3.3 Second quantization

Foundations of second quantization

In first quantization, you write down a wavefunction that depends on the coordinates of every particle: $\Psi(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N)$. For a handful of particles, that's manageable. For $10^{23}$ electrons in a solid, it's not. Second quantization sidesteps this problem by shifting the focus from which particle is where to how many particles occupy each quantum state. Instead of tracking individual coordinates, you work with operators that create or destroy particles in specific states.

This formalism is the standard language for many-body condensed matter physics. It naturally handles indistinguishable particles, enforces the correct quantum statistics, and provides the starting point for techniques like perturbation theory, Feynman diagrams, and Green's functions.

Occupation number representation

The central idea is to label a quantum state by its occupation numbers: the number of particles in each single-particle state. A state is written as $|n_1, n_2, n_3, \ldots\rangle$, where $n_i$ is the number of particles in single-particle state $i$.

• Replaces the need to write out a full $N$-particle wavefunction with explicit coordinates
• For fermions, each $n_i$ can only be 0 or 1 (Pauli exclusion)
• For bosons, each $n_i$ can be 0, 1, 2, 3, ... with no upper limit
• Particle exchange symmetry (bosons) and antisymmetry (fermions) are built in automatically; you don't need to manually symmetrize or antisymmetrize the wavefunction

Creation and annihilation operators

These operators are the workhorses of the formalism. The creation operator $a_i^\dagger$ adds one particle to state $i$, and the annihilation operator $a_i$ removes one particle from state $i$.

For bosons, the action on an occupation number state is:

$a_i^\dagger |n_i\rangle = \sqrt{n_i + 1}\,|n_i + 1\rangle$

$a_i |n_i\rangle = \sqrt{n_i}\,|n_i - 1\rangle$

The prefactors ($\sqrt{n_i+1}$ and $\sqrt{n_i}$) ensure proper normalization. For fermions, the action is simpler since $n_i \in \{0,1\}$, but you pick up sign factors depending on how many occupied states precede state $i$ in your chosen ordering.

The number operator $\hat{n}_i = a_i^\dagger a_i$ counts the particles in state $i$: it satisfies $\hat{n}_i |n_i\rangle = n_i |n_i\rangle$.

Bosons vs. fermions

The algebraic relations that the operators satisfy encode the particle statistics directly.

Bosons (integer spin): commutation relations

$[a_i, a_j^\dagger] = \delta_{ij}, \quad [a_i, a_j] = 0, \quad [a_i^\dagger, a_j^\dagger] = 0$

These allow unlimited occupation of any state. Phonons in a crystal lattice are a key condensed matter example.

Fermions (half-integer spin): anticommutation relations

$\{a_i, a_j^\dagger\} = \delta_{ij}, \quad \{a_i, a_j\} = 0, \quad \{a_i^\dagger, a_j^\dagger\} = 0$

The relation $\{a_i^\dagger, a_i^\dagger\} = 0$ immediately implies $(a_i^\dagger)^2 = 0$, so you can't put two fermions in the same state. That's the Pauli exclusion principle, enforced algebraically. Electrons are the primary fermionic degrees of freedom in solids.

Fock space

Fock space is the Hilbert space that accommodates states with any number of particles. It's the direct sum of all $N$-particle Hilbert spaces:

$\mathcal{F} = \mathcal{H}_0 \oplus \mathcal{H}_1 \oplus \mathcal{H}_2 \oplus \cdots$

This is essential because many processes in condensed matter (phonon emission, Cooper pair formation) involve changing particle number.

Vacuum state

The vacuum state $|0\rangle$ is the state with zero particles. It's the foundation from which all other states are built.

• Every annihilation operator kills it: $a_i|0\rangle = 0$ for all $i$
• It is normalized: $\langle 0|0\rangle = 1$
• Multi-particle states are constructed by acting with creation operators on $|0\rangle$

For example, a two-particle state with one particle in state 1 and one in state 3 is $a_1^\dagger a_3^\dagger |0\rangle$. For fermions, swapping the order gives a minus sign: $a_3^\dagger a_1^\dagger |0\rangle = -a_1^\dagger a_3^\dagger |0\rangle$.

