Why Fock Space Is Necessary

When a process can connect one particle to that particle plus an additional pair, the state space must contain both alternatives. Fock space supplies this variable-number organization, including exchange statistics. For a charged relativistic species, its one-particle modes include particles and antiparticles; fixed net charge can then contain many total particle numbers. The general construction belongs to Fock Space and Occupation Number. This page applies it to the charge sectors needed for relativistic processes.

Required background. Fock Space and Occupation Number defines the direct sum; Antiparticles fixes species and charges; Why Fixed Particle Number Fails identifies the couplings that require several sectors. Helpful background. Fermionic Fock Space and Particle-Number Superselection Preview explain signs and operational restrictions.

The one-particle input includes both species

Let $\mathcal h_{\rm p}$ contain positive-energy particle modes and $\mathcal h_{\rm a}$ positive-energy antiparticle modes. These labels distinguish species, not positive and negative physical energy. For one charged field, use the one-particle input

$$\mathcal h=\mathcal h_{\rm p}\oplus\mathcal h_{\rm a}.$$

The accepted construction then gives $\mathcal F_{\rm B}(\mathcal h)$ for bosons or $\mathcal F_{\rm F}(\mathcal h)$ for fermions. The $N=0$ sector is the vacuum line, the $N=1$ sector includes either species, and higher sectors include every allowed combination. An unconstrained charged Dirac field has independent particle and antiparticle creation operators. A self-conjugate field instead has the corresponding identification; adding an independent duplicate antiparticle species would overcount.

A state can be described as a sequence of fixed-number wavefunctions, with its squared norm the sum of their squared norms. That wavefunction hierarchy is another representation of the same direct-sum organization. Fock notation makes creation, annihilation, and exchange symmetry easier to handle; it does not replace the usual probability rule.

Fixed charge contains a ladder of total numbers

For nonzero particle charge $q$, let $n_{\rm p},n_{\rm a}$ denote the two occupation totals. Then

$$Q=q(n_{\rm p}-n_{\rm a}), \qquad N_{\rm tot}=n_{\rm p}+n_{\rm a}.$$

Choose a charge sector $Q=kq$, with integer $k$. Its allowed occupation pairs can be parameterized as

$$\begin{aligned} n_{\rm p}&=\max(k,0)+j,\\ n_{\rm a}&=\max(-k,0)+j,\\ N_{\rm tot}&=|k|+2j,\qquad j=0,1,2,\ldots. \end{aligned}$$

In a finite fermionic mode regulator, this list stops when exclusion exhausts the available modes. With sufficiently many modes, it illustrates why a fixed charge does not fix total particle number.

For example, the charge-$q$ sector contains:

Particle count	Antiparticle count	Total count	Interpretation
1	0	1	One particle
2	1	3	One particle plus one pair
3	2	5	One particle plus two pairs

The pair interaction from Why Fixed Particle Number Fails moves between such alternatives without changing the charge. This is a direct sum of possible total occupations, not a tensor product describing all those alternatives as simultaneously present.

For bosons, the charge sector can be written as

$$\mathcal H_{Q=kq} =\bigoplus_{n_{\rm p}-n_{\rm a}=k} \operatorname{Sym}^{n_{\rm p}}\mathcal h_{\rm p} \otimes \operatorname{Sym}^{n_{\rm a}}\mathcal h_{\rm a}.$$

For fermions replace symmetric powers by exterior powers, with a consistent species-ordering convention for the operators. The inner product is positive in every sector. The minus sign in $Q$ is an eigenvalue assignment, not a minus sign in the norm of the antiparticle factor.

Fermionic species require a sign convention

Bosonic Fock space over a direct sum factorizes into ordinary species Fock spaces, and the two species’ creation operators commute. For fermions the operator factorization is graded: creation operators of different species must anticommute.

