Infraparticles and Velocity Superselection

An infraparticle is a charged excitation whose energy–momentum spectrum has a sharp lower boundary but no isolated mass hyperboloid carrying a normalizable Wigner one-particle state. In QED the unavoidable soft-photon cloud replaces the electron pole by continuous threshold weight. Different asymptotic velocities can induce disjoint photon-cloud representations, so a single charged Fock space is not available.

Required background. Particles and one-particle subspaces gives the missing sharp-shell criterion; Araki–Haag detectors and particle weights supplies a particle notion without normalizable sharp states; and massless scattering and radiation fields supplies the soft field.

Helpful background. Resonances, infraparticles, and limits of particle language gives the physical distinction, and dressed states and infrared-finite scattering gives the scattering response.

Spectral signature

For a stable Wigner particle, the Källén–Lehmann measure contains $Z\delta(\mu^2-m^2)$ with $Z>0$. A schematic infraparticle threshold instead has

$$\rho(\mu^2)\sim \theta(\mu^2-m^2)(\mu^2-m^2)^{\beta-1}, \qquad \beta>0,$$

with no delta atom. The propagator has a branch singularity rather than a simple isolated pole. The charged sector can still have a lower mass $m=\inf\sqrt{p^2}$, but $E(H_m^+)=0$ and no normalizable mass eigenvector exists.

This differs from an unstable resonance. A resonance pole is associated with decay and appears on a nonphysical sheet away from the real spectrum. An infraparticle threshold is a real-spectrum consequence of infinitely many arbitrarily soft massless quanta attached to a charged excitation.

A sharp lower boundary is not an eigenvalue. For every finite experimental resolution, the spectral weight may be concentrated in a narrow band above $m^2$ and look particle-like, yet the projection of the exact singleton shell can still vanish. Shrinking the band tests whether an atom is present: its probability tends to a positive limit for a Wigner particle and to zero for a purely continuous infraparticle threshold. This provides a regulator-independent spectral diagnostic.

Under algebraic Gauss-law and asymptotic-field assumptions, a charged state with nonzero long-range flux cannot be an eigenstate of the mass operator; Buchholz 1986, pp. 331–334 gives the no-sharp-mass result. Its scope is a theorem under those assumptions, not a construction of full nonperturbative QED.

Bloch–Nordsieck threshold

The Bloch–Nordsieck approximation replaces recoil/spin details by a charged particle of fixed velocity coupled to soft photons. Summing arbitrarily many soft emissions exponentiates the infrared logarithms. The exclusive zero-photon pole loses finite residue, and the spectral density takes the continuous threshold form above. The original analysis shows why the mean number of emitted soft quanta diverges while the radiated energy can remain finite Bloch and Nordsieck 1937, pp. 54–59.

As the infrared cutoff $\lambda$ is removed, the probability for a fixed finite number of photons tends to zero, while inclusive probabilities at fixed energy resolution can remain finite. Therefore setting an electron LSZ factor $Z_e>0$ at a nonexistent isolated pole is not a normalization convention; it contradicts the spectral measure.

This model diagnostic is developed physically on soft photons and infrared-finite QED. The branch exponent is approximation- and convention-dependent; the robust conclusion here is the absence of a delta atom.

Coherent clouds and velocity superselection

For a charge with asymptotic four-velocity $v$, the leading soft coherent displacement has schematic momentum behavior

$$f_v(\mathbf k,\lambda) \sim e\, \frac{v\cdot\varepsilon_\lambda(\mathbf k)} {v\cdot k}\, \frac{\chi(\mathbf k)}{\sqrt{2|\mathbf k|}}.$$

For $v\neq v'$, the infrared norm

$$\|f_v-f_{v'}\|^2 \sim c(v,v')\int_0^\kappa\frac{\mathrm d|\mathbf k|}{|\mathbf k|}$$

diverges. The overlap of cutoff coherent states behaves as $\exp[-\|f_v-f_{v'}\|^2/2]$ and tends to zero. Thus the limiting representations of the photon Weyl algebra are disjoint: no unitary in one Fock representation changes one asymptotic velocity cloud into the other. Fröhlich, Morchio, and Strocchi derive charged non-Fock sectors and the nontrivial Lorentz action in Fröhlich, Morchio, and Strocchi 1979, pp. 241–284.

“Velocity superselection” is model- and algebra-dependent. Restricting observations to a future light cone or changing the asymptotic algebra can weaken the distinction. It should not be promoted to a universal statement about every charged theory.

Independent checks and failure boundary

The logarithm is dimensionless: the soft measure contributes the radial $\mathrm d|\mathbf k|/|\mathbf k|$. Although the photon number diverges, weighting by photon energy changes the infrared integrand to $\mathrm d|\mathbf k|$, which is finite at zero. This reconciles an infinite soft cloud with finite energy.

If one nevertheless inserts a finite $Z_e$ into LSZ, the amputated external limit assumes a delta atom that the branch spectrum lacks. The resulting charged Fock vector has vanishing overlap with the actual infrared sector as the regulator is removed. Dressed, inclusive, or algebraic observables may survive, but ordinary electron LSZ does not.

Exercises

Show why orthogonality of two coherent clouds follows from the logarithmic norm divergence.

Solution

Normalized coherent states displaced by $f$ and $g$ have overlap magnitude $\exp[-\|f-g\|^2/2]$. If the cutoff norm of $f_v-f_{v'}$ grows like $c\log(\kappa/\lambda)$ with $c>0$, the overlap behaves as $(\lambda/\kappa)^{c/2}$ and tends to zero as $\lambda\downarrow0$.

References

• Bloch, Felix, and Arnold Nordsieck. 1937. “Note on the Radiation Field of the Electron.” Physical Review 52: 54–59. DOI.
• Buchholz, Detlev. 1986. “Gauss’ Law and the Infraparticle Problem.” Physics Letters B 174: 331–334. DOI.
• Fröhlich, Jürg, Giovanni Morchio, and Franco Strocchi. 1979. “Charged Sectors and Scattering States in Quantum Electrodynamics.” Annals of Physics 119: 241–284. DOI.