Many-particle states

Any $N$-particle state can be written by applying $N$ creation operators to the vacuum. For bosons, a general basis state is:

$|n_1, n_2, \ldots\rangle = \prod_i \frac{(a_i^\dagger)^{n_i}}{\sqrt{n_i!}} |0\rangle$

The factorial ensures normalization. For fermions, the ordering of operators matters because of the anticommutation relations, and each $n_i$ is restricted to 0 or 1.

These states automatically carry the correct symmetry (symmetric for bosons, antisymmetric for fermions), which is one of the main advantages over first quantization.

Basis states

The set of all occupation number states $|n_1, n_2, \ldots\rangle$ forms a complete orthonormal basis for Fock space:

• Orthonormality: $\langle n_1', n_2', \ldots | n_1, n_2, \ldots \rangle = \prod_i \delta_{n_i', n_i}$
• Completeness: any state in Fock space can be expanded as a superposition of these basis states
• Matrix elements of operators between basis states reduce to algebraic manipulations of creation and annihilation operators, which is far simpler than evaluating multi-dimensional integrals over particle coordinates

Operator formalism

Physical observables (energy, momentum, density) are expressed in terms of creation and annihilation operators. A one-body operator like kinetic energy takes the general form:

$\hat{O}_1 = \sum_{ij} \langle i | O | j \rangle \, a_i^\dagger a_j$

A two-body operator like the Coulomb interaction becomes:

$\hat{O}_2 = \frac{1}{2}\sum_{ijkl} \langle ij | V | kl \rangle \, a_i^\dagger a_j^\dagger a_l a_k$

Note the reversed order of $a_l a_k$ in the two-body term; this matters for getting the signs right with fermions.

Normal ordering

Normal ordering means rearranging an operator product so that all creation operators stand to the left of all annihilation operators. This is denoted by colons: $:\!\hat{O}\!:$

Why bother? The vacuum expectation value of any normal-ordered product is zero: $\langle 0|:\!\hat{O}\!:|0\rangle = 0$. This is useful because it isolates the "interesting" part of an expectation value from trivial vacuum contributions.

When you normal-order fermionic operators, each swap of two operators introduces a minus sign. The extra terms generated during reordering are called contractions, and they connect directly to Wick's theorem.

Wick's theorem

Wick's theorem provides a systematic recipe for evaluating expectation values of products of field operators. It states that a time-ordered product of operators can be decomposed into a sum of all possible normal-ordered terms with all possible contractions:

$T[\hat{A}\hat{B}\hat{C}\hat{D}\cdots] = \sum_{\text{all contractions}} (\text{normal-ordered terms with contractions})$

A contraction of two operators is defined as the difference between their time-ordered and normal-ordered products. For free fields, a contraction equals the free propagator (Green's function).

This theorem is the algebraic engine behind Feynman diagrams: each contraction corresponds to a propagator line, and the sum over all contractions generates all diagrams at a given order in perturbation theory.

Commutation and anticommutation relations

These relations are the axioms of the entire formalism. Everything else follows from them.

Bosons (commutators, denoted $[\cdot,\cdot]$):

$[a_i, a_j^\dagger] = \delta_{ij}, \quad [a_i, a_j] = 0, \quad [a_i^\dagger, a_j^\dagger] = 0$

Fermions (anticommutators, denoted $\{\cdot,\cdot\}$):

$\{a_i, a_j^\dagger\} = \delta_{ij}, \quad \{a_i, a_j\} = 0, \quad \{a_i^\dagger, a_j^\dagger\} = 0$

The $\delta_{ij}$ on the right side means operators for different states commute (or anticommute) freely. Operators for the same state have nontrivial algebra, and that's what enforces the correct counting and statistics.

Applications in condensed matter

Harmonic oscillator systems

The quantum harmonic oscillator is where most students first encounter creation and annihilation operators, and the same algebra carries over directly to lattice vibrations.

In a crystal, atoms vibrate around equilibrium positions. These vibrations can be decomposed into normal modes, and each mode is quantized as an independent harmonic oscillator. The quanta of vibration are phonons. The Hamiltonian for a set of phonon modes is:

$H = \sum_{\mathbf{q},s} \hbar\omega_{\mathbf{q},s}\left(a_{\mathbf{q},s}^\dagger a_{\mathbf{q},s} + \frac{1}{2}\right)$

where $\mathbf{q}$ labels the wavevector and $s$ the polarization branch. This compact form makes it straightforward to calculate thermodynamic quantities like the specific heat (Debye model) and thermal conductivity.

