Fock Space and Occupation Number

Fock space is the Hilbert space for systems whose description includes different particle-number sectors. It packages the vacuum, one-particle states, two-particle states, and all higher sectors into one direct sum while building bosonic or fermionic exchange symmetry into every fixed-number sector.

For a one-particle Hilbert space $\mathcal h$,

$$\begin{aligned} \mathcal F_+(\mathcal h) &= \bigoplus_{N=0}^{\infty} \operatorname{Sym}^N\mathcal h, \\ \mathcal F_-(\mathcal h) &= \bigoplus_{N=0}^{\infty} \bigwedge^N\mathcal h. \end{aligned}$$

The plus sign denotes bosons and the minus sign denotes fermions. Occupation-number notation then replaces explicit sums over particle-label permutations by a list of how many particles occupy each mode.

Chapter Map

Task	Canonical page	Main output
translate modes into occupation tuples	Occupation-Number Basis	bosonic and fermionic basis labels and first-quantized mappings
construct symmetric number sectors	Bosonic Fock Space	direct sum of symmetric tensor powers
construct antisymmetric number sectors	Fermionic Fock Space	direct sum of exterior powers and finite-dimensional termination
understand the no-particle vector	Vacuum State	vacuum sector, annihilation property, and ground-state distinction
work with definite occupations	Number States	normalized one-mode and many-mode eigenstates
interpret and change mode bases	Mode Occupations	position, momentum, energy, spin-orbital, and local modes
separate conservation from superselection	Particle-Number Superselection Preview	sector coherence, reference frames, and cautious operational claims
practice with representative systems	Fock Space Examples	bosons, fermions, photons, oscillator modes, and Hubbard sites

Direct Sum, Not One Giant Tensor Product

Fixed-$N$ particle states belong to an $N$-particle sector $\mathcal H_N^\pm$. Different values of $N$ are alternatives, so Fock space uses a direct sum:

$$\mathcal F_\pm(\mathcal h) = \mathcal H_0 \oplus \mathcal H_1 \oplus \mathcal H_2^\pm \oplus\cdots.$$

A Fock-space vector is a sector sequence

$$\lvert\Psi\rangle = \bigoplus_{N=0}^{\infty} \lvert\psi_N\rangle, \qquad \sum_{N=0}^{\infty} \lVert\psi_N\rVert^2<\infty.$$

The probability of obtaining particle number $N$ is $\lVert\psi_N\rVert^2$ when the full vector is normalized.

By contrast, tensor products combine degrees of freedom or subsystems that coexist. Within each fixed-$N$ sector, tensor products and symmetry projections construct the many-particle states. Across different values of $N$, the direct sum organizes mutually orthogonal sectors.

Question	Construction
spin and position of one particle coexist	tensor product
two distinguishable subsystems coexist	tensor product
zero-, one-, and two-particle possibilities	direct sum
all bosonic fixed-number sectors together	bosonic Fock-space direct sum

The Vacuum Sector

The zero-particle sector is one dimensional:

$$\mathcal H_0 \cong \mathbb C.$$

Its normalized basis vector is the vacuum $\lvert0\rangle$. It is not the zero vector:

$$\langle0|0\rangle=1, \qquad \lVert\mathbf0\rVert=0.$$

The vacuum means no particles relative to the chosen particle and mode description. It is not automatically a zero-energy state. A Hamiltonian may assign vacuum energy, include chemical-potential conventions, or use a reference state whose excitations define a different effective vacuum.

In nonrelativistic many-body quantum mechanics the empty Fock vacuum is often straightforward. Relativistic QFT vacua can carry richer structure, and that preview should not be imported uncritically into every finite-mode problem.

Choose a One-Particle Mode Basis

Let $\{\lvert r\rangle\}$ be an orthonormal basis of $\mathcal h$. A mode may represent a spatial orbital, momentum, energy eigenstate, spin-orbital, lattice site with internal state, polarization, or another complete one-particle basis element.

An occupation-number basis vector is written

$$\lvert n_1,n_2,n_3,\ldots\rangle,$$

where

$$N = \sum_r n_r.$$

For ordinary Fock space over a countable basis, a basis vector has finite total particle number and therefore only finitely many nonzero occupations.

