Second Quantization: Bridge to QFT

Second quantization is the cleanest nonrelativistic doorway into quantum field theory. It replaces explicit labeled-particle wavefunctions with mode occupation, Fock space, creation and annihilation operators, and field-operator notation.

This page is a bridge, not a QFT derivation. The goal is to identify which structures ordinary quantum mechanics has already taught, and where relativistic field theory changes the problem.

What Quantum Mechanics Has Already Taught

By the time one reaches Fock space, ordinary quantum mechanics has already supplied several core ingredients:

Hilbert spaces describe quantum states.
Tensor products describe composite systems.
Identical bosons and fermions live in symmetric or antisymmetric sectors.
Occupation-number states record which modes are occupied.
Creation and annihilation operators change those mode occupations.
Bosonic modes obey commutation relations.
Fermionic modes obey anticommutation relations.

In this language, the central object is no longer “particle 1 at one coordinate and particle 2 at another coordinate.” It is a state of modes:

\lvert n_1,n_2,\ldots\rangle.

That shift is already field-theoretic in spirit. Fields are naturally described by modes, and particles become excitations of those modes.

Fock Space and Variable Particle Number

For a one-particle Hilbert space $\mathcal h$ , the bosonic and fermionic Fock spaces are

\mathcal F_B(\mathcal h) = \bigoplus_{N=0}^{\infty} \operatorname{Sym}^N\mathcal h, \qquad \mathcal F_F(\mathcal h) = \bigoplus_{N=0}^{\infty} \wedge^N\mathcal h.

Each summand is a fixed-particle-number sector. The direct sum lets one describe states with definite particle number and states that are superpositions of different particle numbers.

The total number operator is

N = \sum_i N_i.

If a Hamiltonian satisfies

[H,N]=0,

then particle number is conserved by the time evolution generated by $H$ . Many nonrelativistic many-body Hamiltonians have this property. But the Fock-space framework itself does not require fixed particle number; it only provides the Hilbert space and operators needed to discuss it.

This is one reason Fock space is a natural preparation for QFT. Relativistic field theory must allow processes in which the number of particles of a given type is not fixed, subject to conservation laws such as charge, energy, momentum, and angular momentum.

Field Operators as Mode Operators in Position Space

Choose an orthonormal one-particle basis $\{\varphi_i(\mathbf x)\}$ . A nonrelativistic annihilation field operator may be expanded as

\psi(\mathbf x) = \sum_i \varphi_i(\mathbf x)a_i,

with creation field

\psi^\dagger(\mathbf x) = \sum_i \varphi_i^*(\mathbf x)a_i^\dagger

for bosonic modes. For fermions one uses fermionic operators $c_i,c_i^\dagger$ instead, with the same mode-expansion idea.

These field operators are not ordinary wavefunctions. They are operator-valued distributions that remove or add an excitation localized near $\mathbf x$ , in the sense defined by their smearing against test functions or basis modes.

For bosons, the mode commutation relations imply

[\psi(\mathbf x),\psi^\dagger(\mathbf y)] = \delta(\mathbf x-\mathbf y)

in the continuum normalization. For fermions, the corresponding relation is

\{\psi(\mathbf x),\psi^\dagger(\mathbf y)\} = \delta(\mathbf x-\mathbf y).

This is the bridge from discrete mode operators to local field language. The basis dependence, inverse formulas, completeness relation, momentum modes, and spinful variants are developed in Mode Expansions, while the distributional meaning of $\psi(\mathbf x)$ is developed in Field Operators.

Operators in Field Language

A one-body operator $A$ with matrix elements

A_{ij} = \langle\varphi_i\vert A\vert\varphi_j\rangle

has the second-quantized form

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

for bosons, with the analogous fermionic expression using $c_i^\dagger c_j$ .

In position-space field notation, the same idea is written schematically as

\widehat A = \int d^3x\, \psi^\dagger(\mathbf x) A \psi(\mathbf x),

where $A$ acts on the one-particle variable. For example, a one-particle Hamiltonian $h$ gives a many-particle term

\widehat H_1 = \int d^3x\, \psi^\dagger(\mathbf x) h \psi(\mathbf x).

A two-body interaction has the typical form

\widehat V = \frac12 \int d^3x\,d^3y\, \psi^\dagger(\mathbf x) \psi^\dagger(\mathbf y) V(\mathbf x,\mathbf y) \psi(\mathbf y) \psi(\mathbf x),

with fermionic signs handled by operator ordering. This formula belongs to nonrelativistic many-body quantum mechanics as much as to QFT-inspired notation.

Harmonic Oscillator Modes and Free Fields

The quantum harmonic oscillator prepares the most important free-field pattern. A single oscillator has

H = \hbar\omega \left( a^\dagger a+\frac12 \right), \qquad [a,a^\dagger]=I.

