Multiparticle States, Statistics, and Fock Organization

Once a one-particle Hilbert space $\mathcal H_1$ has been specified, ordinary identical-particle kinematics in $3+1$ dimensions organizes $n$ bosons in the symmetric power $\operatorname{Sym}^n\mathcal H_1$ and $n$ fermions in the antisymmetric power $\bigwedge^n\mathcal H_1$. This fixed-$n$ statement comes before any choice of field components or ladder operators. A Bose or Fermi Fock space adds the vacuum sector, takes the Hilbert direct sum over every $n$, and supplies creation and annihilation maps between adjacent sectors. It is therefore Fock space over a chosen $\mathcal H_1$, not a universal Hilbert space that every exact QFT must use.

This page constructs those sectors, fixes their normalization, and shows the first scalar and spinor examples. It takes ordinary Bose or Fermi exchange statistics as an input. The hypotheses that connect this choice to spin, and the existence of interacting in- or out-particle spaces, are separate questions.

Required background. One-Particle States: Mass, Spin, and Relativistic Normalization supplies $\mathcal H_1$, its Poincaré action, and the invariant inner product.

Helpful background. Direct Sums, Tensor Products, and Index Structure supplies the tensor-product and direct-sum constructions. Exterior and Graded Algebra, Grassmann Variables, and Berezin Integration supplies the graded-sign conventions; Grassmann integration itself is not used here.

The one-particle space and fixed-species setting

Work first with one stable species of mass $m$. A one-particle packet $f\in\mathcal H_1$ has components $f_\sigma(\mathbf p)$ and inner product

$$\begin{aligned} \langle f|g\rangle_1 &= \sum_\sigma \int \mathrm d\Pi_{\mathbf p}\, f_\sigma(\mathbf p)^* g_\sigma(\mathbf p),\\ \mathrm d\Pi_{\mathbf p} &= \frac{\mathrm d^3\mathbf p} {(2\pi)^3\,2E_{\mathbf p}},\\ E_{\mathbf p} &= \sqrt{\mathbf p^2+m^2}. \end{aligned}$$

The label $z=(\mathbf p,\sigma)$ below includes momentum and every spin or helicity label. When an internal label is part of the one-particle space, a permutation moves that label together with all the others; it does not exchange momentum slots while leaving spin behind.

Fixed-particle-number sectors

Before imposing identical-particle statistics, $n$ labeled copies occupy $\mathcal H_1^{\otimes n}$. The permutation group $\mathfrak S_n$ acts by

$$\begin{aligned} \mathsf R(\pi) \bigl( f_1\otimes\cdots\otimes f_n \bigr) &= f_{\pi^{-1}(1)}\\ &\quad\otimes\cdots\otimes f_{\pi^{-1}(n)}. \end{aligned}$$

Define the two characters

$$\chi_+(\pi)=1, \qquad \chi_-(\pi)=\operatorname{sgn}(\pi),$$

and the symmetrizing or antisymmetrizing operator

$$\mathsf P_\pm^{(n)} = \frac{1}{n!} \sum_{\pi\in\mathfrak S_n} \chi_\pm(\pi)\, \mathsf R(\pi).$$

Because $\mathsf R(\pi)^\dagger=\mathsf R(\pi^{-1})$ and the characters are real, $\mathsf P_\pm^{(n)}$ is self-adjoint. In its square, every permutation $\tau$ occurs as $\pi\rho$ for exactly $n!$ pairs $(\pi,\rho)$, while $\chi_\pm(\pi)\chi_\pm(\rho)=\chi_\pm(\tau)$. Hence

$$\left(\mathsf P_\pm^{(n)}\right)^\dagger = \mathsf P_\pm^{(n)}, \qquad \left(\mathsf P_\pm^{(n)}\right)^2 = \mathsf P_\pm^{(n)}.$$

These are orthogonal projectors, and the two ordinary identical-particle sectors are

$$\begin{aligned} \mathcal H_n^+ &= \operatorname{Ran}\mathsf P_+^{(n)} = \operatorname{Sym}^n\mathcal H_1,\\ \mathcal H_n^- &= \operatorname{Ran}\mathsf P_-^{(n)} = \bigwedge\nolimits^n\mathcal H_1. \end{aligned}$$

Equivalently, their wave functions obey

$$\begin{aligned} \psi_n \bigl( z_{\pi(1)},\ldots,z_{\pi(n)} \bigr) &= \chi_\pm(\pi)\\ &\quad\times \psi_n(z_1,\ldots,z_n). \end{aligned}$$

The plus choice defines bosonic exchange and the minus choice fermionic exchange. Weinberg constructs these relativistic multiparticle sectors and their permutation normalization in Weinberg 1995, § 4.1, pp. 170–172.

