Second-Quantized Bosons and Fermions

Second quantization encodes indistinguishable particles by operator-valued fields whose equal-time commutators or anticommutators fix statistics. Once the one-particle basis and its normalization are declared, number operators, one-body dynamics, interactions, and local observables follow without attaching particle labels to identical quanta. A standard construction with the same normalization logic is given in Fetter and Walecka 2003, ch. 1.

Required background. Galilean Fields, Scales, and Low-Energy Degrees of Freedom fixes nonrelativistic normalization and mass; Canonical Quantization: Algebra, Representation, and State fixes the algebra–representation distinction; Fock Space, Vacuum, and Particle Number supplies the Fock construction.

Helpful background. Multiparticle States, Statistics, and Fock Organization reviews symmetrized and antisymmetrized sectors.

Field operators and normalization

For species or spin index $\alpha$, define at equal time

$$[\psi_\alpha(\mathbf x),\psi_\beta^\dagger(\mathbf y)]_\zeta =\delta_{\alpha\beta}\delta^{(d)}(\mathbf x-\mathbf y), \qquad [A,B]_\zeta\equiv AB-\zeta BA,$$

with $\zeta=+1$ for bosons and $\zeta=-1$ for fermions. The remaining equal-time brackets vanish. For fermions this notation gives the anticommutator. These are operator relations on a dense finite-particle domain; products at the same point require a regulator.

In a periodic box of volume $V=L^d$,

$$\psi_\alpha(\mathbf x) =\frac{1}{\sqrt V}\sum_{\mathbf k} e^{i\mathbf k\cdot\mathbf x}a_{\mathbf k\alpha}, \qquad [a_{\mathbf k\alpha},a^\dagger_{\mathbf k'\beta}]_\zeta =\delta_{\mathbf k\mathbf k'}\delta_{\alpha\beta}.$$

As $V\to\infty$,

$$\frac1V\sum_{\mathbf k}\longrightarrow \int\frac{\mathrm d^d k}{(2\pi)^d}, \qquad (2\pi)^d\delta^{(d)}(\mathbf k-\mathbf k') \longleftrightarrow V\delta_{\mathbf k\mathbf k'}.$$

The factors of $V$ are fixed by the equal-time delta function. They are not cosmetic: an error here changes densities and matrix elements.

Number and Hamiltonian operators

The local density and total number are

$$n_\alpha(\mathbf x)=\psi_\alpha^\dagger\psi_\alpha, \qquad N=\sum_\alpha\int\mathrm d^d x\,n_\alpha(\mathbf x).$$

Using the canonical algebra,

$$[N,\psi_\alpha]=-\psi_\alpha, \qquad [N,\psi_\alpha^\dagger]=+\psi_\alpha^\dagger,$$

for either statistics. A general one-body operator $h$ becomes

$$H_1=\sum_{\alpha\beta} \int\mathrm d^d x\,\mathrm d^d y\, \psi_\alpha^\dagger(\mathbf x) h_{\alpha\beta}(\mathbf x,\mathbf y) \psi_\beta(\mathbf y).$$

For a symmetric two-body potential,

$$H_2=\frac12\sum_{\alpha\beta} \int\mathrm d^d x\,\mathrm d^d y\, \psi_\alpha^\dagger(\mathbf x) \psi_\beta^\dagger(\mathbf y) V_{\alpha\beta}(\mathbf x-\mathbf y) \psi_\beta(\mathbf y) \psi_\alpha(\mathbf x).$$

The factor $1/2$ avoids counting an unordered pair twice. Operator ordering is part of the definition. For a zero-range potential the expression is regulated and its coupling is matched to scattering data; $V(0)$ is not a physical parameter.

Normal ordering, denoted $:{\cdots}:$, moves creation operators left with the fermionic signs required by the algebra. It subtracts contractions relative to a named reference vacuum or state. At finite density, “normal ordered” is ambiguous unless that reference is stated.

