Second Quantization: Bridge to QFT

Second quantization expresses many-particle quantum mechanics through occupations and operators that add or remove quanta. It does not quantize the same system a second time: on each fixed-number sector, it reproduces the original symmetric or antisymmetric wavefunction theory.

Nonrelativistic field theory is already quantum field theory. The further step to relativistic QFT adds Lorentz symmetry, local spacetime fields, and causal constraints. Fock-space methods provide useful tools for that step without making the two settings identical.

Helpful background. Fock Space gives particle sectors, and Creation and Annihilation Operators supplies the mode algebra.

Particle sectors and field descriptions

An occupation state $|n_1,n_2,\ldots\rangle$ specifies excitations of complete one-particle modes. The same mode language describes atoms in traps, photons in cavities, and phonons in a lattice. Their dispersion, conserved quantities, and effective range of validity differ; the operator notation does not erase those physical differences.

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.

Hamiltonians expressed through fields

A nonrelativistic kinetic-plus-potential operator becomes

$$H_1=\int d^3x\,\psi^\dagger(\mathbf x) \left(-\frac{\hbar^2\nabla^2}{2m}+U(\mathbf x)\right)\psi(\mathbf x).$$

The field expression acts on each particle and preserves total number. One-Body Operators establishes this equivalence; Two-Body Operators adds pair interactions using four fields. Number-changing effective terms are also possible in nonrelativistic models, so their presence alone does not establish relativistic physics.

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.

Relativistic locality and particle interpretation

The equal-time nonrelativistic relation $[\psi(\mathbf x),\psi^\dagger(\mathbf y)]=\delta^{(3)}(\mathbf x-\mathbf y)$ does not impose a relativistic light cone. A free Schrödinger wavepacket has tails at arbitrarily large distances at any positive time.

Relativistic microcausality instead requires local observables to commute at spacelike separation. Bosonic fields commute there and fermionic fields obey the corresponding graded anticommutation condition; local even fermionic observables commute. A real scalar field contains both particle annihilation and creation pieces, while a charged relativistic field combines particle annihilation with antiparticle creation. It is not simply the nonrelativistic annihilation field with a different name.

Free or asymptotic particles often admit a Fock description. An interacting infinite-volume theory need not have a single global identification with the free-particle Fock representation. Condensed-matter thermodynamic limits can also require inequivalent representations; this is not a uniquely relativistic subtlety.

From relativistic dispersion to slow particles

A massive free relativistic mode has

$$E_{\mathbf k}=\sqrt{m^2c^4+\hbar^2c^2|\mathbf k|^2}.$$

For $\hbar|\mathbf k|\ll mc$, subtracting the rest energy gives $E_{\mathbf k}-mc^2\approx\hbar^2|\mathbf k|^2/(2m)$. This is the Schrödinger dispersion, but it is a low-energy expansion. Restricting to a particle sector also assumes energies and interactions insufficient to produce particle–antiparticle pairs. The rest-energy subtraction is a constant within a fixed-N sector and an $mc^2N$ term when comparing sectors.

Canonical Quantization introduces the field-quantization framework. Nonrelativistic Field Theory develops the Schrödinger-field action and contact matching. For collective fields, imaginary time, and emergent relativistic behavior, continue to Why Many-Body QM Leads to QFT.

Exercises

1. 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.

2. 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.

3. **First relativistic correction.** Expand the free dispersion through fourth order in $p/(mc)$, where $p=\hbar|\mathbf k|$. What relative correction does it give to the nonrelativistic kinetic energy?

Solution

Using $\sqrt{1+x}=1+x/2-x^2/8+O(x^3)$,

$$E-mc^2=\frac{p^2}{2m}-\frac{p^4}{8m^3c^2}+O\!\left(\frac{p^6}{m^5c^4}\right).$$

The first correction is a fraction $-p^2/(4m^2c^2)$ of $p^2/(2m)$. It is small in the stated slow-particle regime; the truncated polynomial is not a reliable high-momentum Hamiltonian.

References

• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• M. Srednicki, Quantum Field Theory, Cambridge University Press, 2007.
• 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.