Antiparticle Decoupling

A particle-only description is controlled when its prepared states remain close to a chosen particle sector for the process and time interval under study. A gap of order $2mc^2$ is a useful scale, but it does not alone prove decoupling. Static spectral invariance, slow transport of an instantaneous subspace, and exclusion of real pair channels in a quantum field theory are different statements. This page specifies their relation and the assumptions needed to use a nonrelativistic effective description.

Required background. Nonrelativistic Limit sets the comparison and time window; Hamiltonian Form defines spectral branches; Dirac in Electromagnetic Fields provides background-field evolution.

Helpful background. Effective Hamiltonians explains stationary elimination; Adiabatic Approximation owns the slow-transport estimates; Pair Creation explains when mode mixing has a particle-count interpretation.

Which particle projector is being retained?

Use $m>0$. There are at least three projectors one might otherwise confuse:

Projector	What it selects	When it is invariant
$P_{\rm upper}=\operatorname{diag}(I_2,0)$	Upper components in the Dirac basis	Generally not invariant, even for a moving free particle
$P_+^{(0)}$	Positive spectral branch of the free Dirac Hamiltonian	Under free evolution
$P_+(H)$	An isolated particle spectral band of the actual static Hamiltonian	Under that static Hamiltonian

For backgrounds, “particle band” requires an identified spectral window and a choice connected to the intended asymptotic or reference sector. It is not invariant under an arbitrary relabeling of an absolute energy zero by the rule “keep every numerically positive eigenvalue.” A strong field can also close the relevant gap or destroy the chosen isolated-band description.

Let $P=P_+(H)$ be a spectral projection of a static self-adjoint $H$ and $Q=I-P$. Spectral calculus gives

$$Qe^{-iHt/\hbar}P=0.$$

If a normalized initial state has preparation error $\epsilon_0=\|Q\psi(0)\|$, its outside-sector norm remains $\epsilon_0$ exactly. For a state prepared in the free sector, a sufficient mismatch bound is

$$\|Q\psi(0)\|\leq\|Q P_+^{(0)}\|_{\rm op}.$$

Thus a small free-versus-dressed mismatch is a preparation issue; it is not continuous production under a static Hamiltonian. Likewise, a nonzero lower Dirac component is not an antiparticle population. The stationary reduction explicitly reconstructs such components even in a positive-energy eigenstate.

A moving Dirac basis produces a coupling

For $H(t)$, the instantaneous spectral projector $P(t)$ need not be transported by the exact dynamics. If $\psi'=U(t)\psi$, then

$$H'=UHU^\dagger+i\hbar\dot U U^\dagger.$$

Even when the first term is block diagonal at each time, the second can mix its blocks. This is the dynamical term missed by diagonalizing independent snapshots.

Consider one momentum and spin channel in a uniform electric background, with a fixed kinetic-momentum direction. Set $M=mc^2$ and write

$$H(t)=c\pi(t)\sigma_x+M\sigma_z,$$ $$E_\pi=\sqrt{M^2+c^2\pi^2}.$$

Define $\theta(t)=\arctan[c\pi(t)/M]$ and $U(t)=e^{i\theta(t)\sigma_y/2}$. The transformed Hamiltonian is exactly

$$H'=E_\pi\sigma_z-\frac{\hbar\dot\theta}{2}\sigma_y, \qquad \dot\theta=\frac{cM\dot\pi}{E_\pi^2}.$$

The instantaneous gap is $2E_\pi$. If $\dot\pi=q\mathcal E(t)$, the magnitude of the off-diagonal connection divided by that gap is

$$\eta(t)=\frac{\hbar cM|q\mathcal E(t)|}{4E_\pi^3}.$$

Near rest this becomes $|\mathcal E|/(4\mathcal E_{\rm crit})$, where $\mathcal E_{\rm crit}=m^2c^3/(|q|\hbar)$. It is a useful local small parameter for this specified mode. It is not a vacuum pair-production probability or an all-duration error bound.

Define interaction-picture coefficients in this moving basis by

$$\begin{aligned} \psi'(t)={}&c_+(t)e^{-i\varphi(t)}|+\rangle\\ &+c_-(t)e^{i\varphi(t)}|-\rangle, \end{aligned}$$ $$\varphi(t)=\frac1\hbar\int_0^t E_\pi(u)\,du,$$

where $|\pm\rangle$ are the $\sigma_z$ eigenvectors. For an initially positive instantaneous mode, first-order transition theory gives the opposite-branch amplitude

$$c_-(t)\simeq-\frac12\int_0^t \dot\theta(t')e^{-2i\varphi(t')}\,dt'.$$

Both the magnitude of the coupling and its accumulated phase matter. Slow smooth driving permits cancellation; a drive with frequency content near the gap can accumulate a transition despite a small instantaneous coupling. The integral is a leading approximation, not a substitute for solving the coupled equations when it becomes large.