This projection is compatible with Poincaré symmetry. The tensor-product action $U_1(g)^{\otimes n}$ applies the same one-particle transformation to every factor, so it commutes with every $\mathsf R(\pi)$ and therefore with $\mathsf P_\pm^{(n)}$. Each projected sector is invariant. In this free or asymptotic tensor-product representation, the translation generator is additive:

$$P_{(n)}^\mu = \sum_{j=1}^n \mathbf 1\otimes\cdots\otimes \underset{j\text{th factor}}{P^\mu} \otimes\cdots\otimes\mathbf 1.$$

Thus a generalized momentum ket has total momentum $p_1^\mu+\cdots+p_n^\mu$. Symmetrization changes exchange properties, not the sum of one-particle momenta.

Relativistic normalization and exchange terms

Let $|z\rangle$ denote the covariantly normalized generalized one-particle ket from the preceding page, and write

$$\int\mathrm d\nu(z) \equiv \sum_\sigma \int\mathrm d\Pi_{\mathbf p}.$$

A useful generalized $n$-particle ket is

$$\begin{aligned} |z_1,\ldots,z_n\rangle_\pm &= \sqrt{n!}\, \mathsf P_\pm^{(n)}\\ &\quad\times \bigl( |z_1\rangle\otimes\cdots\otimes|z_n\rangle \bigr). \end{aligned}$$

Its inner product is not a single ordered product. The projector produces every exchange contraction:

$$\begin{aligned} &{}_\pm\langle z'_1,\ldots,z'_n|\\ &\qquad z_1,\ldots,z_n\rangle_\pm &= \sum_{\pi\in\mathfrak S_n} \chi_\pm(\pi)\\ &\quad\times \prod_{j=1}^n \langle z'_j|z_{\pi(j)}\rangle. \end{aligned}$$

If $\psi_n$ already has the corresponding exchange symmetry, its vector and norm are

$$\begin{aligned} |\psi_n\rangle &= \frac{1}{\sqrt{n!}} \int \prod_{j=1}^n\mathrm d\nu(z_j)\\ &\quad\times \psi_n(z_1,\ldots,z_n)\\ &\quad\times |z_1,\ldots,z_n\rangle_\pm,\\ \lVert\psi_n\rVert^2 &= \int \prod_{j=1}^n\mathrm d\nu(z_j)\\ &\quad\times \left|\psi_n(z_1,\ldots,z_n)\right|^2. \end{aligned}$$

The factor $1/\sqrt{n!}$ compensates the $n!$ terms produced when the permutation-sum inner product is contracted with a wave function of matching exchange symmetry. It is not a universal factor to attach to every product of creation operators: the correct normalization depends on the overlaps of the occupied one-particle packets. Discrete repeated-mode occupation has the separate product of factorials derived below. Coleman makes the bosonic permutation factors and the norm of a general many-particle state explicit in Coleman 2019, § 2.1, pp. 17–20.

Both cited treatments begin with delta-normalized momentum kets. The site convention instead uses

$$|\mathbf p,\sigma\rangle_{\mathrm{cov}} = \sqrt{(2\pi)^3\,2E_{\mathbf p}}\, |\mathbf p,\sigma\rangle_\delta.$$

On an $n$-particle ket the normalization conversion is the product of these $n$ square-root factors. The exchange sum is unchanged because that product is permutation invariant, and the packet norm above is the convention-independent check. Coleman carries out the one-particle rescaling and its many-particle Lorentz transformation in Coleman 2019, § 2.4, pp. 29–30.