One concrete convention orders particle modes before antiparticle modes. On a tensor-product realization of the two species spaces, write

$$a_r^\dagger=a_{r,{\rm p}}^\dagger\otimes I, \qquad b_s^\dagger=(-1)^{N_{\rm p}}\otimes b_{s,{\rm a}}^\dagger.$$

The particle-number parity anticommutes with $a_{r,{\rm p}}^\dagger$, so $\{a_r^\dagger,b_s^\dagger\}=0$. Using $I\otimes b_{s,{\rm a}}^\dagger$ instead would make the species commute and lose the required fermionic signs.

The ordering convention does not change measurable probabilities when used consistently. It makes the notation $\mathcal F_{\rm F}(\mathcal h_{\rm p}\oplus\mathcal h_{\rm a})$ safer than an unqualified product of two independently commuting operator systems.

Charge conservation and sector coherence

A vector

$$|\Psi\rangle=c_0|0\rangle+c_2a^\dagger b^\dagger|0\rangle, \qquad |c_0|^2+|c_2|^2=1,$$

has charge zero in both components. It has $\langle N_{\rm tot}\rangle=2|c_2|^2$, but its total number is not definite unless one coefficient vanishes. Electric-charge superselection alone cannot remove a relative phase between two vectors of the same charge. Both states also have even fermion parity.

For the mode ordering just chosen, the charge-preserving pair observable $O=a^\dagger b^\dagger+ba$ has

$$\langle O\rangle=2\operatorname{Re}(c_2^*c_0).$$

It distinguishes some relative phases that number measurements alone cannot distinguish. Whether such an observable is accessible in a particular experiment depends on its preparation and coupling; the algebra does not guarantee a laboratory measurement protocol. The canonical superselection page treats those operational questions.

Fock space does not by itself define an interacting theory

The construction is exact for free fields and is the natural organization of suitable asymptotic particle states. It does not, by itself, specify the Hamiltonian, the physical interacting vacuum, a renormalized quantum field, or the existence of scattering limits.

For a finite regulated background problem, one can often follow the field modes between defined in and out particle bases. In infinite systems, a transformation that mixes creation and annihilation operators need not be implementable by a single unitary operator on the chosen Fock space. Convergence and infrared conditions matter. The simple empty free vacuum is not automatically the ground state of an interacting Hamiltonian.

These qualifications explain the scope of “necessary” here: a model of number-changing processes must contain the different particle alternatives, and Fock space is the standard free or asymptotic realization. This is not a theorem that every interacting relativistic theory has a global free-particle Fock representation. LSZ Preview states the stable-particle and scattering assumptions needed for its own construction.

Exercises

1. List the first three total numbers allowed in the charge $Q=-2q$ sector.

Solution

The occupation pairs are $(n_{\rm p},n_{\rm a})=(0,2),(1,3),(2,4)$, with totals $2,4,6$. They are allowed only when sufficient modes exist; for fermions no individual mode can be occupied twice.

2. Verify the cross-species anticommutation in the tensor-product realization.

Solution

Let $\Pi_{\rm p}=(-1)^{N_{\rm p}}$. Because $\Pi_{\rm p}a^\dagger=-a^\dagger\Pi_{\rm p}$,

$$\{a^\dagger\otimes I,\Pi_{\rm p}\otimes b^\dagger\} =(a^\dagger\Pi_{\rm p}+\Pi_{\rm p}a^\dagger) \otimes b^\dagger=0.$$

The parity factor supplies the missing sign.

3. Find the total-number variance in the vacuum–pair state above. Can it vanish while both sectors are populated?

Solution

Writing $w=|c_2|^2$, one has $\langle N^2\rangle=4w$ and $\operatorname{Var}N=4w(1-w)$. It vanishes only at $w=0$ or $w=1$. The charge variance is zero for every $w$ because both components belong to the same charge sector.

References

• Fetter, Alexander L., and John Dirk Walecka. Quantum Theory of Many-Particle Systems. McGraw–Hill, 1971; Dover reprint, 2003. Occupation-number spaces and fermionic operators.
• Tong, David. Lectures on Quantum Field Theory. University of Cambridge, 2006, sections 2.5 and 5.2. Charged scalar fields; fermionic particle states.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995, chapters 3–5. Multiparticle asymptotic states and field operators.