Fock space
Introduction
Fock space is a fundamental concept in quantum mechanics and quantum field theory, providing a framework for describing quantum states with a variable number of particles. Named after the Soviet physicist Vladimir Fock, it extends the notion of a Hilbert space to accommodate systems where the particle number is not fixed, such as in quantum fields. This mathematical structure is crucial for understanding phenomena like particle creation and annihilation, which are central to the Standard Model of particle physics.
Mathematical Structure
Fock space is constructed as a direct sum of tensor products of single-particle Hilbert spaces. For a given single-particle Hilbert space \( \mathcal{H} \), the Fock space \( \mathcal{F}(\mathcal{H}) \) is defined as:
\[ \mathcal{F}(\mathcal{H}) = \bigoplus_{n=0}^{\infty} \mathcal{H}^{\otimes n}, \]
where \( \mathcal{H}^{\otimes n} \) denotes the \( n \)-fold tensor product of \( \mathcal{H} \), and \( \mathcal{H}^{\otimes 0} \) is defined as the complex numbers \( \mathbb{C} \), representing the vacuum state.
Each component \( \mathcal{H}^{\otimes n} \) corresponds to the subspace of states with exactly \( n \) particles. The direct sum allows for the superposition of states with different particle numbers, making Fock space suitable for quantum field theories where particles can be created and destroyed.
Creation and Annihilation Operators
In Fock space, creation and annihilation operators are essential tools for manipulating quantum states. These operators, denoted by \( a^\dagger \) and \( a \), respectively, act on Fock space to add or remove particles. For a state \( |\psi\rangle \) in the \( n \)-particle subspace, the action of these operators is given by:
\[ a^\dagger |\psi\rangle = \sqrt{n+1} |\psi_{n+1}\rangle, \] \[ a |\psi\rangle = \sqrt{n} |\psi_{n-1}\rangle, \]
where \( |\psi_{n+1}\rangle \) and \( |\psi_{n-1}\rangle \) are states in the \( (n+1) \)- and \( (n-1) \)-particle subspaces, respectively. These operators satisfy the canonical commutation relations:
\[ [a, a^\dagger] = 1, \] \[ [a, a] = [a^\dagger, a^\dagger] = 0. \]
These relations are fundamental in the algebraic formulation of quantum mechanics and are used extensively in quantum field theory.
Bosonic and Fermionic Fock Spaces
Fock space can be specialized for systems of indistinguishable particles, leading to the concepts of bosonic and fermionic Fock spaces. Bosons, which obey Bose-Einstein statistics, are described by symmetric Fock spaces, where the states are invariant under the exchange of particles. The bosonic Fock space is constructed using symmetric tensor products:
\[ \mathcal{F}_B(\mathcal{H}) = \bigoplus_{n=0}^{\infty} \text{Sym}(\mathcal{H}^{\otimes n}). \]
Fermions, on the other hand, obey Fermi-Dirac statistics and are described by antisymmetric Fock spaces. The antisymmetry is a consequence of the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state. The fermionic Fock space is constructed using antisymmetric tensor products:
\[ \mathcal{F}_F(\mathcal{H}) = \bigoplus_{n=0}^{\infty} \text{Alt}(\mathcal{H}^{\otimes n}). \]
The creation and annihilation operators for fermions satisfy the anticommutation relations:
\[ \{a, a^\dagger\} = 1, \] \[ \{a, a\} = \{a^\dagger, a^\dagger\} = 0. \]
Applications in Quantum Field Theory
Fock space is indispensable in quantum field theory (QFT), where fields are quantized, and particles are treated as excitations of these fields. In QFT, the vacuum state of the Fock space represents the ground state of the field, and particle states are created by applying creation operators to the vacuum.
One of the key applications of Fock space in QFT is in the formulation of perturbation theory, where interactions between particles are treated as small perturbations to the free field theory. The Fock space formalism allows for the systematic calculation of scattering amplitudes and cross-sections, which are essential for making predictions in particle physics.
Fock space also plays a crucial role in the second quantization formalism, where fields are treated as operators acting on Fock space. This approach provides a natural framework for incorporating interactions and is used extensively in the study of quantum electrodynamics (QED) and quantum chromodynamics (QCD).
Fock Space in Many-Body Physics
Beyond quantum field theory, Fock space is also used in the study of many-body systems, such as in condensed matter physics. In these systems, the Fock space formalism provides a convenient way to describe collective excitations, such as quasiparticles and phonons, which arise from the interactions between particles.
In the context of superconductivity, the Fock space approach is used to describe the pairing of electrons into Cooper pairs, leading to the formation of a superconducting state. The BCS theory of superconductivity, named after Bardeen, Cooper, and Schrieffer, employs Fock space to account for the macroscopic quantum coherence observed in superconductors.
Mathematical Properties and Extensions
Fock space possesses several important mathematical properties that make it a versatile tool in quantum mechanics. It is a Hilbert space, ensuring that it has a well-defined inner product and is complete with respect to this inner product. This allows for the rigorous treatment of quantum states and operators within the framework of functional analysis.
Extensions of the Fock space concept include the Segal-Bargmann space, which provides a holomorphic representation of Fock space, and the Fock representation in C*-algebra theory, which generalizes Fock space to infinite-dimensional systems.