Fock space

Fock space is a fundamental construction in mathematical physics and quantum field theory that provides the appropriate Hilbert space for describing systems with a variable number of identical particles. It was introduced by the Soviet physicist Vladimir Fock in the context of quantum electrodynamics and the second quantization formalism. The space is built as the direct sum of the tensor products of single-particle Hilbert spaces, systematically accounting for the bosonic or fermionic nature of the particles through symmetric or antisymmetric subspaces. This structure is essential for the formulation of many-body theory, quantum statistical mechanics, and the operator algebra of creation and annihilation operators.

Definition and construction

The construction begins with a single-particle Hilbert space \(\mathcal{H}\), often corresponding to the state space of one particle, such as \(L^2(\mathbb{R}^3)\) for spatial wavefunctions. The Fock space \(\mathcal{F}(\mathcal{H})\) is defined as the completed direct sum of the particle number sectors: \(\mathcal{F}(\mathcal{H}) = \bigoplus_{n=0}^{\infty} \mathcal{H}^{\otimes n}\), where \(\mathcal{H}^{\otimes 0}\) is a one-dimensional space representing the vacuum state with zero particles. For bosonic systems, one uses the symmetric tensor product, yielding the bosonic Fock space \(\mathcal{F}_\text{sym}(\mathcal{H})\), which is crucial in describing photons and phonons. For fermionic systems, such as electrons, one uses the antisymmetric tensor product, yielding the fermionic Fock space \(\mathcal{F}_\text{asym}(\mathcal{H})\), which enforces the Pauli exclusion principle. This construction naturally accommodates superpositions of states with different particle numbers, a key feature of relativistic quantum field theory.

Mathematical properties

Fock spaces are separable Hilbert spaces when the single-particle space is separable, and they carry a natural inner product induced from \(\mathcal{H}\). They possess a graded algebra structure corresponding to the particle number, with the vacuum expectation value serving as a cyclic vector. The action of creation and annihilation operators on these spaces satisfies canonical commutation relations for bosons or canonical anticommutation relations for fermions, forming representations of the CCR algebra and CAR algebra, respectively. Important operators defined on Fock space include the number operator, which counts particles, and Hamiltonian operators expressed in second-quantized form. The Stone–von Neumann theorem guarantees the essential uniqueness of these representations for systems with finitely many degrees of freedom, though subtleties arise in quantum field theory.

Physical applications

Fock space is the central mathematical framework for second quantization, where fields are treated as operators acting on states of indefinite particle number. In quantum electrodynamics, the photon field is expanded in terms of creation and annihilation operators on a bosonic Fock space, facilitating calculations of scattering amplitudes and Feynman diagrams. The description of condensed matter physics phenomena, such as superconductivity via the BCS theory or the fractional quantum Hall effect, relies heavily on fermionic Fock spaces. In quantum optics, the bosonic Fock space models states of the electromagnetic field, including coherent states and squeezed states. Furthermore, algebraic quantum field theory uses Fock space constructions to study the Wightman axioms and the Haag's theorem.

Relation to other spaces

Fock space is closely related to several other important mathematical structures. It can be viewed as the Hilbert space completion of the tensor algebra or the exterior algebra over the single-particle space. The bosonic Fock space is isomorphic to a space of holomorphic functions, leading to its identification with the Segal–Bargmann space or the reproducing kernel Hilbert space associated with the Gaussian measure. In probability theory, the bosonic Fock space has a deep connection with the Wiener–Itô chaos expansion and the Malliavin calculus. The fermionic Fock space is fundamentally linked to the Clifford algebra and the theory of spinors. In constructive quantum field theory, interactions are often defined by perturbing the free Fock space Hamiltonian, as seen in the Yukawa interaction or the phi-four theory.

Special cases and examples

A fundamental example is the Fock space built from \(\mathcal{H} = \mathbb{C}\), which is essentially the Hilbert space of the quantum harmonic oscillator, with its basis states \(|n\rangle\) labeled by occupation number. The bosonic Fock space for a single mode is used to describe a quantum harmonic oscillator or a single mode of the radiation field. For a system of discrete modes, such as in a lattice model, the Fock space is a tensor product of single-mode spaces. The antisymmetric Fock space for a finite-dimensional single-particle space is simply the full exterior algebra, which is finite-dimensional. In quantum chemistry, the fermionic Fock space with a basis of molecular orbitals is the setting for Hartree–Fock and post-Hartree–Fock methods. The concept also extends to supersymmetry, where one considers a Z2-graded Fock space combining both bosonic and fermionic sectors.

Category:Functional analysis Category:Quantum mechanics Category:Mathematical physics