A free bosonic field decomposed into normal modes behaves like a collection of independent oscillators:

H_0 = \sum_{\mathbf k} \hbar\omega_{\mathbf k} \left( a_{\mathbf k}^\dagger a_{\mathbf k} +\frac12 \right)

in a finite-volume normalization. In infinite volume, the sum becomes an integral and Kronecker deltas become Dirac deltas.

This is why oscillator ladder operators become the language of photons, phonons, and free relativistic particles. Interactions then couple modes and make the theory much richer than a list of independent oscillators.

Why Relativistic Locality Changes the Story

Nonrelativistic second quantization is not yet full QFT. It usually assumes:

a preferred time parameter;
a chosen one-particle Hilbert space;
fixed particle species as input;
Galilean rather than Lorentz symmetry;
no requirement of microcausality at spacelike separation.

Relativistic QFT changes the arena. Fields are local objects on spacetime, Lorentz symmetry constrains their transformation laws, and locality is expressed through commutation or anticommutation conditions at spacelike separation. Particle number is generally not a fundamental fixed input; particles are excitations of fields and are most sharply defined in free or asymptotic regimes.

This is the conceptual boundary. Fock space is a powerful bridge, but QFT is not merely “ordinary quantum mechanics with more particles.” It is a local quantum theory of fields.

Where QFT Begins

For the specifically many-body route through correlations, collective fields, imaginary time, coarse-graining, and emergent relativistic behavior, see Why Many-Body QM Leads to QFT. The regulated Schrödinger-field action, number current, contact-interaction matching, and exact-versus-effective boundary are developed in Nonrelativistic Field Theory from Many-Body QM.

The natural continuation is:

canonical quantization of a free scalar field;
mode expansions and normalization conventions;
relativistic field commutators and microcausality;
path-integral generating functionals;
normal ordering, Wick’s theorem preview, and perturbation theory;
propagators, Feynman rules, and the S-matrix.

Ordinary quantum mechanics contributes the grammar: Hilbert spaces, operators, symmetries, oscillators, path integrals, and scattering. QFT changes the nouns: fields become primary, particles become excitations, and locality is built into the operator or path-integral structure.

Common Mistakes

Thinking second quantization means quantizing a theory twice.
Treating nonrelativistic Fock space as already equivalent to relativistic QFT.
Forgetting that field operators are operator-valued distributions, not ordinary wavefunctions.
Assuming particle number is always conserved in a field theory.
Ignoring the difference between finite-volume sums and continuum delta-function normalizations.
Treating interacting QFT particle states as if they were always simple occupation-number states at all times.

Cross-Links

References

A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971.
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, Volume I, Cambridge University Press, 1995.
A. Zee, Quantum Field Theory in a Nutshell, 2nd ed., Princeton University Press, 2010.
M. Srednicki, Quantum Field Theory, Cambridge University Press, 2007.

Exercises

Why is Fock space useful before one studies relativistic QFT?

Solution

Fock space is useful whenever particle number or excitation number is naturally described by mode occupations. This includes nonrelativistic many-body systems, photons in optical modes, phonons in solids, and cold atoms in trap modes. Relativity is not required for occupation-number notation; it becomes indispensable in relativistic QFT because particle number need not be fixed.

Let $\psi(x)=\sum_i\varphi_i(x)a_i$ and $\psi^\dagger(y)=\sum_j\varphi_j^*(y)a_j^\dagger$ for bosonic modes. Show how the field commutator gives a delta function.

Solution

Using $[a_i,a_j^\dagger]=\delta_{ij}$ ,

[\psi(x),\psi^\dagger(y)] = \sum_{ij} \varphi_i(x)\varphi_j^*(y) [a_i,a_j^\dagger] = \sum_i \varphi_i(x)\varphi_i^*(y).

For a complete orthonormal basis in the continuum position representation,

\sum_i \varphi_i(x)\varphi_i^*(y) = \delta(x-y).

Explain why $[H,N]=0$ expresses particle-number conservation.

Solution

In the Heisenberg picture,

\frac{dN}{dt} = \frac{i}{\hbar}[H,N]

when $N$ has no explicit time dependence. If $[H,N]=0$ , then $dN/dt=0$ , so the total number operator is conserved by the time evolution.

Why is a quantum of a field mode not the same concept as the particle in a one-particle harmonic oscillator potential?

Solution

In the one-particle harmonic oscillator, $\lvert n\rangle$ labels energy eigenstates of one particle in a potential. The ladder operator changes the oscillator excitation level, not the number of particles in the world. In field or Fock-space language, $\lvert n\rangle$ labels $n$ quanta occupying one mode. The algebra is similar, but the Hilbert-space interpretation is different.