What an adiabatic estimate actually controls

The canonical adiabatic workflow uses a fixed path $H(s)$, $0\leq s\leq1$, traversed as $s=t/T$. To state a concrete sufficient bound, consider a bounded $C^2$ Hamiltonian with an isolated nondegenerate eigenvalue and gap $\Delta(s)>0$. Let primes mean $s$ derivatives and define

$$\begin{aligned} \mathcal A(s)=\frac{\hbar}{T}\Bigg\{ &\frac{\|H'(0)\|}{\Delta(0)^2} +\frac{\|H'(s)\|}{\Delta(s)^2}\\ &+\int_0^s\left[ \frac{\|H''(u)\|}{\Delta(u)^2} +\frac{7\|H'(u)\|^2}{\Delta(u)^3} \right]du\Bigg\}. \end{aligned}$$

Theorem 3 of Jansen, Ruskai, and Seiler supplies this rank-one bound on transported-projector error and hence on leakage amplitude from an initially prepared eigenstate. The endpoint terms are added, not subtracted. An initial mismatch $\epsilon_0$ gives the conservative outside-sector probability bound

$$P_{\rm leak}(s) \leq \min\{1,[\epsilon_0+\mathcal A(s)]^2\}.$$

This statement controls leakage, not the full accumulated phase of the retained state. Isolated bands with degeneracy need the corresponding band version. Unbounded continuum Dirac operators require appropriate domain and resolvent hypotheses, or a regulator with estimates uniform in its removal. The finite-dimensional formula is not a proof for every electromagnetic potential.

The fixed-path qualification is consequential. For example, a weak carrier near frequency $2mc^2/\hbar$ retained for many cycles is not made adiabatic merely by extending the observation duration. In the scaled variable $s=t/T$, its derivatives grow with $T$. The resonant and rotating-state examples illustrate why a gap and a small pointwise ratio alone cannot control an arbitrarily long protocol.

Virtual effects and real antiparticles

In a low-energy quantum field theory, eliminating a hard scale can leave virtual effects in local operator coefficients even when no real pair channel is open. These coefficients are not occupations of the eliminated spinor components. Conversely, a small kinetic energy for an external electron does not exclude production if another photon or background supplies the missing energy. One must inspect the whole process.

At early and late times with well-defined particle bases, background mode mixing can be converted into occupations by quantization and the initial-state prescription on Pair Creation. An instantaneous one-body leakage norm, computed with a chosen $P(t)$, is not automatically that asymptotic occupation number.

Nor does “nonrelativistic” always mean “no antiparticle field.” An electron–positron bound system has both species moving slowly. An effective theory can keep both as low-energy degrees of freedom while integrating out hard fluctuations and describing annihilation by matched operators. The NRQED preview uses this distinction when defining what is retained.

For a proposed particle-only calculation, specify the projector and preparation, the relevant gap, the background frequency and variation scales, the observation interval, and the available real channels. These are the quantities that make the approximation assessable.

Exercises

Free components versus a free sector. A normalized positive-energy free Dirac plane-wave spinor has momentum $\mathbf p\ne0$. Must its lower two components vanish? Does it leak into the negative spectral branch?

Solution

The lower component is proportional to $c\,\boldsymbol\sigma\cdot\mathbf p/(E_{\mathbf p}+mc^2)$ and is generally nonzero. The complete spinor is nevertheless in the positive spectral subspace and evolves there exactly. Component size and spectral leakage are different questions.

A preparation budget. A regulated calculation has $\epsilon_0=0.01$ and a valid adiabatic amplitude bound $\mathcal A=0.02$. What outside-sector probability is guaranteed, and does the bound imply the actual error equals that value?

Solution

The conservative bound is $P_{\rm leak}\leq(0.01+0.02)^2=9\times10^{-4}$. Errors can interfere or be much smaller; the bound is an upper guarantee under the stated assumptions.

The instantaneous connection. Verify the sign of $i\hbar\dot U U^\dagger$ for $U=e^{i\theta\sigma_y/2}$. Why does static diagonalization miss this term?

Solution

$\dot U U^\dagger=i\dot\theta\sigma_y/2$, so multiplication by $i\hbar$ gives $-\hbar\dot\theta\sigma_y/2$. A static eigenvalue problem contains $UHU^\dagger$ only. Differentiating the moving state $U(t)\psi(t)$ produces the additional connection.

References

• Foldy, Leslie L., and Siegfried A. Wouthuysen. “On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit.” Physical Review 78, 29–36 (1950). doi:10.1103/PhysRev.78.29. Particle-sector representation and transformed observables.
• Gavrilov, S. P., and D. M. Gitman. “Vacuum Instability in External Fields.” Physical Review D 53, 7162–7175 (1996). doi:10.1103/PhysRevD.53.7162. Asymptotic field quantization and particle counts.
• Jansen, Sabine, Mary-Beth Ruskai, and Ruedi Seiler. “Bounds for the Adiabatic Approximation with Applications to Quantum Computation.” Journal of Mathematical Physics 48, 102111 (2007). doi:10.1063/1.2798382; author manuscript. Fixed-path gap and derivative estimates, including theorem 3.
• Thaller, Bernd. The Dirac Equation. Springer (1992). doi:10.1007/978-3-662-02753-0. Dirac spectral subspaces and operator-domain conditions.