The mode labels are basis dependent. A change of one-particle basis changes the occupation amplitudes, even though the abstract Fock-space state does not change.

Bosonic Occupations

For bosons,

$$n_r\in\{0,1,2,\ldots\}.$$

Any number of bosons may occupy the same mode. A normalized bosonic number state may be generated from the vacuum by

$$\lvert n_1,n_2,\ldots\rangle = \prod_r \frac{(a_r^\dagger)^{n_r}} {\sqrt{n_r!}} \lvert0\rangle,$$

where only finitely many factors differ from the identity.

For $M$ modes and fixed total number $N$, the dimension is the stars-and-bars count

$$\dim\operatorname{Sym}^N(\mathbb C^M) = \binom{M+N-1}{N}.$$

This growth is much smaller than listing all $M^N$ slot assignments and then removing permutation redundancy.

Fermionic Occupations

For fermions,

$$n_r\in\{0,1\}.$$

An ordering convention for modes must be fixed. A fermionic number state is then

$$\lvert n_1,n_2,\ldots\rangle = (c_1^\dagger)^{n_1} (c_2^\dagger)^{n_2} \cdots \lvert0\rangle.$$

Changing the creation-operator order can change the sign. The occupation pattern identifies the ray only together with a consistent ordering convention.

For $M$ one-particle modes,

$$\dim\bigwedge^N\mathbb C^M = \binom{M}{N}.$$

The sector vanishes for $N>M$, and the full fermionic Fock space has dimension

$$\dim\mathcal F_-(\mathbb C^M) = \sum_{N=0}^{M}\binom{M}{N} = 2^M.$$

This is Pauli exclusion encoded in the basis itself.

Mapping First-Quantized States to Occupations

Suppose $\lvert u\rangle$ and $\lvert v\rangle$ are orthonormal modes.

Two bosons in $u$ correspond to

$$\lvert2_u,0_v\rangle \longleftrightarrow \lvert u\rangle\otimes\lvert u\rangle.$$

One boson in each mode corresponds to the symmetric state

$$\begin{aligned} \lvert1_u,1_v\rangle \longleftrightarrow \frac{1}{\sqrt2} \bigl(& \lvert u\rangle_1\lvert v\rangle_2 \\ &+ \lvert v\rangle_1\lvert u\rangle_2 \bigr). \end{aligned}$$

For fermions, one particle in each mode corresponds to

$$\begin{aligned} \lvert1_u,1_v\rangle \longleftrightarrow \frac{1}{\sqrt2} \bigl(& \lvert u\rangle_1\lvert v\rangle_2 \\ &- \lvert v\rangle_1\lvert u\rangle_2 \bigr). \end{aligned}$$

There is no fermionic state $\lvert2_u\rangle$ for an ordinary single-particle mode $u$.

Number Operators

The occupation operator for mode $r$ is

$$\hat n_r = a_r^\dagger a_r$$

for bosons, or $c_r^\dagger c_r$ for fermions. The total number operator is

$$\hat N = \sum_r \hat n_r.$$

Number states satisfy

$$\begin{aligned} \hat n_r \lvert n_1,n_2,\ldots\rangle &= n_r\lvert n_1,n_2,\ldots\rangle, \\ \hat N \lvert n_1,n_2,\ldots\rangle &= N\lvert n_1,n_2,\ldots\rangle. \end{aligned}$$

The operator algebra and domain questions belong to Number Operators. Here the eigenvalues provide the occupation labels.

Mode Basis Changes

Let two orthonormal one-particle bases be related by

$$\lvert\alpha\rangle = \sum_r U_{\alpha r}\lvert r\rangle, \qquad U^\dagger U=I.$$

The corresponding creation operators transform linearly:

$$b_\alpha^\dagger = \sum_r U_{\alpha r}a_r^\dagger.$$

A state with definite occupations in the $r$ basis may be a superposition of occupation patterns in the $\alpha$ basis. Total particle number remains fixed under a number-preserving basis change.

Mode dependence is physically useful rather than pathological. Position-localized modes answer different questions from momentum modes or energy eigenmodes. The basis must match the preparation, Hamiltonian, detectors, or subsystem access relevant to the problem.

Number Conservation and Superselection

If

$$[H,\hat N]=0,$$

then time evolution preserves each number sector. This is particle-number conservation.