Adding all sectors: Bose and Fermi Fock space

Set $\mathcal F_\pm^{(0)}=\mathbb C\Omega$, where $\Omega=(1,0,0,\ldots)$ is the normalized no-particle vector. This symbol names the zero-particle sector, not the full vacuum-representation Hilbert space denoted $\mathcal H_0$ on the preceding vacuum page. The completed Hilbert direct sums are

$$\begin{aligned} \mathcal F_+(\mathcal H_1) &= \mathbb C\Omega \oplus \bigoplus_{n=1}^{\infty} \operatorname{Sym}^n\mathcal H_1,\\ \mathcal F_-(\mathcal H_1) &= \mathbb C\Omega \oplus \bigoplus_{n=1}^{\infty} \bigwedge\nolimits^n\mathcal H_1 \end{aligned}$$

are the Bose and Fermi Fock spaces over $\mathcal H_1$. A vector is a sequence

$$\begin{aligned} \Psi &= (\psi_0,\psi_1,\psi_2,\ldots),\\ \lVert\Psi\rVert^2 &= \sum_{n=0}^{\infty} \lVert\psi_n\rVert^2 < \infty. \end{aligned}$$

The Hilbert direct sum makes different particle numbers orthogonal. It does not say that particle number is conserved or superselected.

For $f\in\mathcal H_1$, define a creation map on the finite-particle domain by

$$a_\pm^\dagger(f)\psi_n = \sqrt{n+1}\, \mathsf P_\pm^{(n+1)} \bigl( f\otimes\psi_n \bigr),$$

and let $a_\pm(f)$ be its adjoint. The creator is linear in $f$ and the annihilator is antilinear. They satisfy

$$\begin{aligned} \left[ a_+(f),a_+^\dagger(g) \right] &= \langle f|g\rangle_1\mathbf 1,\\ \left[ a_+^\dagger(f),a_+^\dagger(g) \right] &= 0,\\ \left\{ a_-(f),a_-^\dagger(g) \right\} &= \langle f|g\rangle_1\mathbf 1,\\ \left\{ a_-^\dagger(f),a_-^\dagger(g) \right\} &= 0. \end{aligned}$$

The first row is the canonical commutation relation (CCR), and the second the canonical anticommutation relation (CAR). Repeated creation gives

$$\begin{aligned} a_\pm^\dagger(f_1)\cdots a_\pm^\dagger(f_n)\Omega &= \sqrt{n!}\, \mathsf P_\pm^{(n)}\\ &\quad\times \bigl( f_1\otimes\cdots\otimes f_n \bigr). \end{aligned}$$

This proves directly that the ladder-operator and projected-tensor descriptions agree. Weinberg derives the creation and annihilation action and the CCR/CAR algebra from normalized multiparticle states in Weinberg 1995, § 4.2, pp. 173–175.

There is a domain difference worth keeping visible. Bosonic $a_+^\dagger(f)$ and $a_+(f)$ are unbounded, although their polynomials are well defined on the dense finite-particle domain. Fermionic smeared creators and annihilators are bounded, with norm $\lVert f\rVert$, but their sharp-momentum symbols are still operator-valued distributions rather than ordinary operators. The next page develops that distributional distinction.

Occupation numbers and factorials

Choose a countable orthonormal basis $\{e_j\}$ of $\mathcal H_1$. A normalized bosonic occupation state is

$$|n_1,n_2,\ldots\rangle_+ = \prod_j \frac{ \left[ a_+^\dagger(e_j) \right]^{n_j} }{ \sqrt{n_j!} }\, \Omega,$$

where $n_j\in\{0,1,2,\ldots\}$ and only finitely many $n_j$ are nonzero. For fermions one fixes an order of the modes and uses the same expression with $n_j\in\{0,1\}$; the factorials are then all one. Changing the chosen fermionic order can change basis-vector signs but not physical inner products.

The number operator of this Fock representation is

$$N\Psi = (0,\psi_1,2\psi_2,3\psi_3,\ldots),$$

on the domain where $\sum_n n^2\lVert\psi_n\rVert^2<\infty$. Mode occupations are $N_j=a_\pm^\dagger(e_j)a_\pm(e_j)$. For bosons $N_j$ can be any nonnegative integer; for fermions the CAR imply $N_j^2=N_j$, so its eigenvalues are $0$ and $1$. Coleman develops the bosonic box-normalized occupation basis and its continuum creator algebra in Coleman 2019, §§ 2.2–2.4, pp. 20–28. The fermionic restriction follows from the CAR in Weinberg 1995, § 4.2, p. 174.

An exact momentum is not a normalizable basis vector in infinite volume, so $a^\dagger(\mathbf p)a(\mathbf p)$ should not be treated as an ordinary mode-number operator there. Use a finite box, a countable packet basis, or smeared operators. This is the occupation-number version of the same $\delta^{(3)}(0)$ warning that applies to sharp one-particle kets.