Electron gas models

The simplest model of electrons in a metal treats them as a gas of non-interacting fermions in a uniform positive background (the jellium model). In second quantization, the free-electron Hamiltonian is:

$H_0 = \sum_{\mathbf{k},\sigma} \epsilon_{\mathbf{k}} \, c_{\mathbf{k}\sigma}^\dagger c_{\mathbf{k}\sigma}$

where $\epsilon_{\mathbf{k}} = \hbar^2 k^2 / 2m$ and $\sigma$ is the spin index. Adding the Coulomb interaction introduces two-body terms, and the second-quantized framework lets you treat these perturbatively.

This setup is the starting point for studying:

• Screening (how electrons rearrange to shield a charge)
• Plasmons (collective charge oscillations)
• Fermi liquid theory (Landau's description of interacting fermions as dressed quasiparticles)

Phonon interactions

Electrons in a solid don't just interact with each other; they also couple to lattice vibrations. The electron-phonon interaction in second quantization takes the form:

$H_{e\text{-}ph} = \sum_{\mathbf{k},\mathbf{q},\sigma} g_{\mathbf{q}} \, c_{\mathbf{k}+\mathbf{q},\sigma}^\dagger c_{\mathbf{k},\sigma}(a_{\mathbf{q}} + a_{-\mathbf{q}}^\dagger)$

Here $g_{\mathbf{q}}$ is the coupling constant. This single expression captures processes where an electron absorbs or emits a phonon, changing its momentum by $\mathbf{q}$.

This interaction is responsible for:

• Electrical resistivity at finite temperature (electron-phonon scattering)
• Conventional superconductivity (phonon-mediated Cooper pairing, as in BCS theory)
• Polaron formation (an electron "dressed" by a cloud of virtual phonons)

Second quantization of fields

So far, the discussion has used discrete state labels ($i$, $\mathbf{k}$). Second quantization extends naturally to continuous fields by promoting the wavefunction itself to an operator.

Quantum field theory connection

You define field operators $\hat{\psi}(\mathbf{r})$ and $\hat{\psi}^\dagger(\mathbf{r})$ that annihilate and create a particle at position $\mathbf{r}$. These are related to the discrete operators by:

$\hat{\psi}(\mathbf{r}) = \sum_i \phi_i(\mathbf{r})\, a_i$

where $\phi_i(\mathbf{r})$ are single-particle wavefunctions. The field operators satisfy:

• Bosons: $[\hat{\psi}(\mathbf{r}), \hat{\psi}^\dagger(\mathbf{r}')] = \delta(\mathbf{r} - \mathbf{r}')$
• Fermions: $\{\hat{\psi}(\mathbf{r}), \hat{\psi}^\dagger(\mathbf{r}')\} = \delta(\mathbf{r} - \mathbf{r}')$

This is the bridge to quantum field theory. Particles are treated as excitations of the underlying field, and the same formalism describes both condensed matter systems and relativistic particle physics (with appropriate modifications).

Scalar fields

A scalar field describes spinless particles or collective excitations. In condensed matter, the relevant examples include:

• The order parameter field in Bose-Einstein condensates and superfluids
• Effective low-energy descriptions of phonon modes

The field operator $\hat{\phi}(\mathbf{r})$ and its conjugate $\hat{\phi}^\dagger(\mathbf{r})$ serve as the continuous-space analogs of $a_i$ and $a_i^\dagger$. Scalar field theory is the simplest quantum field theory and serves as a pedagogical entry point before tackling spinor or gauge fields.

Electromagnetic fields

Quantizing the electromagnetic field means expanding it in modes (plane waves in free space, or cavity modes in confined geometries) and assigning each mode its own creation and annihilation operators:

$\hat{\mathbf{A}}(\mathbf{r}) = \sum_{\mathbf{k},\lambda} \sqrt{\frac{\hbar}{2\epsilon_0 \omega_k V}} \left( a_{\mathbf{k}\lambda} e^{i\mathbf{k}\cdot\mathbf{r}} \hat{\epsilon}_{\mathbf{k}\lambda} + a_{\mathbf{k}\lambda}^\dagger e^{-i\mathbf{k}\cdot\mathbf{r}} \hat{\epsilon}_{\mathbf{k}\lambda} \right)$

Each mode behaves as a quantum harmonic oscillator, and the quanta are photons. This quantized description is essential for:

