Massless Scattering and Radiation Fields

Massless scattering cannot simply reuse the massive Haag–Ruelle proof: the light cone touches multiparticle continua, so a massless shell need not be isolated by a gap. For neutral local observables in four-dimensional theories with suitable propagation and Huygens-type properties, null or large-time limits can nevertheless construct asymptotic radiation fields. Charged fields with Gauss-law tails lie outside this neutral theorem.

Required background. Clustering, vacuum uniqueness, and mass-gap implications shows what is lost at zero mass; particles and one-particle subspaces gives the spectral target; Haag–Ruelle construction gives the massive comparison; and hyperbolic equations and causal propagators supplies null propagation.

Helpful background. Soft theorems gives the amplitude application, while retarded, advanced, and spectral correlators supplies causal boundary values.

Why the massive proof fails

For $m>0$, the one-particle shell $p^2=m^2$ may be separated from the rest of the mass spectrum. For $m=0$, two parallel future null momenta have a null sum:

$$p^2=q^2=0,\quad p\parallel q \quad\Longrightarrow\quad (p+q)^2=0.$$

Thus one- and multiparticle spectral support meet on the light cone. A narrow invariant-mass cutoff cannot isolate one photon from collinear multiphoton configurations. Massive stationary-phase estimates also use subluminal, separated velocity supports, whereas every massless packet moves at unit speed and is distinguished primarily by direction.

This obstruction does not say massless particles are absent. It says the massive convergence proof is unavailable without replacement hypotheses.

Radiation fields at null infinity

For a local observable $A$ whose massless spectral component has suitable regularity, one averages translated observables along outgoing light rays. Schematically, in four spacetime dimensions,

$$A^{\mathrm{out}}(u,\mathbf n) =\underset{r\to\infty}{\operatorname{w-lim}}\, r\,A(u+r,r\mathbf n), \qquad |\mathbf n|=1,$$

after smearing in retarded time $u$ and angle $\mathbf n$. The limit is weak or distributional on a bounded-energy domain; the unsmeared point expression is only notation. Huygens propagation and locality make incoming and outgoing null shells separate, while energy bounds control the limit.

The resulting asymptotic fields carry energy–momentum on $H_0^+=\{p:p^2=0,p^0>0\}$ and satisfy free massless commutation relations on their scattering domain. Buchholz constructs a complete collision theory for the stated class of massless bosons in Buchholz 1977, pp. 147–173. The dimension and locality assumptions matter: sharp Huygens propagation is special, and long-range fields require separate treatment.

The theorem therefore replaces, rather than repairs, massive shell isolation. Its input is sufficiently regular null propagation of local observables, and its output is a smeared weak asymptotic field on bounded-energy vectors. It does not follow merely from the set-theoretic presence of the light cone in the spectrum. Collinear multiparticle support remains there; the propagation estimate is what separates the radiative contribution in the limit.

Free Maxwell observable algebra

Use the gauge-invariant field strength $F_{\mu\nu}$ rather than an indefinite-metric vector potential. Its vacuum two-point function has support on the positive light cone and transverse tensor structure. Smearing $F$ with compactly supported test two-forms gives local observables on the physical photon Fock space. Since the free Maxwell theory already has free dynamics, its outgoing radiation limit equals the corresponding positive-frequency asymptotic field.

For a smooth angular packet $g(\mathbf n)$ and retarded-time packet $k(u)$, the radiation observable

$$\mathcal F^{\mathrm{out}}(k,g) =\lim_{r\to\infty} r\int\mathrm du\,\mathrm d\Omega(\mathbf n)\, k(u)g(\mathbf n) F(u+r,r\mathbf n)$$

selects null momentum directions in $\operatorname{supp}g$. The $r$ factor extracts the radiative $1/r$ term; Coulombic $1/r^2$ flux is different asymptotic data. Fourier support remains $p^0=|\mathbf p|>0$, and angular localization follows because stationary phase pairs $\mathbf n$ with $\mathbf p/|\mathbf p|$.

This is the neutral radiation-field input to dressed states and infrared-finite scattering. It does not yet construct a charged asymptotic state.

An independent check uses the Maxwell stress tensor: a radiative field of order $1/r$ has energy density of order $1/r^2$, whose integral over a sphere of area $4\pi r^2$ gives finite flux. A Coulomb field of order $1/r^2$ gives vanishing radiated flux but nonzero electric flux $\int\mathbf E\cdot\mathrm d\mathbf S$.

Long-range charged fields are outside the theorem

Insert a charged field with a Gauss-law electric tail. It cannot be compactly localized relative to the observable algebra, and its Coulomb component does not have the decay/local commutator behavior assumed for a neutral radiation observable. The asymptotic flux distinguishes charged sectors even when every bounded-region observable looks vacuum-like at large distance.

Consequently a neutral massless-radiation theorem cannot be quoted to construct charged LSZ states. One must instead use charged-sector, dressing, inclusive, or particle-weight methods. The missing hypothesis is localization/decay, not merely a difficult estimate.

Exercises

Why does a two-photon state obstruct isolation of the massless shell even though both photons have positive energy?

Solution

For collinear future null momenta $p=\alpha n$ and $q=\beta n$ with $n^2=0$, their sum is $(\alpha+\beta)n$ and is again null. Hence the two-photon continuum reaches $p^2=0$. Positive energy restricts the cone direction but does not open an invariant-mass gap.

References

• Buchholz, Detlev. 1977. “Collision Theory for Massless Bosons.” Communications in Mathematical Physics 52: 147–173. DOI.
• Duch, Paweł, and Andrzej Herdegen. 2015. “Massless Asymptotic Fields and Haag–Ruelle Theory.” Letters in Mathematical Physics 105: 245–277. DOI. Open manuscript.