Fock Space, Vacuum, and Particle Number

Choosing the future-positive-frequency one-particle space for a free massive real scalar selects both a normalized Minkowski vacuum and a concrete representation of the canonical commutation relations. Its Hilbert space is the symmetric Fock space $\mathcal F_s(\mathcal H_1)=\bigoplus_{n\geq0}\operatorname{Sym}^n\mathcal H_1$: the vacuum is its zero-particle vector, creation operators generate a dense finite-particle subspace, and $N=\mathrm d\Gamma(I)$ has eigenvalue $n$ on the $n$-particle sector. These structures are exact in the selected free representation, but the abstract commutation relations alone do not select them. A unitary change of packet basis within $\mathcal H_1$ preserves the vacuum and total $N$, whereas a change that mixes positive and negative frequencies changes the annihilators, vacuum, and particle count.

The discussion stays in $d$-dimensional Minkowski spacetime with $m>0$ and the standard positive-frequency split. A periodic box with a finite, inversion-symmetric mode set makes occupation products ordinary finite products; the continuum construction is then stated with its operator domains. No claim is made that this free Fock representation is the exact physical Hilbert space of an interacting theory, that curved spacetime has a preferred vacuum, or that a finite Bogoliubov calculation proves inequivalence of representations.

Required background. Quantizing the Real Scalar Field supplies the normalized one-particle space, smeared creators and annihilators, selected Minkowski vacuum, and free Hamiltonian used below.

Helpful background. Vacua, States, and Representations supplies the algebra–state–representation distinction. Multiparticle States, Statistics, and Fock Organization supplies the general symmetric-sector kinematics and normalization that are specialized here to one real scalar field.

Symmetric Fock space over the scalar one-particle space

The preceding construction gives the positive-energy mass-shell space

$$\begin{aligned} \mathcal H_1 &= L^2(\mathcal H_m^+,\mathrm d\Pi_{\mathbf p}), \\ \mathrm d\Pi_{\mathbf p} &= \frac{\mathrm d^{d-1}\mathbf p} {(2\pi)^{d-1}2E_{\mathbf p}} . \end{aligned}$$

For $f,g\in\mathcal H_1$, the convention is that $a(f)$ is antilinear, $a^\dagger(f)$ is linear, and

$$[a(f),a^\dagger(g)] = \langle f|g\rangle_1 .$$

The scalar symmetric Fock space is the Hilbert direct sum

$$\begin{aligned} \mathcal F_s(\mathcal H_1) &= \bigoplus_{n=0}^{\infty}\mathcal F_s^{(n)}, \\ \mathcal F_s^{(0)} &= \mathbb C|0\rangle, \\ \mathcal F_s^{(n)} &= \operatorname{Sym}^n\mathcal H_1 \qquad (n\geq1). \end{aligned}$$

Thus a Fock vector is a sequence $\Psi=(\psi_0,\psi_1,\ldots)$ with $\psi_n\in\mathcal F_s^{(n)}$ and

$$\|\Psi\|^2 = \sum_{n=0}^{\infty}\|\psi_n\|^2 < \infty .$$

The direct sum is the Fock space. It becomes the selected Fock representation when the scalar commutation relations act on it through the operators $a(f)$ and $a^\dagger(f)$ and the chosen state is represented by $|0\rangle$. This distinction matters because an abstract Hilbert-space isomorphism does not by itself intertwine two representations of the field algebra.

The vacuum and the represented creators obey

$$a(f)|0\rangle=0, \qquad \|0\rangle\|=1.$$

Let $\mathcal D_{\mathrm{fin}}$ denote the Fock vectors with only finitely many nonzero sector components:

$$\mathcal D_{\mathrm{fin}} = \left\{ \Psi:\psi_n=0\ \text{eventually} \right\},$$

It is a dense invariant working domain. Finite linear combinations of vectors

$$a^\dagger(f_1)\cdots a^\dagger(f_n)|0\rangle$$

are dense in the full Fock space, so the vacuum is cyclic for the global polynomial algebra generated by these represented operators. This is not a claim about cyclicity for every local algebra, nor a uniqueness statement about vacua in other representations.