• Spontaneous and stimulated emission in solids
• Cavity quantum electrodynamics (coupling atoms or quantum dots to photon modes)
• The Lamb shift and vacuum fluctuation effects

Many-body systems

Density operators

The particle density operator in second quantization is:

$\hat{\rho}(\mathbf{r}) = \hat{\psi}^\dagger(\mathbf{r})\hat{\psi}(\mathbf{r})$

Its expectation value $\langle \hat{\rho}(\mathbf{r}) \rangle$ gives the average particle density at position $\mathbf{r}$. Fluctuations in the density operator, $\delta\hat{\rho} = \hat{\rho} - \langle\hat{\rho}\rangle$, describe density waves and are central to the study of:

• Charge density waves in low-dimensional materials
• Friedel oscillations around impurities in metals
• The foundation of density functional theory (DFT), which maps the many-body problem onto an effective single-particle problem using the electron density as the fundamental variable

Correlation functions

Correlation functions quantify how the presence of a particle at one point affects the probability of finding another particle elsewhere (or at a later time).

The two-point correlation function (or pair correlation function) is:

$C(\mathbf{r}, \mathbf{r}') = \langle \hat{\psi}^\dagger(\mathbf{r})\hat{\psi}^\dagger(\mathbf{r}')\hat{\psi}(\mathbf{r}')\hat{\psi}(\mathbf{r}) \rangle$

• In a non-interacting Fermi gas, this reveals the exchange hole: fermions avoid each other due to antisymmetry, even without interactions
• Higher-order correlation functions probe more complex ordering, such as superconducting pairing correlations or magnetic order
• Off-diagonal long-range order in the one-body density matrix signals Bose-Einstein condensation or superfluidity

Green's functions

Green's functions (propagators) are among the most powerful tools in many-body physics. The single-particle Green's function is defined as:

$G(\mathbf{r},t;\mathbf{r}',t') = -i\langle T[\hat{\psi}(\mathbf{r},t)\hat{\psi}^\dagger(\mathbf{r}',t')] \rangle$

where $T$ denotes time ordering. Physically, this describes the amplitude for a particle created at $(\mathbf{r}',t')$ to propagate to $(\mathbf{r},t)$.

Green's functions encode:

• The spectral function $A(\mathbf{k},\omega)$, which is what angle-resolved photoemission (ARPES) measures
• Quasiparticle lifetimes and energies through the poles and widths of $G$
• Self-energy corrections $\Sigma$, which capture the effects of interactions via Dyson's equation: $G = G_0 + G_0 \Sigma G$

Symmetries and conservation laws

Symmetries in the Hamiltonian lead to conserved quantities (Noether's theorem). Second quantization provides a natural way to express both the symmetry transformations and the associated conservation laws.

Particle number conservation

If the Hamiltonian commutes with the total number operator $\hat{N} = \sum_i a_i^\dagger a_i$, then total particle number is conserved. This corresponds to a global U(1) symmetry: the physics is unchanged if you multiply all creation operators by $e^{i\theta}$ and all annihilation operators by $e^{-i\theta}$.

Particle number conservation can be broken in mean-field treatments of superconductivity (BCS theory), where the order parameter $\langle a_{\mathbf{k}\uparrow} a_{-\mathbf{k}\downarrow} \rangle \neq 0$ mixes states with different particle numbers. This spontaneous U(1) symmetry breaking is a defining feature of the superconducting state.

Angular momentum

Angular momentum operators are expressed in second quantization by labeling single-particle states with angular momentum quantum numbers $(l, m)$ or $(j, m_j)$ for spin-orbit coupled systems.

• Orbital angular momentum arises from the spatial part of the wavefunction
• Spin angular momentum is carried by the internal degree of freedom ($\uparrow, \downarrow$ for spin-1/2 fermions)
• Spin-orbit coupling, which mixes these two, is crucial in topological insulators and heavy-fermion materials
• Selection rules for transitions between states follow directly from the commutation relations of angular momentum operators with the interaction Hamiltonian

Parity

Parity is the symmetry under spatial inversion: $\mathbf{r} \to -\mathbf{r}$. In second quantization, the parity operator transforms field operators as $\hat{\psi}(\mathbf{r}) \to \hat{\psi}(-\mathbf{r})$.

