning modern treatments of condensed matter physics, particle physics, and quantum optics.
Second quantization originated in the late 1920s and early 1930s as attempts to reconcile quantum mechanics with systems of many identical particles and to quantize the electromagnetic field. Early contributions came from Paul Dirac's quantization of the radiation field and from Pascual Jordan, Werner Heisenberg, and Ettore Majorana on many-particle operator methods. The formalism was later integrated into quantum field theory by figures such as Richard Feynman and Julian Schwinger and became central to many-body physics through the work of Ludwig Faddeev, Felix Bloch, and others. Second quantization replaced cumbersome symmetrization of wavefunctions with algebraic relations on operators, enabling systematic use of perturbation theory, diagrammatic methods (e.g., Feynman diagrams), and non-perturbative techniques like renormalization group analysis.
The core objects in second quantization are creation (a^†) and annihilation (a) operators that add or remove quanta in specified single-particle modes. For bosons these operators satisfy canonical commutation relations (CCR) [a_i, a_j^†] = δ_{ij}, while for fermions they satisfy canonical anticommutation relations (CAR) {c_i, c_j^†} = δ_{ij}, implementing the Pauli exclusion principle. The operators act on a Fock space and are used to express Hamiltonians compactly; for example, a two-body interaction can be written as sum_{ijkl} V_{ijkl} a_i^† a_j^† a_k a_l. The formalism naturally encodes particle indistinguishability and exchange symmetry, replacing the explicit symmetrization or antisymmetrization of wavefunctions found in first quantization. Creation and annihilation operators also provide the algebraic foundation for constructing conserved quantities such as total particle number operator N = sum_i a_i^† a_i and for deriving Green's functions used in many-body perturbation theory.
Fock space is the Hilbert space direct sum of n-particle subspaces, permitting a variable number of particles. It is constructed from a single-particle Hilbert space H as F(H) = ⊕_{n=0}^∞ S_n H^{⊗ n} with symmetrization S_n for bosons or antisymmetrization for fermions. Basis states in Fock space are often given by occupation-number representations |n_1, n_2, ...⟩ labeled by eigenvalues of number operators. This representation facilitates second-quantized descriptions of many-electron systems in atoms and molecules (as in quantum chemistry), electronic models such as the Hubbard model, and phonons in crystal lattices. Fock-space techniques also underlie computational methods including Configuration interaction, Coupled cluster theory, and Density functional theory extensions to variable particle number.
In the continuum limit, mode-labeled creation and annihilation operators are combined into field operators ψ(x) and ψ^†(x), which are operator-valued distributions on spacetime and obey CCR or CAR at equal times. Field operators provide a local description of quantum excitations and are the basic variables in quantum field theory and in quantum descriptions of the electromagnetic field (quantum electrodynamics). The quantized fields serve as the starting point for deriving equations of motion (via the Heisenberg picture), functional integrals, and operator-product expansions. They link to classical field theory through procedures like canonical quantization and path integral quantization, and are essential for describing processes such as particle creation and annihilation in scattering theory.
Second quantization is ubiquitous in describing interacting many-body systems: electrons in solids (leading to theories of superconductivity, Fermi liquid theory, and collective excitations), ultracold atomic gases in Bose–Einstein condensates and Fermi gas experiments, and lattice models studied in condensed matter physics. In quantum optics, second quantization provides the framework for photons as bosonic quanta, enabling analysis of coherent states, squeezed states, and phenomena in cavity quantum electrodynamics and quantum information experiments. Practical computational frameworks that rely on second quantization include Green's function methods, Diagrammatic Monte Carlo, and Dynamical mean field theory.
First quantization treats classical particle coordinates promoted to operators on a fixed-particle-number Hilbert space, while second quantization elevates fields or occupation numbers to operators on Fock space, naturally handling variable particle number and indistinguishability. Canonical quantization procedures relate classical canonical variables to operator algebras; for fields this yields second-quantized operator relations. Alternative approaches such as the path integral formulation (Feynman path integrals) offer equivalent descriptions and facilitate nonperturbative and semiclassical analyses. Connections between first and second quantization are explicit in methods like the Slater determinant construction for fermions, which can be represented compactly in occupation-number notation.
The rigorous underpinning of second quantization involves functional analysis, operator algebras, and representation theory. The CCR and CAR algebras are C*-algebras with distinct representation theories; examples include the Weyl algebra for bosons and the CAR algebra for fermions. Fock representations and quasi-free states are central in constructive approaches to quantum statistical mechanics and in studying thermodynamic limits. Techniques from spectral theory, the theory of unbounded operators, and renormalization are applied to Hamiltonians expressed in second-quantized form. Mathematical physics research connects these structures to results such as the Haag–Kastler axioms, Osterwalder–Schrader reconstruction theorem, and rigorous treatments of models like the Pauli–Fierz Hamiltonian and Hubbard model.
Category:Quantum mechanics Category:Quantum field theory Category:Many-body physics