Why Fixed Particle Number Fails

A fixed-particle description fails when the dynamics transfers amplitude out of its chosen particle-number sector by an amount relevant to the prediction. Charge conservation does not prevent this: a particle and antiparticle can be created together without changing net charge. Relativity alone does not force every system to produce particles. Free sectors and controlled low-energy descriptions can remain useful. The issue is dynamical closure under the interactions and preparations actually being modeled.

Required background. Antiparticles distinguishes charge from total number; Fock Space and Occupation Number supplies number sectors. Helpful background. Negative-Energy Solutions explains one-body energy projections, and When Relativistic QM Is Useful sets approximation criteria.

Closure of a chosen number sector

Let $P_N$ project onto total particle number $N$ in a specified free or asymptotic Fock description. Start with a unit state $P_N|\psi\rangle=|\psi\rangle$. Exact closure means

$$(I-P_N)e^{-iHt/\hbar}P_N=0$$

for all times, equivalently on an interval containing $t=0$. Closure at an isolated revival time would not suffice. For a finite regulated system with Hermitian $H$, this is equivalent to $[H,P_N]=0$, or to vanishing off-diagonal coupling $(I-P_N)HP_N=0$. Hermiticity also removes the reverse coupling. In the continuum, the corresponding statement requires compatible operator domains and a projection reducing $H$, equivalently commuting with its unitary evolution; a formal commutator on unspecified vectors is insufficient.

If all number sectors are preserved, total number is conserved. A particular sector or selected state can remain invariant even when other sectors mix. Conversely, conserving the expectation of some other generator says nothing by itself about closure of $P_N$.

For $|\psi\rangle$ in the domain of self-adjoint $H$, strong differentiability at $t=0$ gives the leakage amplitude

$$(I-P_N)e^{-iHt/\hbar}|\psi\rangle =-\frac{it}{\hbar}(I-P_N)H|\psi\rangle+o(t).$$

Its probability is

$$P_{\rm leak}(t) =\frac{t^2}{\hbar^2} \|(I-P_N)H|\psi\rangle\|^2+o(t^2).$$

This identifies a state-dependent short-time diagnostic. It is not a long-time decay law or a scattering cross section. Small initial leakage does not ensure small leakage after resonant driving or long evolution.

A charge-conserving pair interaction

Consider a finite two-mode fermionic model. Let $a^\dagger$ create a particle of charge $q$, and $b^\dagger$ an antiparticle of charge $-q$. All modes obey the canonical anticommutation relations. Define

$$N=N_a+N_b,\qquad Q=q(N_a-N_b),$$

and

$$\begin{aligned} H(t)&=E_aN_a+E_bN_b+H_{\rm pair}(t),\\ H_{\rm pair}(t)&=g(t)a^\dagger b^\dagger+g(t)^*ba. \end{aligned}$$

Here $E_a,E_b>0$ and $g$ has units of energy. This is an illustrative regulated interaction with a specified external drive, not a derivation of a complete relativistic field theory. One may switch $g$ on for a finite interval and off again so the initial and final occupations refer to the same uncoupled modes.

The number commutators are

$$[N,a^\dagger b^\dagger]=2a^\dagger b^\dagger, \qquad [N,ba]=-2ba.$$

Both pair operators commute with $Q$. Therefore

$$[Q,H(t)]=0, \qquad [N,H(t)]=2g\,a^\dagger b^\dagger-2g^*ba.$$

The interaction preserves charge but changes total number by two. For the bare vacuum,

$$H_{\rm pair}|0\rangle=g\,a^\dagger b^\dagger|0\rangle,$$

and the created pair state has norm one. For a constant coupling just after a specified switch-on, the leakage from $N=0$ starts as $|g|^2t^2/\hbar^2+o(t^2)$. The probability is a property of this prepared finite-mode model; it is not the vacuum-decay rate of an eternal electromagnetic field. The bare vacuum need not be the ground state while the coupling is on.

