Fock Space and Occupation Number

Fock space collects states with different particle numbers into one Hilbert space. Each fixed-number sector already respects bosonic or fermionic exchange symmetry. Occupation-number notation then describes how many particles occupy each complete one-particle mode.

Helpful background. Identical particles explains exchange symmetry. Direct sums versus tensor products distinguishes alternative number sectors from coexisting degrees of freedom.

Enter this chapter

Task	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

A sum of number sectors

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

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

The direct sum is a Hilbert-space sum: a vector $\Psi=\bigoplus_N\psi_N$ must satisfy $\sum_N\|\psi_N\|^2<\infty$. For a normalized state, $\|\psi_N\|^2$ is the probability of measuring total number $N$. This normalization condition does not guarantee a finite mean number; that additionally requires $\sum_N N\|\psi_N\|^2<\infty$.

The vacuum spans the one-dimensional $N=0$ sector. It has norm 1 and is distinct from the zero vector. Whether it is the ground state depends on the Hamiltonian and the meaning of the counted particles or excitations.

Occupations depend on the mode basis

For orthonormal modes, bosonic occupations are $n_r=0,1,2,\ldots$; fermionic occupations are $n_r=0,1$. With $M$ modes and fixed total number $N$, the numbers of basis states are

$$D_B=\binom{M+N-1}{N},\qquad D_F=\binom{M}{N}.$$

The fermionic sector vanishes for $N>M$. The full finite-mode fermionic Fock space has dimension $2^M$, whereas the bosonic space remains infinite dimensional unless an occupation cutoff is imposed.

A mode is a chosen one-particle state, such as a spin-orbital or a momentum-polarization pair. A state with definite occupation in one basis can be a superposition in another. For example, one particle in the symmetric combination of modes 1 and 2 has state

$$\frac{|1,0\rangle+|0,1\rangle}{\sqrt2}$$

in the original occupation basis. A number-preserving unitary mode change leaves total number unchanged. A Bogoliubov transformation mixing creation and annihilation operators is a different construction and can change the reference vacuum.

Conservation is different from superselection

If the Hamiltonian preserves the eigenspaces of $\hat N$, a state initially in one number sector stays there. This dynamical conservation does not by itself prohibit all inter-sector coherences. Whether their relative phases can be measured also depends on the observables and available reference systems.

Likewise, a fluctuating number does not imply a coherent superposition. A thermal mixture can distribute probabilities over number sectors without any measurable phase between them. Particle-number superselection separates these possibilities, including charge, parity, and local restrictions.

Next: operators acting on occupation states

Creation, Annihilation, and Second Quantization constructs the operators that connect sectors and builds many-body Hamiltonians from them. Keep a fixed mode order for fermions: amplitudes acquire signs when creation operators are reordered. Before studying dynamics, use Fock Space Examples to practice translating between occupation tuples and physical states.

Exercises

Count and list. How many states describe two bosons in two modes? How many describe two fermions in the same two complete modes?

Solution

The bosonic states are $|2,0\rangle$, $|1,1\rangle$, and $|0,2\rangle$, so $D_B=\binom32=3$. The only fermionic occupation is $|1,1\rangle$, so $D_F=\binom22=1$. Additional spin labels would constitute additional modes and change the count.

A mixture and a superposition. Compare $\rho=(|0\rangle\langle0|+|1\rangle\langle1|)/2$ with $|\psi\rangle=(|0\rangle+|1\rangle)/\sqrt2$ for a single bosonic mode. What does a number measurement distinguish?

Solution

Both give probabilities 1/2 for $N=0$ and $N=1$, so number measurements alone do not distinguish them. The superposition has off-diagonal matrix elements $\langle0|\rho_\psi|1\rangle=1/2$, while the mixture has none. Detecting that coherence requires an allowed phase-sensitive measurement and, where relevant, a phase reference.

References

• Bruus, H., and Flensberg, K., Many-Body Quantum Theory in Condensed Matter Physics, Oxford University Press (2004).
• Fetter, A. L., and Walecka, J. D., Quantum Theory of Many-Particle Systems, Dover (2003).
• Scully, M. O., and Zubairy, M. S., Quantum Optics, Cambridge University Press (1997).