• States in crystals with inversion symmetry can be classified as even or odd parity
• Parity determines optical selection rules (which transitions are allowed or forbidden)
• Broken inversion symmetry in a crystal enables phenomena like ferroelectricity (spontaneous electric polarization) and Rashba spin-orbit coupling (momentum-dependent spin splitting at surfaces or interfaces)

Computational techniques

Diagrammatic methods

Feynman diagrams translate the algebra of second quantization into a visual language. Each element of a diagram corresponds to a specific mathematical object:

1. Propagator lines (straight for electrons, wavy for photons, dashed for phonons) represent free Green's functions $G_0$
2. Vertices represent interaction matrix elements
3. Internal lines are summed/integrated over all intermediate states and times
4. External lines represent the incoming/outgoing particles of the process you're calculating

Dyson's equation resums an infinite series of diagrams to give the full (dressed) Green's function. Vertex corrections account for how interactions modify the coupling between particles and external probes. The diagrammatic approach turns a complicated many-body problem into a systematic, order-by-order calculation.

Perturbation theory

When interactions are weak compared to the kinetic energy, you can expand observables as a power series in the interaction strength.

1. Start with the exactly solvable non-interacting Hamiltonian $H_0$
2. Treat the interaction $H_1$ as a perturbation
3. Use Wick's theorem to evaluate each term in the expansion as a sum of contractions
4. Organize the terms using Feynman diagrams
5. Resum selected classes of diagrams (e.g., ring diagrams for the random phase approximation)

Perturbation theory works well for weakly correlated systems like simple metals. It breaks down for strongly correlated materials (Mott insulators, heavy fermions), where non-perturbative methods are needed.

Numerical simulations

For systems where perturbation theory fails, numerical methods formulated in the second-quantized framework become essential:

• Exact diagonalization: construct the full Hamiltonian matrix in Fock space and diagonalize it. Limited to small systems (tens of sites) due to exponential growth of the Hilbert space.
• Quantum Monte Carlo (QMC): stochastically sample many-body configurations. Powerful for bosonic systems, but fermionic systems suffer from the sign problem (oscillating contributions that make statistical averaging difficult).
• Density matrix renormalization group (DMRG): a variational method based on matrix product states, extremely effective for one-dimensional and quasi-one-dimensional systems. Captures strong correlations that perturbation theory misses.

Limitations and extensions

Beyond second quantization

Standard second quantization assumes you can expand in a fixed single-particle basis and treat interactions perturbatively. This fails for strongly correlated systems where interaction energies are comparable to or larger than kinetic energies.

Approaches that go beyond the standard framework include:

• Dynamical mean-field theory (DMFT): maps the lattice problem onto a self-consistent impurity problem, capturing local correlations exactly
• Functional renormalization group (fRG): a non-perturbative flow equation approach that tracks how effective interactions evolve as high-energy modes are integrated out
• Tensor network methods: represent many-body states as networks of tensors, generalizing DMRG to higher dimensions
• Topological considerations: Berry phases and Chern numbers classify quantum states in ways that aren't captured by standard perturbative second quantization

Relativistic considerations

Standard second quantization in condensed matter is non-relativistic. Extending it to relativistic systems requires:

• Replacing the Schrödinger field with the Dirac field, whose operators create both particles and antiparticles
• Dealing with negative-energy states and the filled Dirac sea (or equivalently, antiparticle creation)
• Accounting for vacuum polarization and pair creation processes

In condensed matter, relativistic-like physics appears in systems such as graphene (where electrons near the Dirac points obey a 2D massless Dirac equation) and topological surface states. Connections between condensed matter and high-energy physics, such as the AdS/CFT correspondence applied to strongly correlated electron systems, are an active area of research.

Non-equilibrium systems

Equilibrium second quantization uses imaginary-time (Matsubara) formalism. For systems driven out of equilibrium, you need the Keldysh formalism, which works on a closed time contour in real time.

• The Keldysh contour has a forward branch and a backward branch, leading to a $2 \times 2$ matrix structure for Green's functions
• This framework handles quantum transport (current through a nanoscale junction), pump-probe spectroscopy, and ultrafast dynamics
• Quantum quenches (sudden changes in Hamiltonian parameters) and thermalization in isolated quantum systems are studied using non-equilibrium Green's functions
• Connections to open quantum systems (Lindblad master equations) and quantum information theory are areas of active development