For several species, collect all bosonic one-particle spaces into $\mathcal H_B=\bigoplus_r\mathcal H_{1,r}$ and all fermionic spaces into $\mathcal H_F=\bigoplus_s\mathcal H_{1,s}$. A convenient total organization is

$$\mathcal F_+(\mathcal H_B) \otimes \mathcal F_-(\mathcal H_F).$$

Exchange symmetry is physically required only for identical particles. Taking creators for distinct fermion species to anticommute is a consistent ordering convention, useful when symmetries mix species; it does not make the species identical. Weinberg explains this convention and its relation to species ordering in Weinberg 1995, § 4.1, pp. 171–172.

First application: scalar and spinor packets

Let $f$ and $g$ be normalized one-particle packets and set $\alpha=\langle f|g\rangle_1$. For a free real scalar, use the symmetric Fock space and write

$$|f,g\rangle_B = a_+^\dagger(f) a_+^\dagger(g)\Omega.$$

For one fixed positive-energy spin-$\tfrac12$ species, let each packet include its momentum and spin label, use the antisymmetric Fock space, and write

$$|f,g\rangle_F = a_-^\dagger(f) a_-^\dagger(g)\Omega.$$

Commuting or anticommuting the annihilators back to the vacuum gives

$$\begin{aligned} \lVert |f,g\rangle_B\rVert^2 &= 1+|\alpha|^2,\\ \lVert |f,g\rangle_F\rVert^2 &= 1-|\alpha|^2. \end{aligned}$$

These two equations expose both the normalization factor and the physical exchange effect:

Packets	Bose sector	Fermi sector
$f\perp g$	Norm one	Norm one
$g=f$	Divide $[a_+^\dagger(f)]^2\Omega$ by $\sqrt2$	The state vanishes
General $\alpha$	Divide by $\sqrt{1+\lvert\alpha\rvert^2}$	Divide by $\sqrt{1-\lvert\alpha\rvert^2}$ if $\lvert\alpha\rvert<1$

For $n$ packets the corresponding norm is the permanent of the one-particle overlap matrix for bosons and its determinant for fermions. The two-packet calculation already shows why a blanket $1/\sqrt{n!}$ rule is wrong: factorials arise for repeated occupation of the same bosonic mode, while products of orthogonal normalized modes need no extra factor.

In either sector, the free total four-momentum is the second-quantized one-particle generator,

$$\left. \mathrm d\Gamma(P^\mu) \right|_{\mathcal H_n^\pm} = \sum_{j=1}^n P_j^\mu.$$

Thus exchange symmetry changes the allowed wave functions but preserves the additive free-particle spectrum. Coleman 2019, § 2.4, pp. 26–30 constructs this scalar Fock realization, while Weinberg 1995, §§ 4.1–4.2, pp. 170–176 treats both Bose and Fermi sectors and their additive operators.

The scalar and spinor assignments were inputs to this application. A spinor index alone does not prove Fermi statistics, and the antisymmetric tensor power is not a proof of the spin–statistics connection. The Spin–Statistics Connection gives the later theorem map with its covariance, locality, positive-energy, positivity, dimension, and field hypotheses. Fock Space, Vacuum, and Particle Number constructs the scalar model in detail; Canonical Quantization of the Free Dirac Field supplies the spinor mode expansion, particle and antiparticle operators, Hamiltonian, charge, and vacuum.

What is kinematic, and what depends on a representation?

Statement	What has been assumed
$\mathcal H_1^{\otimes n}$ and its permutation action	A particular one-particle Hilbert space has already been selected
$\operatorname{Sym}^n\mathcal H_1$ or $\bigwedge^n\mathcal H_1$	Ordinary Bose or Fermi exchange statistics has been chosen
$\mathbb C\Omega\oplus\bigoplus_{n\geq1}\mathcal H_n^\pm$	Every finite particle number is included and the sum is completed
$\Omega$, $a^\dagger$, $a$, and $N$	The Fock representation over that $\mathcal H_1$ has been selected
$[H,N]=0$	An additional dynamical particle-number conservation law holds
Exact interacting states form this Fock space	A further representation or scattering theorem is needed

For the standard free scalar, selecting the Minkowski vacuum and its positive-frequency one-particle space leads to the symmetric Fock representation. The conceptual order established on Vacua, States, and Representations matters: the algebra and selected state determine a representation; one does not assume that every admissible state is already a vector on one universal Fock space.

