Second quantization

The change of bookkeeping that makes many-body theory tractable: stop labelling which particle occupies which orbital — a label that is unphysical for identical particles anyway — and count only how many particles sit in each mode. The antisymmetry (or symmetry) under exchange, which in first quantization is an awkward constraint imposed by hand on the Slater determinant, becomes an algebraic identity — the (anti)commutation relations of creation and annihilation operators. Every operator, one-body or two-body, is then a normal-ordered polynomial in these, and Wick’s theorem turns expectation values into sums of contractions. This is the language in which Green’s functions, BdG diagonalization, and essentially all of interacting electronic structure are written.

Indistinguishability and the exchange dichotomy

For $N$ identical particles the Hamiltonian is symmetric under any permutation $P\in S_N$ of the particle labels, $[H,P]=0$, so physical states must carry a one-dimensional irrep of $S_N$. Only two such irreps exist for every $N$ — the trivial and the sign (alternating) representation — and the symmetrization postulate selects them: physical states are either totally symmetric or totally antisymmetric under transposition of any two particles,

$$P|\Psi ⟩=|\Psi ⟩(\text{bosons}),P|\Psi ⟩=\mathrm{sgn}(P)|\Psi ⟩(\text{fermions}).$$

Higher-dimensional (“parastatistics”) irreps of $S_N$ are excluded in $d\ge 3$; the spin–statistics theorem ties the two surviving cases to spin — integer-spin fields quantize with commutators and are bosonic, half-integer-spin fields with anticommutators and are fermionic — a result of relativistic QFT (microcausality + positivity of energy) that we take as input here. The antisymmetry of fermions is the Pauli exclusion principle: a state with two fermions in the same single-particle mode is its own negative, hence zero.

Slater determinants and permanents

Pick an orthonormal single-particle basis $\{|{\varphi }_i⟩\}$ (spin-orbitals: spatial orbital times spin). A properly antisymmetric $N$-fermion state built from orbitals ${\varphi }_{i_1},\dots ,{\varphi }_{i_N}$ is the Slater determinant

$$\Psi (x_1,\dots ,x_N)=\frac{1}{\sqrt{N!}}\left|\begin{array}{ccc}{\varphi }_{i_1}(x_1)& \cdots & {\varphi }_{i_N}(x_1)\\ ⋮& \ddots & ⋮\\ {\varphi }_{i_1}(x_N)& \cdots & {\varphi }_{i_N}(x_N)\end{array}\right|=\frac{1}{\sqrt{N!}}\sum _{P\in S_N}\mathrm{sgn}(P)\prod _{n=1}^N{\varphi }_{i_{P(n)}}(x_n),$$

where $x_n=({\mathbf{r}}_n,{\sigma }_n)$ collects space and spin. The determinant is the antisymmetrizer applied to a Hartree product: two equal columns (the same orbital twice) make it vanish — Pauli — and swapping two rows (two particles) flips the sign — antisymmetry. For bosons the same construction with the permanent (the determinant with all signs $+1$) gives the totally symmetric state, divided by $\sqrt{N!{\prod }_i{n}_i!}$ to account for multiply occupied modes. The single Slater determinant is the exact ground state of any non-interacting fermion Hamiltonian and the variational ansatz of Hartree–Fock; the permanent plays the analogous role for free bosons. The clumsiness of carrying $N!$ terms and re-antisymmetrizing every product is exactly what second quantization removes.

Fock space and occupation numbers

Rather than fix $N$, assemble all sectors into one Fock space, the direct sum of the (anti)symmetrized $N$-particle Hilbert spaces,

$${\mathcal{F}}_{\pm }=\underset{N=0}{\overset{\infty }{⨁}}{\mathcal{H}}_N^{\pm },{\mathcal{H}}_N^{\pm }={\mathcal{S}}_{\pm }\left({\mathcal{H}}_1^{\otimes N}\right),$$

with ${\mathcal{S}}_{+}$ the symmetrizer, ${\mathcal{S}}_{-}$ the antisymmetrizer, and ${\mathcal{H}}_0=\mathbb{C}$ the one-dimensional vacuum sector spanned by $|0⟩$ (normalized, $⟨0|0⟩=1$; it is not the zero vector). A basis of $\mathcal{F}$ is the occupation-number representation: list how many particles occupy each mode,

$$|n_1,n_2,n_3,\dots ⟩,n_i\in \{\begin{array}{ll}\{0,1,2,\dots \}& \text{bosons},\\ \{0,1\}& \text{fermions (Pauli)}.\end{array}$$

