Second Quantization

Second quantization is the occupation-number and operator language for identical-particle quantum mechanics and quantum fields. It replaces explicit symmetrization or antisymmetrization of coordinate wavefunctions with bosonic commutators or fermionic anticommutators on Fock space.

The name is historical. In nonrelativistic many-body theory, one is usually rewriting an already quantum $N$-particle theory in a representation that allows variable occupation. In canonical field theory, one instead begins with a classical field phase space and quantizes its modes. Those routes meet in similar creation, annihilation, and Fock-space structures, but they are not the same logical operation.

Prerequisites

Before using this bridge, be comfortable with:

• one-particle Hilbert spaces and orthonormal bases;
• tensor products and permutation symmetry;
• identical bosons and fermions;
• the harmonic oscillator’s ladder algebra;
• direct sums and occupation-number notation;
• commutators, anticommutators, and operator ordering.

The canonical many-particle construction is developed in Second Quantization: Bridge to QFT. This page emphasizes the dictionary and its field-theory boundary.

From Fixed Particle Number to Fock Space

Let $\mathcal h$ be the one-particle Hilbert space. The unsymmetrized $N$-particle space is $\mathcal h^{\otimes N}$. Identical particles occupy its symmetric or antisymmetric subspace:

$$\mathcal h_+^{(N)} =\operatorname{Sym}^N\mathcal h, \qquad \mathcal h_-^{(N)} =\bigwedge^N\mathcal h.$$

Bosonic and fermionic Fock spaces collect all particle-number sectors:

$$\mathcal F_\pm(\mathcal h) = \bigoplus_{N=0}^{\infty} \mathcal h_\pm^{(N)}.$$

The $N=0$ sector is one dimensional:

$$\mathcal h_\pm^{(0)} \cong\mathbb C,$$

and its normalized basis vector is the vacuum $\lvert0\rangle$.

Fock space does not require the Hamiltonian to change particle number. It provides a common arena in which number-conserving and number-changing operators can both be represented.

Mode Operators

Choose an ordered orthonormal basis $\lbrace\lvert i\rangle\rbrace$ of $\mathcal h$.

Bosons

Bosonic operators satisfy

$$[a_i,a_j^\dagger] =\delta_{ij}I, \qquad [a_i,a_j] =[a_i^\dagger,a_j^\dagger]=0.$$

Their occupation basis is

$$\lvert n_1,n_2,\ldots\rangle = \prod_i \frac{(a_i^\dagger)^{n_i}}{\sqrt{n_i!}} \lvert0\rangle, \qquad n_i=0,1,2,\ldots.$$

The action on mode $j$ is

$$\begin{aligned} a_j\lvert\ldots,n_j,\ldots\rangle &= \sqrt{n_j}\, \lvert\ldots,n_j-1,\ldots\rangle, \\ a_j^\dagger\lvert\ldots,n_j,\ldots\rangle &= \sqrt{n_j+1}\, \lvert\ldots,n_j+1,\ldots\rangle. \end{aligned}$$

Fermions

Fermionic operators satisfy

$$\lbrace c_i,c_j^\dagger\rbrace =\delta_{ij}I, \qquad \lbrace c_i,c_j\rbrace =\lbrace c_i^\dagger,c_j^\dagger\rbrace=0.$$

Because $(c_i^\dagger)^2=0$, each occupation is $n_i=0$ or $1$. Fix the basis ordering through

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

Then

$$\begin{aligned} c_j\lvert\boldsymbol n\rangle &= (-1)^{\sum_{i<j}n_i} n_j \lvert n_1,\ldots,0_j,\ldots\rangle, \\ c_j^\dagger\lvert\boldsymbol n\rangle &= (-1)^{\sum_{i<j}n_i} (1-n_j) \lvert n_1,\ldots,1_j,\ldots\rangle. \end{aligned}$$

The sign is not an optional convention once the mode order has been fixed. Changing the mode order changes basis-state phases and must be propagated consistently through every operator.

