Second Quantization and Fock Space

Occupation numbers are the working variables of essentially all quantitative many-body physics. Condensed-matter theory writes the Hubbard model, superconductivity, and the fractional quantum Hall effect entirely in $a^\dagger,a$; quantum optics counts photons in cavity modes with the same operators; nuclear and quantum-chemistry codes evaluate energies as $\sum\langle i|\hat t|j\rangle a_i^\dagger a_j+\tfrac12\sum\langle ij|\hat v|kl\rangle a_i^\dagger a_j^\dagger a_l a_k$, never symmetrizing a wavefunction by hand. In quantum computing, the Jordan-Wigner transformation that first appeared in 1928 is the standard way to map fermionic Hamiltonians onto qubits so that molecular ground-state energies can be estimated on a quantum processor. And the moment the mode label becomes a continuous field coordinate, the same operators become the field operators $\hat\psi(\mathbf x)$ whose relativistic quantization — imposing precisely equations (3) and (4) on $\hat\psi$ — builds quantum field theory.

Picture the single-particle Hilbert space as a row of pigeonholes — the modes, labelled $i$ (they might be momentum states, oscillator levels, atomic orbitals). A Fock state is nothing but a tally of how many particles sit in each hole: $|n_1,n_2,\dots\rangle$. The creation operator $a_i^\dagger$ drops one particle into hole $i$; the annihilation operator $a_i$ takes one out. Because identical particles have no individual identity, the labels on the particles were always fiction; the only physical content is the occupation numbers, and Fock space is the space that tracks exactly that and nothing more. This is why the notation is so much lighter than a symmetrized wavefunction: it never writes down the fiction it would have to symmetrize away. The whole difference between bosons and fermions is one sign in how two creation operators slide past each other. For bosons they commute, so you can stack any number into a single hole. For fermions they anticommute, so trying to put a second one into an occupied hole gives $a_i^\dagger a_i^\dagger|0\rangle=-a_i^\dagger a_i^\dagger|0\rangle$, which can only be zero. That minus sign is the Pauli exclusion principle — the same content the Pauli Exclusion Principle article derives from the vanishing of an antisymmetric wavefunction, here compressed into $(a_i^\dagger)^2=0$.

Each bosonic mode is an independent harmonic oscillator: $a_i^\dagger$ raises its excitation by one, and a many-boson state is just a product of oscillators sitting in various excited levels. That is the reason "particle" and "field quantum" become the same word — a quantum of a field mode is one unit of oscillator excitation. The punchline of the whole formalism is that occupancy is quantized and capped differently for the two statistics: for bosons the ladder runs forever, for fermions it stops at one.

Operators translate from first to second quantization by one rule: take the single-particle matrix element and attach $a^\dagger a$ for a one-body operator, or $a^\dagger a^\dagger a a$ for a two-body one. The structure $a_i^\dagger a_j$ reads literally "remove a particle from mode $j$, put it in mode $i$" — a transition amplitude — and it is the same for every particle number, which is exactly the labour the old formalism could not save.

The starting object is the state space itself. First quantization gave us one $N$-particle space; here we take the direct sum over all $N$, so that a state can be a superposition of different particle numbers — indispensable the moment particles can be created or destroyed. The vacuum is the $N=0$ summand, a genuine normalized state, not "nothing".

where: $\mathcal F_\pm$ — bosonic ($+$) or fermionic ($-$) Fock space; $\mathcal H_N^{(\pm)}$ — the symmetric/antisymmetric $N$-particle subspace; $|0\rangle$ — the vacuum. Reading it physically: a general Fock state can have components in several $N$-sectors at once, which is precisely what a first-quantized wavefunction with fixed $N$ can never do — this is the structural reason the formalism can describe emission, absorption, and pair creation. States are built by stacking creation operators on the vacuum. The normalization factor is inherited from the oscillator ladder of Creation and Annihilation Operators, which owns the relation $a^\dagger|n\rangle=\sqrt{n+1}\,|n+1\rangle$.

where: $n_i$ — the occupation of mode $i$ (any nonnegative integer for bosons; $0$ or $1$ for fermions); the $1/\sqrt{n_i!}$ makes $\langle n_i|n_i\rangle=1$. Reading it physically: this is the occupation-number basis. For a single mode it is literally the oscillator number state $|n\rangle$; the many-mode Fock state is a tensor product of such oscillators, one per pigeonhole. The statistics is one algebraic relation, differing only by a sign. Bosons commute:

where: $[A,B]=AB-BA$; $\delta_{ij}$ is the Kronecker delta over modes. Reading it physically: the $[a_i^\dagger,a_j^\dagger]=0$ line says the order in which you create bosons does not matter, so the state is automatically symmetric. Fermions anticommute:

where: $\{A,B\}=AB+BA$. Reading it physically: swapping two fermion creations flips the sign, so the state is automatically antisymmetric — a Slater determinant without the determinant. Setting $i=j$ in the last relation gives $2(a_i^\dagger)^2=0$, the Pauli principle. The particle number is measured by the number operator, and single-particle observables become bilinears in the operators. These two — plus the isomorphism proved below — are the working machinery of every many-body calculation.

where: $\hat N$ — Hermitian total-number operator with eigenvalues $\sum_i n_i$; $\hat n_i=a_i^\dagger a_i$ — the occupation operator for mode $i$; $\langle i|\hat t|j\rangle$ — the first-quantized matrix element of a single-particle operator $\hat t$. Reading it physically: $\hat T$ has the same form in every $N$-sector, so one expression handles one particle or a million — the compression that makes second quantization worth the abstraction.