The choice of which orbitals to occupy that a Slater determinant or permanent encoded is now just the list of $n_i$; the antisymmetry is no longer in the label of the basis vector but, as we now see, in the operators that move between sectors.

Creation and annihilation operators

Define ${\hat{a}}_i^{†}$ (${\hat{b}}_i^{†}$) to adjoin a particle in mode $i$ and ${\hat{a}}_i$ (${\hat{b}}_i$) its adjoint, removing one; they map ${\mathcal{H}}_N\leftrightarrow {\mathcal{H}}_{N\pm 1}$, so they carry the dynamics between sectors of $\mathcal{F}$. The entire statistics is encoded in their algebra. For fermions, the canonical anticommutation relations (CAR), with $\{A,B\}\equiv AB+BA$,

$$\{{\hat{a}}_i,{\hat{a}}_j^{†}\}={\delta }_{ij},\{{\hat{a}}_i,{\hat{a}}_j\}=0,\{{\hat{a}}_i^{†},{\hat{a}}_j^{†}\}=0.$$

The case $i=j$ of the last identity gives $({\hat{a}}_i^{†}{)}^2=0$ — Pauli as one line of algebra — and the sign in ${\hat{a}}_i^{†}{\hat{a}}_j^{†}=-{\hat{a}}_j^{†}{\hat{a}}_i^{†}$ is precisely the determinant’s antisymmetry, now built into the operators. Acting on occupation kets (with the Jordan–Wigner sign string fixing the order),

$${\hat{a}}_i^{†}|\dots n_i\dots ⟩=(-1{)}^{{\sum }_{j<i}{n}_j}(1-n_i)|\dots n_i+1\dots ⟩,{\hat{a}}_i|\dots n_i\dots ⟩=(-1{)}^{{\sum }_{j<i}{n}_j}{n}_i|\dots n_i-1\dots ⟩.$$

For bosons, the canonical commutation relations (CCR), $[A,B]\equiv AB-BA$,

$$[{\hat{b}}_i,{\hat{b}}_j^{†}]={\delta }_{ij},[{\hat{b}}_i,{\hat{b}}_j]=0,[{\hat{b}}_i^{†},{\hat{b}}_j^{†}]=0,$$

so each mode is an independent harmonic oscillator and ${\hat{b}}_i^{†}|n_i⟩=\sqrt{n_i+1}|n_i+1⟩$, ${\hat{b}}_i|n_i⟩=\sqrt{n_i}|n_i-1⟩$ — no upper bound on $n_i$. In both cases the number operator ${\hat{n}}_i={\hat{a}}_i^{†}{\hat{a}}_i$ (or ${\hat{b}}_i^{†}{\hat{b}}_i$) reads off the occupation, ${\hat{n}}_i|\dots n_i\dots ⟩=n_i|\dots n_i\dots ⟩$, and $\hat{N}={\sum }_i{\hat{n}}_i$ counts total particles. Every state is reached from the vacuum, which is annihilated by every ${\hat{a}}_i$ (${\hat{b}}_i$):

$$|n_1,n_2,\dots ⟩=\prod _i\frac{({\hat{a}}_i^{†}{)}^{n_i}}{\sqrt{n_i!}}|0⟩,{\hat{a}}_i|0⟩=0.$$

(For fermions the product must be taken in a fixed mode order, $i=1,2,\dots$, to pin down the overall sign.) This is the same bosonic algebra that, applied to spin-deviation operators, generates the Holstein–Primakoff bosons of linear spin-wave theory.