On $\psi_n\in\mathcal F_s^{(n)}$, the sector estimates

$$\begin{aligned} \|a^\dagger(f)\psi_n\| &\leq \sqrt{n+1}\,\|f\|\,\|\psi_n\|, \\ \|a(f)\psi_n\| &\leq \sqrt n\,\|f\|\,\|\psi_n\| \end{aligned}$$

show both the adjacent-sector action and the growth that makes creators and annihilators unbounded on the full Fock space. For a normalized packet $\|f\|=1$, the repeated-mode vector

$$|n_f\rangle = \frac{[a^\dagger(f)]^n}{\sqrt{n!}}|0\rangle$$

has unit norm. The factorial follows recursively from $[a(f),a^\dagger(f)]=1$; it is not a universal normalization for products of nonorthogonal packets. The direct-sum construction, cyclic vacuum, and occupation-number organization are developed in Coleman 2019, §§ 2.1–2.4, pp. 17–30.

The regulated scalar occupation basis

First retain the predecessor’s periodic box and finite inversion-symmetric set of momenta $K_\Lambda$. Its oscillator algebra is

$$[b_{\mathbf k},b^\dagger_{\mathbf l}] = \delta_{\mathbf k\mathbf l}, \qquad b_{\mathbf k}|0\rangle=0 .$$

For occupation numbers $n_{\mathbf k}\in\mathbb N_0$, define

$$\begin{aligned} |\{n_{\mathbf k}\}\rangle &= \prod_{\mathbf k\in K_\Lambda} \frac{(b^\dagger_{\mathbf k})^{n_{\mathbf k}}} {\sqrt{n_{\mathbf k}!}} |0\rangle . \end{aligned}$$

Oscillators at distinct momenta commute, while for each mode

$$\langle0| b_{\mathbf k}^{\,n} (b^\dagger_{\mathbf k})^n |0\rangle = n! .$$

Consequently the displayed product states are normalized and mutually orthogonal. With

$$\begin{aligned} N_{\mathbf k} &= b^\dagger_{\mathbf k}b_{\mathbf k}, \\ N_\Lambda &= \sum_{\mathbf k\in K_\Lambda}N_{\mathbf k}, \end{aligned}$$

the commutator $[N_{\mathbf k},b^\dagger_{\mathbf l}] =\delta_{\mathbf k\mathbf l}b^\dagger_{\mathbf l}$ gives

$$\begin{aligned} N_{\mathbf k}|\{n_{\mathbf l}\}\rangle &= n_{\mathbf k}|\{n_{\mathbf l}\}\rangle, \\ N_\Lambda|\{n_{\mathbf l}\}\rangle &= \left( \sum_{\mathbf k\in K_\Lambda}n_{\mathbf k} \right) |\{n_{\mathbf l}\}\rangle . \end{aligned}$$

The already-derived regulated Hamiltonian becomes diagonal in the same basis:

$$\begin{aligned} H_\Lambda &= E_{0,\Lambda} + \sum_{\mathbf k\in K_\Lambda} \omega_{\mathbf k}N_{\mathbf k}, \\ H_{\mathrm{exc},\Lambda} &\equiv H_\Lambda-E_{0,\Lambda}. \end{aligned}$$

It follows directly that

$$[H_{\mathrm{exc},\Lambda},N_\Lambda]=0.$$

This commutator is a property of the free Hamiltonian, not part of the definition of Fock space. The vacuum term $E_{0,\Lambda}$ has also been kept: changing its treatment belongs to Normal Ordering and Vacuum Terms.

A finite spatial box alone still has countably many momentum modes. The additional finite set $K_\Lambda$ is what makes every product and sum above an ordinary finite-mode oscillator calculation.

