Dressed, Inclusive, and Algebraic Infrared Observables

Infrared finiteness is a property of a specified observable and limiting procedure, not of a bare charged Fock amplitude. Coherent dressing changes the asymptotic state space, inclusive summation changes the measured probability by tracing over unresolved radiation, and algebraic detectors use asymptotic observables without requiring a global charged Fock vector. These constructions can agree on selected soft factors while remaining mathematically inequivalent.

Required background. Massless scattering and radiation fields supplies neutral photons; infraparticles and velocity superselection supplies the charged obstruction; and Gauss-law charges and infrared sectors fixes the long-range state space.

Helpful background. Dressed states and infrared-finite scattering gives the coherent construction; Bloch–Nordsieck and KLN cancellation gives inclusive probabilities; and infrared and collinear safety gives the measurement criterion.

Three different infrared objects

Coherently dressed asymptotic states

For an asymptotic charged momentum $p$, a Faddeev–Kulish-type state has the schematic form

$$|p\rangle_{\mathrm{dr},\lambda} =W_\lambda(p)|p\rangle, \qquad W_\lambda(p) =\exp\!\left[ \int_{|\mathbf k|>\lambda}\mathrm d\mu_0(k)\, \frac{e\,p\cdot\varepsilon(k)}{p\cdot k} \bigl(a^\dagger(k)-a(k)\bigr) \right].$$

The coherent cloud matches the long-range current of the charge. Matrix elements between consistently dressed states can have a finite regulator limit after Coulomb phases and hard/soft separation are treated coherently. But $W_\lambda(p)$ generally has no unitary limit on ordinary photon Fock space: the limit belongs to an infrared representation selected by the asymptotic charges and velocities. The original construction and its modified asymptotic dynamics are in Kulish and Faddeev 1970, pp. 745–757.

Inclusive probabilities

An inclusive experiment fixes an energy/angular resolution and sums probabilities over unresolved final radiation:

$$P_{\Delta E}(\beta\leftarrow\alpha) =\sum_{X:\,E_X<\Delta E} \left|\langle\beta;X|S_\lambda|\alpha\rangle\right|^2.$$

Real and virtual soft singularities cancel for an appropriately degenerate set of initial and final states. The result depends on $\Delta E$ and on the measurement function; it is a probability or density-matrix trace, not a pure-state amplitude. Bloch–Nordsieck establishes the soft-photon mechanism in Bloch and Nordsieck 1937, pp. 54–59, while the general treatment of degenerate systems is Lee and Nauenberg 1964, pp. B1549–B1562.

Algebraic asymptotic observables

Araki–Haag counters and particle weights ask for large-time limits of local or almost-local observables on bounded-energy states. They can encode inclusive energy–momentum flow without choosing a charged point field or a photon-number basis. Their limits are positive functionals on an asymptotic observable algebra. This is neither a coherent vector in a fixed Fock representation nor an explicit sum over a detector’s unobserved Fock states.

Soft-photon QED comparison

Regulate QED with photon mass or energy cutoff $\lambda$. A conventional exclusive charged Fock amplitude has no nonzero regulator-independent limit. With matched coherent clouds, a dressed transition amplitude can remain finite in the soft approximation and in perturbative constructions for which the asymptotic dynamics is controlled. At fixed resolution $\Delta E>0$, the inclusive probability can also remain finite as $\lambda\downarrow0$ after summing all required degenerate channels.

Thus both a dressed amplitude and an inclusive probability can be infrared finite, but they answer different questions:

• the dressed amplitude compares pure asymptotic states in momentum/charge-dependent infrared representations;
• the inclusive probability describes a finite-resolution measurement after tracing over unobserved radiation;
• an algebraic counter is a positive asymptotic functional whose state representation may be disjoint from either convenient Fock realization.

This is the exact distinction needed on dressed states and infrared-finite scattering. Equality of their leading logarithms is a consistency check, not an isomorphism of state spaces.

An independent check tracks the two limits. The inclusive result is meaningful at fixed $\Delta E$ followed by $\lambda\to0$; subsequently sending $\Delta E\to0$ recovers the exclusive question and may drive the probability to zero. A dressing instead fixes the asymptotic soft profile before removing $\lambda$. Interchanging these operations changes the object.

Failure of a false equivalence

Suppose an inclusive density-matrix trace and a pure dressed amplitude yield the same leading soft exponent. The trace discards phase coherence between unresolved sectors and depends on the measurement resolution. The dressed amplitude retains phases but changes the representation and depends on a cloud prescription. No unitary equivalence follows from one matching coefficient.

Subleading soft terms, collinear degeneracies, recoil, large-gauge charges, and finite detector resolution can distinguish the prescriptions. Any claimed equivalence must specify an observable algebra, state map, regulator, order of limits, and accuracy. Without those data, the strongest licensed statement is agreement of a selected perturbative soft factor.

Exercises

Why can $P_{\Delta E}$ be finite for every fixed $\Delta E>0$ while the probability of emitting exactly zero photons tends to zero?

Solution

The expected number of photons below any fixed soft scale diverges logarithmically as the regulator is removed, so every fixed finite-photon sector receives vanishing probability. The sum over all unresolved multiplicities exponentiates real and virtual contributions and can have a finite total at fixed energy resolution.

References

• Bloch, Felix, and Arnold Nordsieck. 1937. “Note on the Radiation Field of the Electron.” Physical Review 52: 54–59. DOI.
• Kulish, P. P., and L. D. Faddeev. 1970. “Asymptotic Conditions and Infrared Divergences in Quantum Electrodynamics.” Theoretical and Mathematical Physics 4: 745–757. DOI. Open text.
• Lee, T. D., and Michael Nauenberg. 1964. “Degenerate Systems and Mass Singularities.” Physical Review 133: B1549–B1562. DOI.