Field operators

A basis-independent statement is got by superposing modes with their wavefunctions. The field operator annihilates a particle at point $\mathbf{r}$ (suppress spin or carry it as an index),

$$\hat{\psi }(\mathbf{r})=\sum _i{\varphi }_i(\mathbf{r}){\hat{a}}_i,{\hat{\psi }}^{†}(\mathbf{r})=\sum _i{\varphi }_i^{\ast }(\mathbf{r}){\hat{a}}_i^{†}.$$

Completeness ${\sum }_i{\varphi }_i(\mathbf{r}){\varphi }_i^{\ast }({\mathbf{r}}^{\prime })=\delta (\mathbf{r}-{\mathbf{r}}^{\prime })$ promotes the discrete (anti)commutators to local ones,

$$[\hat{\psi }(\mathbf{r}),{\hat{\psi }}^{†}({\mathbf{r}}^{\prime }){]}_{\mp }=\delta (\mathbf{r}-{\mathbf{r}}^{\prime }),[\hat{\psi }(\mathbf{r}),\hat{\psi }({\mathbf{r}}^{\prime }){]}_{\mp }=0,$$

with the lower sign ($\{\cdot ,\cdot \}$) for fermions and the upper ($[\cdot ,\cdot ]$) for bosons. The density operator is $\hat{n}(\mathbf{r})={\hat{\psi }}^{†}(\mathbf{r})\hat{\psi }(\mathbf{r})$ and $\int \hat{n}(\mathbf{r})d\mathbf{r}=\hat{N}$; ${\hat{\psi }}^{†}(\mathbf{r})|0⟩$ is a position eigenstate of one particle. The pair $\hat{\psi },{\hat{\psi }}^{†}$ is the object the path-integral and the equal-time correlator $⟨{\hat{\psi }}^{†}({\mathbf{r}}^{\prime })\hat{\psi }(\mathbf{r})⟩$ — the one-body density matrix, and the equal-time limit of the Green’s function — are built from.

One- and two-body operators

The payoff: any operator that is a sum of identical one- or two-particle terms has a basis-size (not $N$-dependent) second-quantized form. A one-body operator $\hat{O}={\sum }_n\hat{o}(x_n)$ (kinetic energy, external potential, a one-particle observable) becomes

$$\hat{O}=\sum _{ij}⟨i|\hat{o}|j⟩{\hat{a}}_i^{†}{\hat{a}}_j,⟨i|\hat{o}|j⟩=\int {\varphi }_i^{\ast }(x)\hat{o}{\varphi }_j(x)dx,$$

equivalently $\hat{O}=\int {\hat{\psi }}^{†}(\mathbf{r})o(\mathbf{r})\hat{\psi }(\mathbf{r})d\mathbf{r}$ for a local $o$ — “destroy in $j$, create in $i$, weighted by the matrix element.” A two-body operator $\hat{V}=\frac{1}{2}{\sum }_{m\ne n}v(x_m,x_n)$ (the Coulomb interaction) becomes

$$\boxed{\hat{V}=\frac{1}{2}\sum _{ijkl}⟨ij|v|kl⟩{\hat{a}}_i^{†}{\hat{a}}_j^{†}{\hat{a}}_l{\hat{a}}_k}⟨ij|v|kl⟩=\iint {\varphi }_i^{\ast }(x){\varphi }_j^{\ast }(x^{\prime })v(x,x^{\prime }){\varphi }_k(x){\varphi }_l(x^{\prime })dxd{x}^{\prime }.$$

Two points of notation that are perennial sources of sign errors:

• The order of the annihilation operators is reversed relative to the bra-side labels: creation reads $i,j$ (left to right) but annihilation reads $l,k$. Indices $i$ and $k$ share the first coordinate $x$ (the ${\varphi }_i^{\ast }\dots {\varphi }_k$ pair), while $j$ and $l$ share the second $x^{\prime }$; the reversed annihilation order ${\hat{a}}_l{\hat{a}}_k$ is exactly what keeps the physicists’ matrix element $⟨ij|v|kl⟩$ above correct. Keeping the operators normal-ordered (all daggers left) and in this order is what guarantees $\hat{V}|0⟩=0$ and reproduces the right exchange (Fock) term automatically; getting the order wrong silently swaps direct and exchange.
• The same expression with $\hat{a}\to \hat{b}$ holds for bosons; the antisymmetry/symmetry of $⟨ij|v|kl⟩$ under $i\leftrightarrow j$, $k\leftrightarrow l$ is carried entirely by the operator algebra, not imposed on the integral.

