Particle-Number Superselection Preview

Fock space allows vectors with components in several particle-number sectors. A particle-number superselection rule is the stronger statement that relative phases between different particle-number sectors are not accessible under the allowed observables, preparations, or reference frames.

This page is a particle-number preview. The broader symmetry-side sector language is summarized in Superselection Sectors Preview. The key caution is that there are three different statements:

• a state is mathematically written as a superposition of different total particle numbers;
• a Hamiltonian conserves total particle number;
• a physical setting permits, forbids, or hides coherent operations between different number sectors.

Those statements are related, but they are not the same.

Number Sectors in Fock Space

For a one-particle Hilbert space $\mathcal h$, a bosonic or fermionic Fock space is a direct sum of fixed-number sectors:

$$\mathcal F(\mathcal h) = \bigoplus_{N=0}^{\infty}\mathcal H_N.$$

Here $\mathcal H_N$ means $\operatorname{Sym}^N\mathcal h$ for bosons and $\wedge^N\mathcal h$ for fermions. The total number operator has eigenvalue $N$ on $\mathcal H_N$:

$$N_{\mathrm{tot}}\Pi_N = N\Pi_N,$$

where $\Pi_N$ projects onto the $N$-particle sector.

A general Fock-space vector may have the form

$$\lvert\Psi\rangle = c_0\lvert0\rangle + c_1\lvert\psi_1\rangle + c_2\lvert\psi_2\rangle + \cdots,$$

with $\lvert\psi_N\rangle\in\mathcal H_N$. If more than one coefficient is nonzero, the vector is not an eigenvector of total particle number.

The direct-sum formalism permits such a vector. Whether its relative phases are physically meaningful depends on the available operations and reference frames.

Number Conservation

Particle-number conservation is a statement about dynamics. If a Hamiltonian $H$ satisfies

$$[H,N_{\mathrm{tot}}]=0,$$

then time evolution generated by $H$ does not mix different total-number sectors. If the state begins in $\mathcal H_N$, it remains in $\mathcal H_N$.

This is weaker than a superselection rule. A Hamiltonian may conserve particle number, but a laboratory might still prepare a coherent superposition of different number sectors by coupling the system to a reservoir or reference. Conversely, a superselection restriction can say that even if a formal vector is written with several $N$ sectors, no allowed observable can detect the relative phases.

The practical distinction is:

$$\begin{array}{c|c} \text{Statement} & \text{Meaning}\\ \hline [H,N_{\mathrm{tot}}]=0 & \text{the closed-system dynamics preserves }N\\ [A,N_{\mathrm{tot}}]=0\ \text{for allowed }A & \text{allowed measurements cannot change or compare }N\\ \rho=\sum_N\Pi_N\rho\Pi_N & \text{the described state has no coherences between }N\text{ sectors} \end{array}$$

The second and third rows are common ways of modeling a superselection restriction.

Superselection as Inaccessible Coherence

Suppose the allowed observables all commute with total number:

$$[A,N_{\mathrm{tot}}]=0.$$

Then $A$ is block diagonal in the particle-number decomposition:

$$A = \sum_N \Pi_N A\Pi_N.$$

For any density operator $\rho$, define the number-dephased state

$$\mathcal G_N(\rho) = \sum_N \Pi_N\rho\Pi_N.$$

For every number-conserving observable $A$,

$$\operatorname{Tr}(\rho A) = \operatorname{Tr}(\mathcal G_N(\rho)A).$$

Thus the off-diagonal blocks $\Pi_N\rho\Pi_M$ with $N\ne M$ cannot be detected by those observables. In that operational setting, a coherent state across number sectors is indistinguishable from its number-dephased mixture.

This is the minimal operational content of a particle-number superselection rule: relative phases between distinct number sectors are not available unless additional structure supplies a way to compare them.

A One-Mode Example

Consider the formal one-mode state

$$\lvert\psi\rangle = \frac{ \lvert0\rangle+\lvert1\rangle }{\sqrt2}.$$

Its density operator contains off-diagonal terms:

$$\rho = \frac12 \bigl( \lvert0\rangle\langle0\rvert + \lvert0\rangle\langle1\rvert + \lvert1\rangle\langle0\rvert + \lvert1\rangle\langle1\rvert \bigr).$$

Number dephasing removes the coherence:

$$\mathcal G_N(\rho) = \frac12 \bigl( \lvert0\rangle\langle0\rvert + \lvert1\rangle\langle1\rvert \bigr).$$

If the only allowed observables commute with $N$, these two density operators give the same measurement statistics. An observable such as

$$X = \lvert0\rangle\langle1\rvert + \lvert1\rangle\langle0\rvert$$

would distinguish them, but it does not commute with $N$. Whether such an observable is physically available is the superselection question.