The continuum number operator and its domain

The basis-independent continuum definition is the second quantization of the identity on the selected one-particle space:

$$\begin{aligned} N &= \mathrm d\Gamma(I_{\mathcal H_1}), \\ N(\psi_0,\psi_1,\psi_2,\ldots) &= (0,\psi_1,2\psi_2,\ldots). \end{aligned}$$

It is a nonnegative self-adjoint operator on

$$\mathcal D(N) = \left\{ \Psi: \sum_{n=0}^{\infty} n^2\|\psi_n\|^2 < \infty \right\}.$$

Not every normalized Fock vector belongs to this domain. Finite expected particle number requires the weaker quadratic-form condition

$$\Psi\in\mathcal D(N^{1/2}) \quad\Longleftrightarrow\quad \sum_{n=0}^{\infty} n\|\psi_n\|^2 < \infty .$$

On $\mathcal D_{\mathrm{fin}}$,

$$\begin{aligned} [N,a^\dagger(f)] &= a^\dagger(f), \\ [N,a(f)] &= -a(f), \\ N|0\rangle &= 0. \end{aligned}$$

These identities say exactly that creation and annihilation change particle number by one. They also give a quick consistency check on the free Hamiltonian: since $H_{\mathrm{exc}}=\mathrm d\Gamma(E)$ preserves each $\mathcal F_s^{(n)}$, $[H_{\mathrm{exc}},N]=0$ on a common invariant core.

For any countable orthonormal packet basis $\{e_j\}$ of $\mathcal H_1$, the finite-support occupation vectors are

$$\begin{aligned} |\{n_j\}\rangle &= \prod_j \frac{[a^\dagger(e_j)]^{n_j}} {\sqrt{n_j!}} |0\rangle, \\ N|\{n_j\}\rangle &= \left(\sum_j n_j\right)|\{n_j\}\rangle . \end{aligned}$$

Changing this orthonormal basis changes the individual mode occupations but not $N=\mathrm d\Gamma(I)$. Equivalently, its quadratic form on $\mathcal D(N^{1/2})$ is

$$\|N^{1/2}\Psi\|^2 = \sum_j\|a(e_j)\Psi\|^2,$$

and the value is independent of the orthonormal basis used to evaluate the sum.

The familiar expression $\int_{\mathbf p}a^\dagger(\mathbf p)a(\mathbf p)$ is therefore distributional shorthand. A sharp-momentum $a^\dagger(\mathbf p)a(\mathbf p)$ is not an ordinary infinite-volume mode-number operator; packet modes, the regulated box, or $\mathrm d\Gamma(I)$ supply the controlled definitions.

For a real scalar, $N$ is also not a Noether charge. The field contains a creation and an annihilation part and therefore connects adjacent number sectors; interactions can mix those sectors, and their eigenspaces are not automatically superselection sectors. Complex Scalars and Conserved Charge separates particle number from the genuine $U(1)$ charge of a complex field.

Changing the positive-frequency split

A unitary change of orthonormal packets inside the same $\mathcal H_1$ does not alter the positive-frequency subspace. If $U$ is unitary on $\mathcal H_1$, its symmetric second quantization acts sector by sector:

$$\Gamma_s(U) = \bigoplus_{n=0}^{\infty}U^{\otimes_s n}.$$

It fixes $|0\rangle$ and commutes with $N$, so this is a change of one-particle basis, not a new particle concept.

A Bogoliubov change is different because it mixes annihilators and creators. Write $|0_b\rangle\equiv|0\rangle$ for the vacuum satisfying $b_{\mathbf l}|0_b\rangle=0$ for every regulated mode. For a pair of distinct regulated modes $\mathbf k$ and $-\mathbf k$, let

$$\begin{aligned} \widetilde b_{\mathbf k} &= \cosh r\,b_{\mathbf k} + \sinh r\,b^\dagger_{-\mathbf k}, \\ \widetilde b_{-\mathbf k} &= \cosh r\,b_{-\mathbf k} + \sinh r\,b^\dagger_{\mathbf k}. \end{aligned}$$