For quantum fields with infinitely many degrees of freedom, and especially on general curved spacetimes, there need not be a preferred positive-frequency split or a preferred Hilbert-space representation. Hollands and Wald 2015, § 1, arXiv v2 pp. 4–8; § 2.1, pp. 14–19, PDF gives the structural reason that a chosen Fock construction is not automatic or universal.

Limits of the construction

Free theory. When the one-particle dynamics and vacuum have been selected, $\mathcal F_\pm(\mathcal H_1)$ gives the exact free-particle organization. The free Hamiltonian preserves every $n$-particle sector.

A chosen interacting Fock representation. An interaction may be represented on a regulated or perturbative Fock space while mixing its number sectors. Then $N$ remains a useful grading operator but need not commute with the interacting Hamiltonian.

Exact interacting theory. The construction above does not establish that its physical representation is globally equivalent to the free Fock representation. If stable particles and suitable scattering wave operators exist, separate in- and out-Fock spaces can organize asymptotic states. Their existence does not by itself prove asymptotic completeness. In and Out States and Haag’s Theorem: Physical Meaning and Scope develop those additional hypotheses and obstructions.

Resonances and infraparticles may fail to provide the isolated one-particle input assumed at the start; Resonances, Infraparticles, and Limits of Particle Language treats those failures. Braided statistics in lower spatial dimensions, parastatistics, and generalized charge sectors also require structures beyond the symmetric-versus-antisymmetric alternative used here.

Boundaries and handoffs

• One-Particle States supplies the normalized $\mathcal H_1$ on which every construction here depends.
• Quantum Fields as Operator-Valued Distributions explains why sharp-momentum creators and point fields require smearing and domain control.
• Fock Space, Vacuum, and Particle Number develops the chosen free-scalar Fock representation and its number basis.
• Canonical Quantization of the Free Dirac Field realizes the CAR with spinor particle and antiparticle modes.
• The Spin–Statistics Connection determines when relativistic hypotheses connect spin to the exchange choice used here.
• In and Out States develops the construction of interacting asymptotic Fock spaces.

Common pitfalls

“The tensor-product slots label physical particle identities.” They are bookkeeping slots. Permuting two identical particles exchanges every one-particle label, and the projected state is unchanged or changes only by a sign.

“Every product of $n$ creators needs $1/\sqrt{n!}$.” Products of orthogonal normalized modes already have norm one. The factorial appears for repeated bosonic occupation, while a general product is normalized by its permanent or determinant.

“Fermionic antisymmetry acts only on the spatial wave function.” It acts on the complete one-particle label. A spatially symmetric factor can be allowed when a spin or internal factor supplies the compensating antisymmetry.

“Pauli exclusion proves spin–statistics.” Same-mode exclusion follows algebraically from the CAR. Connecting the CAR to half-integer spin requires the additional hypotheses of a spin–statistics theorem.

“Particle-number sectors are superselection sectors.” Creators, annihilators, and fields connect adjacent sectors, and interactions can mix them. Superselection is a statement about the physical observable algebra and its representations.

“A Fock space is the Hilbert space of any QFT.” It is built over specified one-particle and vacuum data. Exact interacting, thermal, curved-spacetime, infraparticle, or generalized-statistics settings can require a different description.

Check your understanding

1. Prove directly that $\mathsf P_\pm^{(n)}$ is a projector and that it commutes with $U_1(g)^{\otimes n}$.

   Answer

   Self-adjointness follows by replacing each permutation with its inverse. In the square, fix $\tau=\pi\rho$; there are $n!$ pairs with that product and $\chi_\pm(\pi)\chi_\pm(\rho)=\chi_\pm(\tau)$, so one factor of $n!$ cancels and $\mathsf P_\pm^{(n)2}=\mathsf P_\pm^{(n)}$. The same one-particle transformation acts in every tensor slot, so applying it before or after a slot permutation gives the same result. It therefore commutes with the sum defining the projector.

2. Let $f$ and $g$ be normalized with $\langle f|g\rangle_1=\alpha$. Compute the norms of the bosonic and fermionic two-packet states, then take $g=f$.

   Answer

   Moving both annihilators through both creators gives one direct contraction and one exchange contraction. The direct term is $1$; the exchange term is $+|\alpha|^2$ for the CCR and $-|\alpha|^2$ for the CAR. Thus the norms squared are $1+|\alpha|^2$ and $1-|\alpha|^2$. When $g=f$, the bosonic norm squared is $2$, so division by $\sqrt2$ normalizes the state, while the fermionic norm is zero and the state vanishes.

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.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.