A general Fock vector may still be written mathematically as

$$\lvert\Psi\rangle = \sum_N c_N\lvert\psi_N\rangle, \qquad \sum_N|c_N|^2=1.$$

Whether relative phases between sectors are observable depends on the theory, allowed observables, conserved charges, and available reference systems. Calling every number-conserving model subject to a fundamental particle-number superselection rule is too strong.

Coherent optical states, condensate phase descriptions, charge superselection, and local particle-number restrictions require different qualifications. Particle-Number Superselection Preview develops those distinctions cautiously.

Number States Are Not All Fock States

An occupation-number basis is complete, but a general state is a superposition or mixture of number states. Even within one fixed-$N$ sector,

$$\lvert\psi_N\rangle = \sum_{\{n_r\}:\,\sum_r n_r=N} C_{n_1n_2\cdots} \lvert n_1,n_2,\ldots\rangle.$$

Superpositions within a sector carry coherence among mode configurations. Mixtures carry statistical uncertainty. A list of mean occupations $\langle\hat n_r\rangle$ generally does not determine the state.

Mode Entanglement Caution

Once a mode decomposition is chosen, mode algebras can define operational subsystems. For two sets of independently controlled modes $L$ and $R$, occupation-number states and their superpositions may be analyzed across the $L|R$ split.

The result can depend on the mode basis and on local particle-number restrictions. A global mode transformation that mixes $L$ and $R$ changes the subsystem decomposition; it is not merely a local basis change within fixed laboratories.

This is the same partition discipline used for identical particles: identify accessible modes, regions, and operations before assigning an entanglement interpretation.

Representative Examples

System	Natural modes	Occupation constraint
photons in two paths	path and polarization modes	bosonic, number may vary
ultracold bosons in a lattice	site and band modes	bosonic, often fixed total number
electrons in orbitals	spin-orbitals	fermionic, zero or one per spin-orbital
Hubbard site	spin-up and spin-down local modes	each fermionic mode has zero or one occupation
oscillator normal modes	normal-mode excitations	bosonic ladder for each mode

The same notation can describe particles, quasiparticles, or excitations. The meaning of the vacuum and number operator must be stated for the effective model.

Bridge to Second Quantization

Fock space supplies the state space. Creation and annihilation operators supply the natural maps between sectors:

$$\begin{aligned} a_r^\dagger: \mathcal H_N^+ &\longrightarrow \mathcal H_{N+1}^+, \\ a_r: \mathcal H_N^+ &\longrightarrow \mathcal H_{N-1}^+, \end{aligned}$$

with analogous fermionic maps. Their commutation or anticommutation relations encode statistics, while second-quantized one- and two-body operators act directly on occupation states.

The chapter guide Creation, Annihilation, and Second Quantization develops this operator construction from the statistics algebra through one- and two-body Hamiltonians. “Second quantization” is historical terminology: it is not a second application of quantization to an already quantized particle, but an operator language for many-body quantum mechanics and, later, quantum fields.

An Analysis Protocol

1. Define the one-particle modes

State the Hilbert space, basis, and physical meaning of each mode label.

2. Choose bosonic or fermionic Fock space

Use symmetric sectors for bosons and exterior sectors for fermions. Record any finite-mode cutoff.

3. Identify the number sector

Decide whether the state has fixed $N$, spans several sectors, or is a mixture across sectors.

4. Fix ordering conventions

For fermions, choose and preserve a mode ordering before translating occupation tuples to operator products or qubit encodings.

5. Check normalization and conserved quantities

Verify coefficient norms, occupation constraints, and whether the Hamiltonian commutes with total number or only with a weaker parity or charge.

6. Match modes to observables

Choose local, momentum, energy, or spin-orbital modes according to the experiment and calculation rather than treating one basis as intrinsically preferred.

Common Mistakes

• Using a tensor product across particle-number sectors. Alternative values of $N$ form a direct sum.
• Calling the vacuum the zero vector. The vacuum is normalized and spans the $N=0$ sector.
• Calling the vacuum automatically the ground state. Energy ordering depends on the Hamiltonian and conventions.
• Allowing fermionic occupation $n_r=2$. A complete fermionic mode has occupation zero or one.
• Ignoring fermionic mode order. Reordering creation operators can change signs.
• Treating occupation numbers as basis independent. Mode occupations change under one-particle basis transformations.
• Equating number conservation with fundamental superselection. Operational access to inter-sector phase requires a separate analysis.
• Assuming mean occupations specify the state. Correlations and coherences remain undetermined.
• Confusing oscillator quanta with material particles. State what the counted excitation represents.