Number Operator

The total number operator is

$$N =\sum_i a_i^\dagger a_i$$

for bosons, or the analogous expression $N=\sum_i c_i^\dagger c_i$ for fermions. It satisfies

$$[N,a_i^\dagger]=a_i^\dagger, \qquad [N,a_i]=-a_i,$$

and likewise for $c_i^\dagger,c_i$. Thus creation and annihilation operators map

$$\mathcal h_\pm^{(N)} \longrightarrow \mathcal h_\pm^{(N\pm1)}.$$

A Hamiltonian conserves total particle number exactly when

$$[H,N]=0.$$

Pairing terms such as $c_i^\dagger c_j^\dagger+\text{h.c.}$ do not conserve $N$, though they may conserve fermion-number parity.

Basis Changes

Suppose a new one-particle basis is

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

The corresponding operators transform as

$$b_\alpha^\dagger =\sum_i U_{i\alpha}a_i^\dagger, \qquad b_\alpha =\sum_i U_{i\alpha}^\ast a_i.$$

Unitarity preserves the canonical algebra:

$$[b_\alpha,b_\beta^\dagger] =\delta_{\alpha\beta}I$$

for bosons, with the analogous fermionic anticommutator. Creation and annihilation operators therefore depend on a chosen mode basis; the underlying many-particle operator does not.

Lifting One-Body Operators

Let $h$ be a one-particle operator with matrix elements

$$h_{ij}=\langle i\vert h\vert j\rangle.$$

On the $N$-particle sector, the corresponding additive operator is

$$h^{(1)}+\cdots+h^{(N)}.$$

Its Fock-space lift is

$$d\Gamma(h) =\sum_{ij} h_{ij}a_i^\dagger a_j$$

for bosons, and

$$d\Gamma(h) =\sum_{ij} h_{ij}c_i^\dagger c_j$$

for fermions. The notation $d\Gamma(h)$ emphasizes that this is the differential second quantization of a one-particle operator.

Important examples are

One-particle operator	Fock-space operator
Identity $I_{\mathcal h}$	Number operator $N=d\Gamma(I_{\mathcal h})$
One-particle Hamiltonian $h$	Free many-body Hamiltonian $\sum_{ij}h_{ij}a_i^\dagger a_j$
Momentum $\boldsymbol p$	Total momentum $\sum_{ij}\langle i\vert\boldsymbol p\vert j\rangle a_i^\dagger a_j$
One-particle symmetry generator $g$	Additive many-body generator $d\Gamma(g)$

Every number-conserving one-body lift commutes with $N$.

Lifting Two-Body Interactions

For a symmetric pair interaction $v$ acting on two one-particle slots, define ordered-basis matrix elements

$$V_{ij;kl} = \langle i,j\vert v\vert k,l\rangle.$$

With unsymmetrized matrix elements and the displayed operator order, the second-quantized interaction is

$$\widehat V =\frac12 \sum_{ijkl} V_{ij;kl} a_i^\dagger a_j^\dagger a_l a_k,$$

or the same expression with $c$ operators for fermions.

The factor $1/2$ removes double counting of particle pairs. Some sources use antisymmetrized fermionic matrix elements, change index order, or absorb combinatorial factors into $V_{ij;kl}$. Copy the matrix-element definition together with the operator formula.

The operator contains two creators and two annihilators, so it conserves total particle number.

Field Operators in Nonrelativistic Many-Body Theory

Let $\varphi_i(\boldsymbol x)=\langle\boldsymbol x\vert i\rangle$. Define

$$\psi(\boldsymbol x) =\sum_i \varphi_i(\boldsymbol x)a_i, \qquad \psi^\dagger(\boldsymbol x) =\sum_i \varphi_i^\ast(\boldsymbol x)a_i^\dagger.$$

For fermions, replace $a_i$ by $c_i$. Completeness of the one-particle basis gives

$$[\psi(\boldsymbol x),\psi^\dagger(\boldsymbol y)] = \delta^{(d)} (\boldsymbol x-\boldsymbol y)I$$

for bosons, and

$$\lbrace \psi(\boldsymbol x),\psi^\dagger(\boldsymbol y) \rbrace = \delta^{(d)} (\boldsymbol x-\boldsymbol y)I$$

for fermions.

The field operator has dimensions

$$[\psi]=L^{-d/2}$$

under this delta-function convention. It annihilates a particle at position $\boldsymbol x$ in a distributional sense; it is not a complex-valued single-particle wavefunction.

