Fock Space Construction

Fock space is a Hilbert completion of finite-particle sectors, together with a systematic way to lift one-particle maps and dynamics. The completion alone does not define an unbounded second-quantized Hamiltonian; its sector domains and square-summability condition must also be specified. This page gives that construction and proves the bosonic exponential-vector factorization.

Required background. Symmetric and Antisymmetric Tensor Products constructs the closed statistics sectors. Many-Body Hilbert Spaces defines their independent-particle generators. The Fock-space reader bridge explains sector probabilities and number moments.

Complete the algebraic sum of particle sectors

Let $\mathcal h$ be a complex Hilbert space. Set $\mathcal h_\pm^{(0)}=\mathbb C$ and, for $n\geq1$,

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

Give the algebraic direct sum of these spaces the orthogonal-sum inner product and complete it:

$$\mathcal F_\pm(\mathcal h) =\left\{\Psi=(\psi_n)_{n\geq0}: \psi_n\in\mathcal h_\pm^{(n)},\quad \sum_n\|\psi_n\|^2<\infty\right\}.$$

The vacuum is $\Omega=(1,0,\ldots)$. Finite-sector truncations converge in norm by square summability, so the finite-particle subspace is dense. This construction is basis-independent. An occupation basis is a subsequent choice of coordinates, described in Bosonic Fock Space and Fermionic Fock Space.

The finite-particle subspace includes arbitrary vectors within each occupied sector. It should not be confused with a finite-dimensional mode cutoff.

Exponential vectors form a total bosonic set

For $f\in\mathcal h$, define the unnormalized exponential vector

$$\varepsilon(f)=\bigoplus_{n=0}^\infty \frac{f^{\otimes n}}{\sqrt{n!}}.$$

The defining series converges because

$$\langle\varepsilon(f),\varepsilon(g)\rangle =\sum_{n=0}^\infty\frac{\langle f,g\rangle^n}{n!} =e^{\langle f,g\rangle}.$$

In particular $\|\varepsilon(f)\|^2=e^{\|f\|^2}$.

Their linear span is dense. If $\Psi$ is orthogonal to every $\varepsilon(f)$, then for fixed $f$ the entire function

$$z\longmapsto\langle\Psi,\varepsilon(zf)\rangle$$

vanishes identically. Its Taylor coefficients give $\langle\psi_n,f^{\otimes n}\rangle=0$ for every $n,f$. Polarization expresses arbitrary symmetric tensors through linear combinations of pure powers, so every $\psi_n$ vanishes. Thus $\Psi=0$.

The totality proof is useful: identities checked on exponential vectors can determine bounded operators uniquely.

Orthogonal mode sums factorize bosonic Fock space

Suppose $\mathcal h=\mathcal h_A\oplus\mathcal h_B$. Define on exponential vectors

$$T\varepsilon(f_A\oplus f_B) =\varepsilon(f_A)\otimes\varepsilon(f_B).$$

The exponential inner-product formula shows that $T$ preserves pairings. The totality result proves that its domain span is dense, and that its range span is dense in the tensor-product Fock space. It therefore extends uniquely to a unitary

$$\mathcal F_+(\mathcal h_A\oplus\mathcal h_B) \simeq\mathcal F_+(\mathcal h_A)\otimes\mathcal F_+(\mathcal h_B).$$

This is a mode decomposition. Within a fixed total-particle sector it sums over all allocations of particle number between the two mode families.

Fermionic Fock spaces also admit a factorization after an ordering is fixed, but odd field operators require a parity grading. The sign issue belongs to CAR and CCR Algebras; the bosonic map above must not be used to infer ordinary commutation of fermion fields.

Multiplicative second quantization preserves contractions

For a contraction $C:\mathcal h\to\mathcal k$, define

$$\Gamma_\pm(C)=\bigoplus_{n=0}^\infty C^{\otimes n} \big|_{\mathcal h_\pm^{(n)}}.$$

The vacuum block is the identity on $\mathbb C$, and $\|C^{\otimes n}\|\leq1$, so $\Gamma_\pm(C)$ is bounded with norm one. Tensor-product composition gives $\Gamma(CD)=\Gamma(C)\Gamma(D)$ when the maps compose, and $\Gamma(C)^*=\Gamma(C^*)$.

If $U$ is unitary, $\Gamma(U)$ is unitary. In the bosonic case $\Gamma(C)\varepsilon(f)=\varepsilon(Cf)$. For a general bounded map with norm greater than one, boundedness of the full direct sum is not automatic.

Additive second quantization has a precise domain

Let $h$ be self-adjoint. On the $n$-particle sector, let $h^{(n)}$ be the self-adjoint sum of its $n$ strongly commuting copies, restricted to the statistics sector; set $h^{(0)}=0$. Define

$$\begin{aligned} d\Gamma(h)\Psi&=(h^{(n)}\psi_n)_{n\geq0},\\ D(d\Gamma(h)) &=\left\{\Psi:\psi_n\in D(h^{(n)}),\quad \sum_n\|h^{(n)}\psi_n\|^2<\infty\right\}. \end{aligned}$$

A direct sum of self-adjoint operators on this domain is self-adjoint. Equivalently, $\Gamma(e^{-ith})$ is a strongly continuous unitary group: prove continuity on finitely many sectors and control the omitted tail by unitarity. Its generator is precisely the displayed operator, so

$$e^{-it\,d\Gamma(h)}=\Gamma(e^{-ith}).$$

For $h=I$, this is the number operator. Its form and operator domains, which differ, are explained in the reader bridge.

Finite-sector vectors built from an operator core of $h$ form a core for $d\Gamma(h)$. One first truncates the sector sum in graph norm, then applies one-particle spectral cutoffs within each remaining sector and approximates by algebraic tensors. Bare Hilbert-norm density would not prove the core property.

Exercises

1. When $\mathcal h=\mathbb C$ and $C$ is multiplication by $c$ with $|c|>1$, is bosonic $\Gamma(C)$ bounded?

Solution

Its action on the normalized $n$-particle vector is multiplication by $c^n$. Those norms are unbounded in $n$, so the direct-sum operator is unbounded. By contrast, one-mode fermionic Fock space has only the vacuum and one-particle blocks.

2. Suppose $h\geq\epsilon I$ with $\epsilon>0$. Relate finite mean energy to finite mean number.

Solution

On each sector $h^{(n)}\geq n\epsilon I$, hence $d\Gamma(h)\geq\epsilon N$ in the quadratic-form sense. Finite mean second-quantized energy therefore implies finite mean number. The implication can fail for gapless $h$, where arbitrarily many particles can occupy sufficiently low-energy modes.

References

• J. Dereziński, “Introduction to representations of the canonical commutation and anticommutation relations,” in Large Coulomb Systems, Lecture Notes in Physics 695, Springer, 2006, pp. 63–143, doi:10.1007/3-540-32579-4_3, author text. — Sections 5.1–5.6.