Global and Local Number

Total particle number and local particle number are different. A state can have fixed total number while having indefinite local number.

For two modes $L$ and $R$,

$$\lvert\psi\rangle = \frac{ \lvert1_L,0_R\rangle + \lvert0_L,1_R\rangle }{\sqrt2}$$

has total particle number $1$. It is not a superposition of different total $N$ sectors. But the local number in the left mode is indefinite: a measurement of $N_L$ gives $0$ or $1$.

This distinction matters for mode entanglement. The same state is nonfactorizable across the $L\vert R$ mode split, but if two distant laboratories are restricted to operations that conserve their local particle numbers, they may be unable to access all of the coherence as an ordinary bipartite entanglement resource.

The detailed entanglement cautions belong to Identical-Particle Entanglement Cautions. The lesson here is narrower: always state whether the restriction concerns total number, local number, charge, parity, or some other conserved quantity.

Coherent States and Phase References

Optical coherent states are often written as superpositions of photon-number states:

$$\lvert\alpha\rangle = e^{-\lvert\alpha\rvert^2/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} \lvert n\rangle.$$

This formula does not by itself settle the superselection question. In quantum optics, phase is usually meaningful relative to a phase reference, local oscillator, pump, or larger experimental frame. If that reference is not included explicitly, the single-mode state may be effectively averaged over a global phase:

$$\rho_{\mathrm{twirl}} = \int_0^{2\pi} \frac{d\theta}{2\pi}\, e^{-i\theta N} \rho e^{i\theta N}.$$

For a coherent state, this phase twirl gives

$$\rho_{\mathrm{twirl}} = e^{-\lvert\alpha\rvert^2} \sum_{n=0}^{\infty} \frac{\lvert\alpha\rvert^{2n}}{n!} \lvert n\rangle\langle n\rvert.$$

The photon-number distribution remains Poissonian, but the off-diagonal number coherences are gone. Interference experiments can still be described consistently because the relevant phase information may be stored relationally between system and reference rather than as an absolute phase of one isolated mode.

This is why one should not say “coherent states are impossible” when discussing superselection. A better statement is: the operational meaning of number-sector coherence depends on the reference frames and observables included in the physical description.

Matter Waves and Mean-Field States

Similar language appears in many-body physics. A Bose-condensed mean-field state, a BCS superconducting state, or a symmetry-broken variational ansatz is often written with indefinite particle number. These states can be excellent approximations or convenient representations even when the underlying closed system has fixed total particle number.

There are several compatible interpretations:

• the system is effectively open to a large reservoir;
• a reference phase is supplied by the surrounding apparatus or condensate;
• the thermodynamic limit makes number and phase descriptions approximately interchangeable for many local observables;
• a number-conserving state can reproduce the same local predictions as a symmetry-broken ansatz within its regime of validity.

The point is not that particle number is arbitrary. The point is that the correct description depends on the physical question and on which degrees of freedom are included as part of the quantum system.

QFT Caveats

In relativistic quantum field theory, particle number is usually not a fundamental conserved quantity. Interactions can create and annihilate particles, and particle concepts are sharpest for free fields or asymptotic scattering states. Conserved charges, such as electric charge, are often more fundamental than particle number.

Superselection in QFT is also tied to locality, gauge constraints, global charges, and the algebra of observables. Charge superselection is a different and deeper subject than the elementary Fock-space question “may I write $\lvert0\rangle+\lvert1\rangle$?”

For this volume, keep the modest lesson:

• Fock space supplies the mathematical room for different number sectors.
• Number conservation is expressed by commutation with $N_{\mathrm{tot}}$.
• Superselection restrictions are restrictions on accessible coherences, operations, or observables.
• Reference frames can convert apparently forbidden absolute coherences into relationally meaningful ones.
• Relativistic QFT replaces simple particle-number intuition with field, charge, locality, and asymptotic-particle concepts.

Common Mistakes

• Treating every Fock-space vector across number sectors as physically preparable in every context.
• Confusing number conservation with a superselection rule.
• Saying a superselection rule makes the forbidden vector mathematically meaningless.
• Ignoring the role of reservoirs, phase references, and relational degrees of freedom.
• Confusing total particle-number superselection with local particle-number restrictions.
• Treating photon number, atom number, quasiparticle number, and electric charge as if they obey the same physical rules.
• Importing QFT charge-superselection language into nonrelativistic Fock space without stating the assumptions.