Coordinate-space operator dictionary

For a one-particle kernel $h(\boldsymbol x,\boldsymbol y)$,

$$\widehat H_1 = \int d^dx\,d^dy\, \psi^\dagger(\boldsymbol x) h(\boldsymbol x,\boldsymbol y) \psi(\boldsymbol y).$$

For a local differential one-particle Hamiltonian,

$$\widehat H_1 = \int d^dx\, \psi^\dagger(\boldsymbol x) \left[ -\frac{\hbar^2\nabla^2}{2m} +U(\boldsymbol x) \right] \psi(\boldsymbol x).$$

A two-body potential has the standard form

$$\begin{aligned} \widehat V =\frac12 \int d^dx\,d^dy\, &\psi^\dagger(\boldsymbol x) \psi^\dagger(\boldsymbol y) V(\boldsymbol x,\boldsymbol y) \\ &\times \psi(\boldsymbol y) \psi(\boldsymbol x). \end{aligned}$$

For fermions, the written order fixes the signs. Equal-point products can also require regularization in continuum models.

Recovering the Many-Particle Wavefunction

Let $\Psi_N(\boldsymbol x_1,\ldots,\boldsymbol x_N)$ be normalized and symmetric for bosons or antisymmetric for fermions. Its Fock-space vector is

$$\begin{aligned} \lvert\Psi_N\rangle =\frac{1}{\sqrt{N!}} \int d^dx_1\cdots d^dx_N\, &\Psi_N(\boldsymbol x_1,\ldots,\boldsymbol x_N) \\ &\times \psi^\dagger(\boldsymbol x_1)\cdots \psi^\dagger(\boldsymbol x_N) \lvert0\rangle. \end{aligned}$$

Conversely,

$$\begin{aligned} \Psi_N(\boldsymbol x_1,\ldots,\boldsymbol x_N) =\frac{1}{\sqrt{N!}} \langle0\vert &\psi(\boldsymbol x_N)\cdots \\ &\psi(\boldsymbol x_1) \vert\Psi_N\rangle. \end{aligned}$$

This equivalence shows what second quantization does: it packages exchange symmetry into the operator algebra. It does not introduce a different empirical theory for a fixed nonrelativistic particle number.

Why the Language Is Powerful

• Exchange symmetry is automatic once the operator algebra is chosen.
• Variable particle number is represented without changing Hilbert spaces.
• Additive observables and few-body interactions have compact formulas.
• Lattice, momentum, orbital, band, and position bases are related by unitary mode transformations.
• Collective excitations and quasiparticles can be introduced by new operator bases.
• Perturbations that create or destroy excitations can be written directly.
• The same algebra prepares the mode language of quantum field theory.

Transition to Quantum Field Theory

Nonrelativistic field operators and relativistic quantum fields look similar, but several new ingredients enter.

Classical-field starting point

For a relativistic field, canonical quantization begins from a classical field and its conjugate momentum. Normal modes become operator modes after imposing commutation or anticommutation relations. The Harmonic Oscillator to Fields card develops the bosonic free-field case.

Particles and antiparticles

A complex relativistic field generally contains distinct particle and antiparticle creation operators. A Dirac field combines electron and positron modes. This structure is not captured by merely allowing the particle number of a nonrelativistic Schrödinger field to vary.

Locality and spin–statistics

Relativistic fields must satisfy locality conditions at spacelike separation. The connection between integer spin and bosonic commutation, and between half-integer spin and fermionic anticommutation, is a QFT theorem with assumptions including Lorentz invariance, locality, and positive energy. In nonrelativistic many-body mechanics, Bose or Fermi statistics is imposed as the particle type.

Renormalized local products

Quantum fields are operator-valued distributions. Products at the same point can be singular, so expressions such as $\phi^4(x)$ or $\overline\psi(x)\psi(x)$ require a regulator and renormalization prescription.

Interacting state spaces

Free Fock space supplies perturbative and asymptotic particle language, but an interacting continuum QFT need not be globally equivalent to the free representation. “Create a particle by acting once with the field” is therefore an approximation or spectral statement, not a universal exact identity.