The full electronic Hamiltonian is then the compact $\hat{H}={\sum }_{ij}{h}_{ij}{\hat{a}}_i^{†}{\hat{a}}_j+\frac{1}{2}{\sum }_{ijkl}⟨ij|v|kl⟩{\hat{a}}_i^{†}{\hat{a}}_j^{†}{\hat{a}}_l{\hat{a}}_k$, with $h_{ij}$ the one-body (kinetic + external) core. These are the operators that reappear verbatim in Hartree–Fock (where the two-body term is mean-field-decoupled into direct and exchange potentials), in GW (where it is resummed into a screened self-energy $\Sigma =iGW$), and in the electron–ion coupling of electron–phonon coupling.

Normal ordering and Wick’s theorem

Two operators commute or anticommute up to a $c$-number; ordering all annihilation operators to the right strips those $c$-numbers off. Normal ordering $:\hat{A}:$ (relative to a chosen vacuum or Fermi sea) places every creation operator to the left of every annihilation operator, inserting the sign of the permutation used for fermions; by construction $⟨0|:\hat{A}:|0⟩=0$. The contraction of two operators, written with an overbrace $\overbrace{AB}$, is the difference between their product and its normal-ordered form,

$$\overbrace{AB}\equiv AB-:AB:=⟨0|AB|0⟩,$$

a $c$-number equal to the vacuum (or Fermi-sea) expectation value; the only nonvanishing fermionic contractions are $⟨0|{\hat{a}}_i{\hat{a}}_j^{†}|0⟩$ (particle) and, with respect to a filled Fermi sea, $⟨{\hat{a}}_j^{†}{\hat{a}}_i⟩$ (hole). Wick’s theorem then states that any product of creation/annihilation (or, in the time-dependent case, field) operators equals its normal-ordered form plus the normal-ordered form of every possible single contraction, double contraction, and so on:

$${\hat{A}}_1{\hat{A}}_2\cdots {\hat{A}}_n=:{\hat{A}}_1\cdots {\hat{A}}_n:+\sum _{\text{single}}:(\text{contracted}):+\sum _{\text{double}}:(\text{contracted}):+\cdots .$$

Taken between the vacuum, only the fully contracted terms survive (every uncontracted normal product has zero vacuum expectation), so a $2n$-point function collapses to a sum over pairings of products of two-point contractions — for fermions with the sign of the pairing permutation. This is the combinatorial engine behind perturbation theory: each contraction is a propagator (a Green’s function line) and each surviving pairing is a Feynman diagram. The time-ordered version (Gell-Mann–Low / Wick for $T$-products) is what generates the diagrammatic expansion of the interacting Green’s function and self-energy directly.

Prerequisites

single-particle quantum mechanics (Hilbert space, orthonormal bases, operators) · the identical-particle (anti)symmetrization postulate · the harmonic-oscillator ladder algebra

Builds toward: Green’s functions · BdG diagonalization

Key references

• Origins of the formalism — P. A. M. Dirac, Proc. R. Soc. A 114, 243 (1927); P. Jordan & O. Klein, Z. Phys. 45, 751 (1927); P. Jordan & E. Wigner, Z. Phys. 47, 631 (1928) (the fermionic anticommutators / Jordan–Wigner transformation); V. Fock, Z. Phys. 75, 622 (1932) (Fock space).
• Wick’s theorem — G. C. Wick, Phys. Rev. 80, 268 (1950).
• Texts — A. L. Fetter & J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, 1971), Ch. 1; J. W. Negele & H. Orland, Quantum Many-Particle Systems (Addison-Wesley, 1988), Ch. 1; A. Szabo & N. S. Ostlund, Modern Quantum Chemistry (Macmillan, 1989), Ch. 1–2 (Slater determinants, the chemists’ two-electron-integral conventions).