Cross-Links

• Occupation-Number Basis
• Superselection Sectors Preview
• Direct Sums versus Tensor Products
• Bosonic Fock Space
• Fermionic Fock Space
• Vacuum State
• Number States
• Mode Occupations
• Two-Mode Entanglement
• Fock Space Examples
• Number Operators
• Creation and Annihilation Operators
• Entanglement Depends on a Decomposition
• Identical-Particle Entanglement Cautions
• LOCC Preview
• Reference Bridge: Second Quantization

References

• G. C. Wick, A. S. Wightman, and E. P. Wigner, “The Intrinsic Parity of Elementary Particles,” Physical Review 88, 101-105, 1952.
• Y. Aharonov and L. Susskind, “Charge Superselection Rule,” Physical Review 155, 1428-1431, 1967.
• S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information,” Reviews of Modern Physics 79, 555-609, 2007.
• D. Giulini, “Superselection Rules,” in Compendium of Quantum Physics, Springer, 2009; arXiv:0710.1516.
• H. M. Wiseman and J. A. Vaccaro, “Entanglement of Indistinguishable Particles Shared between Two Parties,” Physical Review Letters 91, 097902, 2003.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.

Exercises

1. Number-dephased statistics. Let $A$ commute with $N_{\mathrm{tot}}$. Show that $\operatorname{Tr}(\rho A)=\operatorname{Tr}(\mathcal G_N(\rho)A)$, where $\mathcal G_N(\rho)=\sum_N\Pi_N\rho\Pi_N$.

Solution

Since $A$ commutes with $N_{\mathrm{tot}}$, it is block diagonal in the number-sector projectors:

$$A = \sum_N \Pi_N A\Pi_N.$$

Insert the identity $\sum_N\Pi_N=I$ on both sides of $\rho$:

$$\operatorname{Tr}(\rho A) = \sum_{M,N} \operatorname{Tr}(\Pi_M\rho\Pi_N A).$$

The off-diagonal terms with $M\ne N$ vanish because $A$ has no matrix elements between different number sectors. Therefore

$$\operatorname{Tr}(\rho A) = \sum_N \operatorname{Tr}(\Pi_N\rho\Pi_N A) = \operatorname{Tr}(\mathcal G_N(\rho)A).$$

2. Conservation versus superselection. A Hamiltonian satisfies $[H,N_{\mathrm{tot}}]=0$. Does that alone prove that coherent superpositions of different total particle numbers are impossible?

Solution

No. The commutator says that the closed-system time evolution generated by $H$ preserves total particle number. It does not by itself classify all possible preparations, measurements, reservoirs, or phase references. A superselection claim requires a statement about allowed observables, states, or operations, not only about this Hamiltonian.

3. One-mode coherence. For

$$\lvert\psi\rangle = \frac{ \lvert0\rangle+\lvert1\rangle }{\sqrt2},$$

compute $\mathcal G_N(\lvert\psi\rangle\langle\psi\rvert)$.

Solution

The density operator is

$$\rho = \frac12 \bigl( \lvert0\rangle\langle0\rvert + \lvert0\rangle\langle1\rvert + \lvert1\rangle\langle0\rvert + \lvert1\rangle\langle1\rvert \bigr).$$

Number dephasing keeps only blocks with the same total number on bra and ket:

$$\mathcal G_N(\rho) = \frac12 \bigl( \lvert0\rangle\langle0\rvert + \lvert1\rangle\langle1\rvert \bigr).$$

4. Local number. The state

$$\frac{ \lvert1_L,0_R\rangle + \lvert0_L,1_R\rangle }{\sqrt2}$$

has fixed total particle number. Why can local particle-number restrictions still matter?

Solution

The total number is always $1$, but the left and right local occupations are not definite. The state is a coherent superposition of different local-number sectors: left occupied and right empty, or left empty and right occupied. If the local laboratories can only perform operations that commute with their local number operators, some coherence across the $L\vert R$ split may be operationally inaccessible without an additional phase reference or reservoir.

5. Pair creation. Let $d_i$ denote either bosonic or fermionic annihilation operators, and suppose

$$[N_{\mathrm{tot}},d_i^\dagger]=d_i^\dagger.$$

Show that $d_i^\dagger d_j^\dagger$ is not number conserving.

Solution

Use the product rule for commutators:

$$\begin{aligned} [N_{\mathrm{tot}},d_i^\dagger d_j^\dagger] &= [N_{\mathrm{tot}},d_i^\dagger]d_j^\dagger + d_i^\dagger[N_{\mathrm{tot}},d_j^\dagger]\\ &= d_i^\dagger d_j^\dagger + d_i^\dagger d_j^\dagger\\ &= 2d_i^\dagger d_j^\dagger. \end{aligned}$$

The operator creates two quanta, so it changes total particle number by two.