Convention Checklist

Before using a second-quantized formula, record:

1. bosonic commutators or fermionic anticommutators;
2. ordered mode basis, especially for fermions;
3. normalization of continuum modes and delta functions;
4. whether matrix elements are symmetrized or antisymmetrized;
5. index order in two-body coefficients;
6. whether total number, a species charge, or only parity is conserved;
7. normal-ordering convention;
8. whether the field is nonrelativistic, relativistic real, relativistic complex, spinor, or gauge;
9. finite-volume versus continuum normalization.

Common Mistakes

• Describing second quantization as literally quantizing the same theory twice.
• Forgetting the vacuum sector in Fock space.
• Treating creation operators as basis-independent objects without mode labels.
• Reordering fermionic operators without the required sign.
• Using bosonic occupation factors for fermions.
• Combining antisymmetrized matrix elements with a prefactor meant for unsymmetrized elements.
• Calling $\psi(\boldsymbol x)$ a many-particle wavefunction.
• Assuming Fock-space notation implies that the Hamiltonian changes particle number.
• Inferring relativistic spin–statistics from nonrelativistic notation alone.
• Treating products of continuum field operators at one point as automatically finite.

Exercises

Exercise 1: Number conservation

Show that a one-body lift

$$\widehat A=\sum_{ij}A_{ij}a_i^\dagger a_j$$

commutes with $N=\sum_k a_k^\dagger a_k$.

Solution

Use

$$[N,a_i^\dagger]=a_i^\dagger, \qquad [N,a_j]=-a_j.$$

The product rule gives

$$\begin{aligned} [N,a_i^\dagger a_j] &= [N,a_i^\dagger]a_j +a_i^\dagger[N,a_j] \\ &= a_i^\dagger a_j-a_i^\dagger a_j =0. \end{aligned}$$

Therefore every term commutes with $N$, so $[N,\widehat A]=0$. The same argument works for fermions because $N$ and the bilinear are even operators and the ordinary commutator is used.

Exercise 2: Fermionic ordering sign

Using the convention

$$\lvert n_1,n_2,n_3\rangle =(c_1^\dagger)^{n_1} (c_2^\dagger)^{n_2} (c_3^\dagger)^{n_3} \lvert0\rangle,$$

calculate $c_2^\dagger\lvert1,0,1\rangle$.

Solution

One occupied mode lies before mode $2$, so

$$(-1)^{\sum_{i<2}n_i}=(-1)^{n_1}=-1.$$

Because $n_2=0$, creation is allowed:

$$c_2^\dagger\lvert1,0,1\rangle =-\lvert1,1,1\rangle.$$

The minus sign records the reordering needed to place $c_2^\dagger$ after $c_1^\dagger$ in the chosen canonical basis order.

Exercise 3: Basis independence

Verify that a unitary mode transformation preserves the bosonic canonical commutator.

Solution

With

$$b_\alpha =\sum_iU_{i\alpha}^\ast a_i, \qquad b_\beta^\dagger =\sum_jU_{j\beta}a_j^\dagger,$$

one obtains

$$\begin{aligned} [b_\alpha,b_\beta^\dagger] &= \sum_{ij} U_{i\alpha}^\ast U_{j\beta} [a_i,a_j^\dagger] \\ &= \sum_i U_{i\alpha}^\ast U_{i\beta} \\ &= (U^\dagger U)_{\alpha\beta} =\delta_{\alpha\beta}. \end{aligned}$$

The fermionic calculation is identical with anticommutators.

Canonical Links

• Occupation-Number Basis
• Bosonic Fock Space
• Fermionic Fock Space
• Creation and Annihilation Operators
• Bosonic Commutation Relations
• Fermionic Anticommutation Relations
• One-Body Operators
• Two-Body Operators
• Field Operators
• Many-Body Field Operators
• Pauli Exclusion Principle
• From Phase Space to Canonical Quantization
• Quantization vs Classical Limit

Continue in Field Theory

Continue with canonical quantization of scalar and spinor fields, relativistic normalization, particles and antiparticles, Wick’s theorem, interacting fields, and the spin–statistics theorem at QFT.org.

References

• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Addison-Wesley, 1988.
• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• S. Weinberg, The Quantum Theory of Fields, Vol. I, Cambridge University Press, 1995.