Example 1 — the double-occupancy norm. Build the two-boson state in a single mode and find its normalization. Start from $|0\rangle$ and apply $a^\dagger$ twice.

Step 1. First application: $a^\dagger|0\rangle=\sqrt{0+1}\,|1\rangle=|1\rangle$, using the ladder relation from Creation and Annihilation Operators.

Step 2. Second application: $a^\dagger|1\rangle=\sqrt{1+1}\,|2\rangle=\sqrt2\,|2\rangle$. Hence $(a^\dagger)^2|0\rangle=\sqrt2\,|2\rangle$.

Step 3. The norm is $\lVert(a^\dagger)^2|0\rangle\rVert=\sqrt2=\mathbf{1.414214}$, so the normalized two-boson state is $|2\rangle=(a^\dagger)^2|0\rangle/\sqrt{2!}$. This $\sqrt{2!}=\sqrt2$ is exactly the $1/\sqrt{n_i!}$ factor in the Fock-basis definition (2) — omit it and the isomorphism stops preserving norms.

Step 4. Check the number: $\hat N|2\rangle=a^\dagger a\,|2\rangle=2\,|2\rangle$, eigenvalue $2$. The number operator reads off the occupation directly.

Example 2 — a one-body operator, both ways. For a two-mode system with single-particle operator $\hat t=\begin{pmatrix}E_0&t\\ t&E_1\end{pmatrix}$, $E_0=1.3$, $E_1=2.1$, $t=0.7$, evaluate $\langle\hat T\rangle$ on the two-boson state $|1_0,1_1\rangle$ in both formalisms.

Step 1. Second-quantized. $\langle\hat T\rangle=\sum_{ij}\langle i|\hat t|j\rangle\langle a_i^\dagger a_j\rangle$. On $|1_0,1_1\rangle$ the surviving terms are the diagonal occupations $\langle a_0^\dagger a_0\rangle=\langle a_1^\dagger a_1\rangle=1$, so $\langle\hat T\rangle=E_0+E_1$.

Step 2. First-quantized. On the symmetric tensor state the one-body operator is $\hat t\otimes\mathbb 1+\mathbb 1\otimes\hat t$; its expectation on the normalized symmetric combination of $|0\rangle|1\rangle$ is likewise $E_0+E_1$.

Step 3. Numbers. $E_0+E_1=1.3+2.1=\mathbf{3.400000}$ from both routes. The transition amplitude to the double-occupancy state is $\langle2_0|\hat T|1_0,1_1\rangle=\sqrt2\,t=\mathbf{0.989949}$. The two formalisms agree on a diagonal energy and an off-diagonal amplitude — the full matrix, not just the trace.

This article assembles the Fock-space picture from the oscillator algebra of Creation and Annihilation Operators and the exchange argument of the Pauli Exclusion Principle, and proves the isomorphism that makes the occupation-number reformulation legitimate rather than merely convenient. The dynamics it deliberately leaves out — giving a field its own Lagrangian, imposing these (anti)commutators on $\hat\psi(\mathbf x)$, and finding antiparticles as the quanta of a second operator set — is the content of the field theory that follows, Field Quantization of the Klein-Gordon and Dirac Fields. Readers wanting the two-body machinery in full should follow the interacting many-body references.

Provenance. Base article QUANT.RQM.SECQF.1103 (AllQFT "quantum-mechanics" track, Sakurai / Fetter-Walecka). Upgraded to v3 canon. The base stated the operators and relations correctly but proved nothing and gave no numbers; the isomorphism, the $\sqrt2$ double-occupancy norm, the dimension counts $d(d\pm1)/2$, and the one-body operator equivalence ($E_0+E_1=3.4$, transition amplitude $\sqrt2\,t=0.9899$) are new to this version and verified in `verify_aq-qm-second-quantization-fock-space.py`. The base's ordering remark for the two-body interaction ($a_i^\dagger a_j^\dagger a_l a_k$ with reversed annihilation order) is retained and flagged in the FAQ.

Prerequisites. Creation and Annihilation Operators · Indistinguishability of Identical Particles · Pauli Exclusion Principle · Quantum Harmonic Oscillator: Analytic Solution. Leads to. Field Quantization of the Klein-Gordon and Dirac Fields · From Single-Particle to Quantum Fields · interacting many-body theory.