Reading Paths

Core construction: Occupation-Number Basis → Bosonic Fock Space or Fermionic Fock Space → Vacuum State.

Working language: Number States → Mode Occupations → Fock Space Examples.

Interpretive boundary: Particle-Number Superselection Preview → Identical-Particle Entanglement Cautions → Entanglement Depends on a Decomposition.

Operator bridge: Creation, Annihilation, and Second Quantization → Creation and Annihilation Operators → Number Operators → Many-Particle Hamiltonians.

Continuous-variable bridge: Mode Occupations → Continuous Variables and Modes → Mode Decompositions.

References

• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Westview Press, 1998.
• A. Altland and B. Simons, Condensed Matter Field Theory, 3rd ed., Cambridge University Press, 2023.
• H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics, Oxford University Press, 2004.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.
• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer, 2008.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

Exercise 1: Count fixed-number states

How many occupation-number basis states describe (a) three bosons in four modes and (b) three fermions in four modes?

Solution

For bosons,

$$\binom{M+N-1}{N} = \binom{6}{3} =20.$$

For fermions,

$$\binom{M}{N} = \binom{4}{3} =4.$$

The difference comes from unrestricted repeated bosonic occupation versus fermionic occupation numbers restricted to zero or one.

Exercise 2: Direct sum or tensor product

Explain why the sectors $N=0$ and $N=1$ are joined by a direct sum, while spin and spatial degrees of freedom inside the one-particle sector use a tensor product.

Solution

Zero particles and one particle are mutually exclusive alternatives. A state may have amplitudes in both orthogonal sectors, so the combined space is $\mathcal H_0\oplus\mathcal H_1$.

For one particle with spin and position, both degrees of freedom coexist. Its Hilbert space is therefore a tensor product such as $L^2(\mathbb R^3)\otimes\mathbb C^2$.

Exercise 3: Vacuum is not zero

Give two algebraic facts that distinguish the Fock vacuum $\lvert0\rangle$ from the zero vector $\mathbf0$.

Solution

First,

$$\langle0|0\rangle=1, \qquad \langle\mathbf0|\mathbf0\rangle=0.$$

Second, a creation operator produces a nonzero state from the vacuum,

$$a_r^\dagger\lvert0\rangle = \lvert1_r\rangle,$$

whereas every linear operator maps the zero vector to the zero vector.

Exercise 4: One-particle mode change

Let

$$b_+^\dagger = \frac{a_1^\dagger+a_2^\dagger}{\sqrt2}.$$

Express the one-particle state $b_+^\dagger\lvert0\rangle$ in the occupation basis of modes $1$ and $2$.

Solution

Linearity gives

$$\begin{aligned} b_+^\dagger\lvert0\rangle &= \frac{1}{\sqrt2} \left( a_1^\dagger+a_2^\dagger \right) \lvert0\rangle \\ &= \frac{ \lvert1,0\rangle+\lvert0,1\rangle }{\sqrt2}. \end{aligned}$$

The state has definite occupation in the $+$ mode but is a superposition of occupations in the original mode basis.

Exercise 5: Conserved number sectors

Suppose $[H,\hat N]=0$ and $\lvert\psi_N\rangle$ is in the $N$-particle sector. Show that $e^{-iHt/\hbar}\lvert\psi_N\rangle$ remains in that sector.

Solution

Because $H$ commutes with $\hat N$, so does its exponential:

$$\left[ e^{-iHt/\hbar},\hat N \right]=0.$$

Therefore

$$\begin{aligned} \hat N e^{-iHt/\hbar}\lvert\psi_N\rangle &= e^{-iHt/\hbar} \hat N\lvert\psi_N\rangle \\ &= N e^{-iHt/\hbar}\lvert\psi_N\rangle. \end{aligned}$$

The evolved vector remains an eigenvector of $\hat N$ with eigenvalue $N$. This proves conservation; it does not by itself establish a fundamental superselection rule.