Continuum and lattice bases

Let $w_i(\mathbf x)$ be orthonormal localized orbitals. Then

$$\psi_\alpha(\mathbf x)=\sum_iw_i(\mathbf x)c_{i\alpha}+\text{discarded bands}, \qquad [c_{i\alpha},c_{j\beta}^\dagger]_\zeta =\delta_{ij}\delta_{\alpha\beta}.$$

Projection produces matrix elements

$$t_{ij}=\int\mathrm d^d x\, w_i^*(\mathbf x)h_0w_j(\mathbf x),$$

and four-index interactions. A single-site Hubbard parameter is an approximation obtained by retaining selected matrix elements, not an identity. Completeness within the retained band and the gap to discarded bands control the map.

For a simple real-space grid of cell volume $a^d$, the normalization is

$$c_{i\alpha}=a^{d/2}\psi_\alpha(\mathbf x_i), \qquad \sum_i c_{i\alpha}^\dagger c_{i\alpha} \longrightarrow \int\mathrm d^d x\,\psi_\alpha^\dagger\psi_\alpha.$$

This relation provides a quick dimension check for every lattice-to-continuum formula.

A two-site statistics check

Let $i\ne j$. For fermions,

$$c_i^\dagger c_j^\dagger|0\rangle =-c_j^\dagger c_i^\dagger|0\rangle, \qquad (c_i^\dagger)^2|0\rangle=0.$$

For bosons, the exchange sign is positive and multiple occupation is allowed. Both theories use occupation numbers; statistics enters through the algebra, not through labels assigned afterward.

The one-body density matrix

$$\rho^{(1)}_{\alpha\beta}(\mathbf x,\mathbf y) =\langle\psi_\beta^\dagger(\mathbf y)\psi_\alpha(\mathbf x)\rangle$$

is positive semidefinite: for every test function $f$, $\langle A^\dagger A\rangle\ge0$ with $A=\int f_\alpha^*(\mathbf x)\psi_\alpha(\mathbf x)$. Its trace is $\langle N\rangle$. These basis-independent facts are useful checks on numerical density matrices.

Common pitfalls

Losing the basis normalization. Continuum fields have dimension $L^{-d/2}$, while lattice operators are dimensionless. The factor $a^{d/2}$ cannot be dropped.

Using bosonic algebra with a Pauli exclusion rule added by hand. Fermionic exchange and occupancy follow from anticommutation. A separate rule is redundant and often inconsistent.

Treating normal ordering as state independent. Vacuum normal ordering and Fermi-sea normal ordering subtract different contractions. State which one is used before comparing energies or diagrams.

Exercises

Recover the continuum delta function

Use the box expansion to derive $[\psi_\alpha(\mathbf x),\psi_\beta^\dagger(\mathbf y)]_\zeta$.

Solution

Substitution gives

$$[\psi_\alpha(\mathbf x),\psi_\beta^\dagger(\mathbf y)]_\zeta =\frac{\delta_{\alpha\beta}}{V} \sum_{\mathbf k}e^{i\mathbf k\cdot(\mathbf x-\mathbf y)}.$$

The periodic Fourier completeness relation equals the box delta function and tends to $\delta^{(d)}(\mathbf x-\mathbf y)$ in the infinite-volume limit.

Check number conservation of a two-body interaction

Show that the displayed $H_2$ commutes with $N$.

Solution

Using $[N,\psi]=-\psi$ and $[N,\psi^\dagger]=\psi^\dagger$, each term in $H_2$ receives $+1+1-1-1=0$. Therefore $[N,H_2]=0$, independently of statistics and the detailed potential, provided the operator domains and regulator respect the algebra.

Continue

Coherent-State Path Integrals for Many-Body Systems converts normal-ordered operators to a regulated functional integral. From Microscopic Hamiltonians to Continuum Fields projects between bases while retaining matching errors. Densities, Currents, and Nonrelativistic Ward Identities defines the observable currents generated by this algebra.

References

• Fetter, Alexander L., and John Dirk Walecka. Quantum Theory of Many-Particle Systems. Mineola, NY: Dover, 2003; originally published 1971. Publisher record.

Further reading

• Berezin, Felix A. The Method of Second Quantization. New York: Academic Press, 1966. Stable catalog record.
• Negele, John W., and Henri Orland. Quantum Many-Particle Systems. Boca Raton, FL: CRC Press, 2018; originally published 1988. DOI.