Fock Space

Fock space is the Hilbert space for variable particle number. In nonrelativistic many-body quantum mechanics it is a compact way to organize identical particles. In quantum field theory it becomes the natural language for particle excitations of fields, especially in free or asymptotic theories.

Quantum Mechanics Starting Point

Given a one-particle Hilbert space $\mathcal h$, bosonic and fermionic Fock spaces are

$$\mathcal F_B(\mathcal h) = \bigoplus_{N=0}^{\infty} \operatorname{Sym}^N\mathcal h, \qquad \mathcal F_F(\mathcal h) = \bigoplus_{N=0}^{\infty} \wedge^N\mathcal h.$$

The $N=0$ sector is the vacuum sector. A basis can be labeled by mode occupations:

$$\lvert n_1,n_2,\ldots\rangle.$$

For bosons $n_i=0,1,2,\ldots$. For fermions $n_i=0$ or $1$.

Field-Theory Continuation

In a free field theory, each momentum mode behaves like an oscillator. Acting with creation operators on the vacuum builds particle states:

$$\lvert \mathbf p_1,\ldots,\mathbf p_N\rangle \sim a^\dagger_{\mathbf p_1}\cdots a^\dagger_{\mathbf p_N} \lvert 0\rangle.$$

The symbol $\sim$ hides normalization choices. Relativistic continuum normalization, finite-box normalization, and nonrelativistic normalization use different factors.

What Carries Over

• Occupation-number labels.
• Vacuum as the zero-particle state for a chosen representation.
• Creation and annihilation operators.
• Bosonic commutators and fermionic anticommutators.
• Number operators.
• Mode expansions.

What Changes

Field theory changes the status of particle number. In interacting relativistic theories, particle number is usually not conserved. A Fock-space basis is most cleanly tied to free fields and asymptotic scattering states. The interacting vacuum is not merely the empty state of a naive free-particle basis.

Common Mistakes

• Treating Fock space as optional notation rather than the natural Hilbert-space structure for variable particle number.
• Forgetting that the vacuum depends on the Hamiltonian and representation.
• Assuming particle number is always conserved.
• Confusing nonrelativistic field operators with relativistic quantum fields.
• Moving between finite-volume sums and continuum integrals without changing normalization.

Canonical Links

• Bosonic Fock Space
• Fermionic Fock Space
• Occupation-Number Basis
• Occupation-Number Representation in Many-Body Models
• Vacuum State
• Mode Occupations
• Second Quantization
• Harmonic Oscillator to Fields

References

• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
• A. Altland and B. D. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Westview Press, 1995.
• S. Weinberg, The Quantum Theory of Fields, Volume I, Cambridge University Press, 1995.

Exercises

1. Why does a fermionic mode have only two occupation possibilities?

Solution

For a fermionic creation operator, $\{c_i^\dagger,c_i^\dagger\}=0$, so $(c_i^\dagger)^2=0$. Applying the same creation operator twice gives zero. A single mode can therefore be empty or occupied once, but not occupied twice.