The hyperbolic identity gives the same oscillator algebra:

$$\begin{aligned} [\widetilde b_{\mathbf k}, \widetilde b^\dagger_{\mathbf k}] &= \cosh^2r-\sinh^2r = 1, \\ [\widetilde b_{\mathbf k}, \widetilde b_{-\mathbf k}] &= 0. \end{aligned}$$

Nevertheless the old vacuum is not empty according to the transformed number operator:

$$\langle0_b| \widetilde b^\dagger_{\mathbf k} \widetilde b_{\mathbf k} |0_b\rangle = \sinh^2r .$$

For $r\neq0$, the vector annihilated by all $\widetilde b$ operators is not $|0_b\rangle$. This proves that the commutation relations alone do not fix which state is called the vacuum or which operator is called particle number.

With finitely many regulated modes, the transformation is implemented by a unitary squeezing operator. The two Fock constructions are then unitarily equivalent even though their vacuum vectors and particle assignments differ. For infinitely many modes, unitary implementability is an additional theorem-level condition and can fail. More generally, the absence of a preferred vacuum and particle interpretation outside stationary settings is explained in Hollands and Wald 2015, § 1, pp. 7–8, Open PDF. The finite calculation above demonstrates the positive-frequency-split dependence of particle language; it does not prove inequivalence.

Four distinctions keep the conclusion precise:

• $a^\dagger(f)|0\rangle$ is a new vector state in the same representation.
• A unitary packet-basis change within $\mathcal H_1$ preserves the selected positive-frequency split, vacuum, and total number operator.
• Bogoliubov mixing changes the annihilators and particle assignment, even when the transformation is unitarily implementable.
• Inequivalence requires failure of an algebra-intertwining unitary in the infinite system; a nonzero transformed particle count is not enough.

Uses, limits, and continuations

The selected Fock representation provides a controlled language for the free-field vacuum, finite-particle excitations, and the diagonal free Hamiltonian. Several nearby questions require additional structure:

• Normal Ordering and Vacuum Terms studies representation-relative reordering and the regulated vacuum term.
• Coherent States and the Classical Limit uses the same creators and annihilators to construct displaced states and test classical behavior.
• Bogoliubov Transformations and Unitary Implementability develops the infinite-mode implementability criterion and curved-spacetime comparison.
• States, GNS Representations, and Folia develops general representation and sector comparisons.
• In and Out States states the scattering hypotheses needed before asymptotic Fock spaces can be used. An exact interacting theory need not be represented globally by the free vacuum Fock space used here.

Common pitfalls

Treating Fock space and a Fock representation as synonyms. The direct-sum Hilbert space is only part of the data. The represented field algebra and the vacuum vector that realizes the selected state are also needed.

Calling the vacuum universally “no particles.” It has zero eigenvalue for the number operator selected by its positive-frequency split. A different split can assign it nonzero particle content.

Using a finite box as though it contained finitely many modes. A periodic box discretizes momenta but does not impose a UV cutoff. The finite set $K_\Lambda$ was retained whenever ordinary occupation products and sums were used.

Treating a sharp-momentum density as an ordinary operator. In infinite volume, $a(\mathbf p)$ is operator-valued distributional notation. Use normalizable packets, a regulator, or the second-quantized operator.

Identifying free scalar number with charge or superselection. A real scalar has no particle-number $U(1)$ symmetry, and the field connects different number sectors. Free conservation of $N$ does not make those sectors superselected or guarantee conservation after interactions are introduced.

Inferring inequivalence from a Bogoliubov particle count. The expectation $\sinh^2r$ already occurs in a finite, unitarily implementable system. Inequivalence is a separate infinite-mode representation question.