The occupation basis makes the selection rule explicit:

State	Charge	Total number	Pair term connects it to
$\lvert0\rangle$	$0$	$0$	$a^\dagger b^\dagger\lvert0\rangle$
$a^\dagger\lvert0\rangle$	$q$	$1$	No state in this two-mode model
$b^\dagger\lvert0\rangle$	$-q$	$1$	No state in this two-mode model
$a^\dagger b^\dagger\lvert0\rangle$	$0$	$2$	$\lvert0\rangle$

The singly occupied states are protected here by exclusion and the restricted mode set. In a field with additional empty modes, a pair can be created in other modes while an original particle remains. Protection in a two-mode example is not a proof that the full one-particle sector is closed.

Particle number is not the sign of one-body energy

The free Dirac matrices $P_\pm(\mathbf p)$ project a four-component wavefunction onto positive and negative one-body energy sectors. In contrast, $P_N$ here projects a Fock-space state onto an entire $N$-particle sector. They act on different spaces and answer different questions.

A lower component of a positive-energy Dirac spinor is not an extra particle. A classical transition between frequency sectors is not by itself a particle count. To turn mode conversion into production one must supply a quantized field, initial state, and final particle basis. For a background pulse these are often specified by asymptotic in and out modes.

Particle number itself is tied to that mode or asymptotic description. An interacting theory does not generally possess one universal local particle-number operator that is simultaneously conserved and identifies all detector counts. Conserved charges and operationally defined asymptotic particle counts remain distinct.

Why fixed-number approximations can still work

The useful question is whether the omitted channels affect the observable at the required accuracy. For a free theory, number sectors are exactly invariant. For suitably weak, slowly varying fields and energies well below accessible pair channels, an effective fixed-number description can organize small corrections. The gap and time scales, the background’s available work, and the duration of observation all enter that claim.

Pair-Creation Thresholds owns the invariant kinematic conditions. Dirac to Pauli and Foldy–Wouthuysen Expansion show how controlled positive-energy effective dynamics recovers familiar low-energy physics. They do not assert exact closure under arbitrary time-dependent or strong backgrounds.

Even when real additional particles are kinematically unavailable, eliminated sectors can contribute virtual corrections to an effective Hamiltonian. Therefore “no on-shell pair production” does not mean “delete every term associated with the other sector.” The effective description must match the retained observables at its stated order.

Exercises

1. For the pair Hamiltonian, compute $d\langle N\rangle/dt$ and $d\langle Q\rangle/dt$.

Solution

The operators have no explicit time dependence, so

$$\frac{d\langle N\rangle}{dt} =-\frac{2i}{\hbar} \left(g\langle a^\dagger b^\dagger\rangle -g^*\langle ba\rangle\right), \qquad \frac{d\langle Q\rangle}{dt}=0.$$

The first expression is real because the two expectation values are conjugates. In the bare vacuum its first derivative initially vanishes, while the number probability changes at order $t^2$.

2. A state is a superposition of the vacuum and one neutral pair. Does electric-charge superselection alone forbid its relative phase?

Solution

No. Both components have the same charge zero. An electric-charge superselection rule distinguishes different charge sectors, not every different total particle number. Whether a particular phase is accessible depends on the allowed preparations and observables; the pair Hamiltonian supplies an explicit charge-preserving coupling in this model. The broader distinctions belong to Particle-Number Superselection Preview.

3. Does $(I-P_N)H|\psi\rangle=0$ for one state prove that the whole $N$-particle sector is invariant?

Solution

No. It only removes that state’s leading leakage coefficient. The Hamiltonian can first move it to another vector inside the sector that subsequently couples out. Whole-sector closure requires the corresponding operator condition on every vector of the sector, with the necessary domain qualifications.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. doi:10.1017/9781139540940. Particle creation and the field-theory state space.
• Tong, David. Lectures on Quantum Field Theory. University of Cambridge, 2006, sections 2.5 and 5.2. Charged scalar fields; fermionic fields.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge University Press, 1995, chapters 3–5. Asymptotic states, conserved generators, and interaction operators.