Check your understanding

1. Retrieval. Write the symmetric Fock direct sum, identify its vacuum and $n$-particle sector, and state how $N$ acts on that sector.
2. Distinction. Contrast an excited vector state, a unitary change of packet basis, a Bogoliubov change of particle split, and an inequivalent representation. Why is real-scalar number neither a Noether charge nor a superselection label?
3. Normalization and domain. Normalize $(b^\dagger_{\mathbf k})^n|0\rangle$, derive its number eigenvalue, and state $\mathcal D(N)$.
4. Failure mode. Diagnose the claims “every QFT has this Fock space,” “vacuum means universally no particles,” and “$[H,N]=0$ by definition.”
5. Transfer. Verify the two-mode transformed commutators and compute the transformed occupation in the old vacuum.
6. Handoff. Where should one continue for normal ordering, coherent states, infinite-mode implementability, general representation theory, and asymptotic in/out particles?

Answers and repair routes

1. The space is $\mathcal F_s(\mathcal H_1)=\bigoplus_{n\geq0}\mathcal F_s^{(n)}$, with $\mathcal F_s^{(0)}=\mathbb C|0\rangle$ and $\mathcal F_s^{(n)}=\operatorname{Sym}^n\mathcal H_1$. The vacuum is the normalized zero-particle vector annihilated by every $a(f)$, and $N\psi_n=n\psi_n$. Revisit Quantizing the Real Scalar Field if the one-particle normalization is unclear.

2. Applying $a^\dagger(f)$ changes the vector but not the representation. A unitary basis rotation stays inside the same positive-frequency $\mathcal H_1$ and preserves total $N$. Bogoliubov mixing changes the annihilators, vacuum, and particle assignment; only failure of an intertwining unitary makes the infinite-system representations inequivalent. The real field changes number by one, so $N$ is not generated by a field-preserving $U(1)$ symmetry and its eigenspaces are not superselection sectors. Review Vacua, States, and Representations for the state–representation distinction and Complex Scalars and Conserved Charge for the charge comparison.

3. From $\langle0|b_{\mathbf k}^n(b^\dagger_{\mathbf k})^n|0\rangle=n!$, the normalized vector is $(b^\dagger_{\mathbf k})^n|0\rangle/\sqrt{n!}$. Commuting $N_{\mathbf k}$ through the creators gives eigenvalue $n$. In the continuum, $\mathcal D(N)=\{\Psi:\sum_n n^2\|\psi_n\|^2<\infty\}$; normalization alone does not imply this condition. Return to the continuum-number section and compare it with the weaker form domain $\mathcal D(N^{1/2})$.

4. The first statement fails because the abstract algebra need not select a preferred Fock state, and an interacting theory need not have the free vacuum representation. The second suppresses the number operator and positive-frequency split relative to which the vacuum is empty. The third confuses a property of the free Hamiltonian with the kinematic definition of Fock space; interactions can mix number sectors. Repair all three by separating algebra, state, representation, and dynamics.

5. Writing $c=\cosh r$ and $s=\sinh r$ gives $[\widetilde b_{\mathbf k},\widetilde b^\dagger_{\mathbf k}] =c^2-s^2=1$. The cross terms in $[\widetilde b_{\mathbf k},\widetilde b_{-\mathbf k}]$ cancel. Only $s^2 b_{-\mathbf k}b^\dagger_{-\mathbf k}$ survives in the old-vacuum expectation, so $\langle0_b|\widetilde b^\dagger_{\mathbf k} \widetilde b_{\mathbf k}|0_b\rangle=s^2=\sinh^2r$. This changes particle assignment but, for finitely many modes, not the representation’s unitary equivalence class.

6. Continue to Normal Ordering and Vacuum Terms for vacuum reordering, Coherent States and the Classical Limit for displacement states, Bogoliubov Transformations and Unitary Implementability for the infinite-mode criterion, States, GNS Representations, and Folia for general representation theory, and In and Out States for asymptotic particle spaces.

References

• Coleman, Sidney. Lectures of Sidney Coleman on Quantum Field Theory. Edited by Bryan Gin-ge Chen, David Derbes, David Griffiths, Brian Hill, Richard Sohn, and Yuan-Sen Ting. Singapore: World Scientific, 2019. DOI.
• Hollands, Stefan, and Robert M. Wald. “Quantum Fields in Curved Spacetime.” Physics Reports 574 (2015): 1–35. DOI. Open PDF.