Particles, Mass-Shell Spectrum, and One-Particle Subspaces

A stable relativistic particle is a translation-spectral object: a nonzero spectral subspace carried by a positive-energy mass hyperboloid, on which the Poincaré group acts as one or more irreducible Wigner representations. A free-field mode expansion may realize that subspace, but it does not define it. An isolated physical-sheet pole can reveal the same particle when a chosen field couples to it; a resonance pole or threshold branch point cannot replace the Hilbert-space spectral projection.

Required background. Positivity, spectrum, covariance, and locality hypotheses fixes the translation representation; Wightman functions and spectral support gives its correlator signature; and the Wightman reconstruction theorem explains how the representation is recovered.

Helpful background. One-particle states: mass, spin, and normalization gives the physical representation; the Källén–Lehmann representation gives the pole measure; and mode bases and number operators shows why particle language may depend on global structure outside Minkowski vacuum theory.

The one-particle spectral subspace

Let $U(a)=e^{iP\cdot a}$ be the strongly continuous translation representation and $E(\Delta)$ its joint projection-valued measure. The positive-energy condition is $\operatorname{supp}E\subset\overline V_+$. For $m>0$, write

$$H_m^+=\{p:p^2=m^2,\ p^0>0\}.$$

If $H_m^+$ is an isolated component of the joint spectrum and $E(H_m^+)\neq0$, then

$$\mathcal H_1=E(H_m^+)\mathcal H$$

is the one-particle subspace. It is invariant under the Poincaré representation because the shell is Lorentz invariant. The restriction of $U$ decomposes into irreducible positive-energy representations labelled by mass $m$, spin, and any finite or countable multiplicity. Wigner’s classification identifies these irreducible representations in Wigner 1939, pp. 149–204.

Isolation supplies stability against decay: a vector on the shell is an exact eigenvector of the mass operator $M^2=P_\mu P^\mu$. A mass gap below multiparticle continuum is the simplest useful hypothesis, though refined Haag–Ruelle results can replace a global gap by regularity near a stable shell. “Particle” does not require that the entire spectrum be discrete; only the relevant shell must support a projection with the needed separation/regularity.

The projection onto a shell does not by itself select a single species. The restricted Poincaré representation may contain several spins, internal-charge sectors, or multiplicities at the same mass. One must decompose that representation and specify which summands the interpolating operators reach. Conversely, degeneracy does not spoil stability: it changes the multiplicity space, while the common mass shell remains an exact spectral component. This distinction prevents a mass value alone from being mistaken for a complete particle classification.

Pole, residue, and field dependence

For a scalar interpolating field,

$$\rho(\mu^2)=Z\,\delta(\mu^2-m^2)+\rho_c(\mu^2), \qquad Z>0,$$

gives

$$\widetilde W_2(p) =2\pi\theta(p^0)\rho(p^2).$$

The atom is the two-point signature of $E(H_m^+)$. If $\Psi_f=\phi(f)\Omega$, then its projected norm is

$$\|E(H_m^+)\Psi_f\|^2 =Z\int_{H_m^+}|\widetilde f(p)|^2\,\mathrm d\mu_m(p).$$

The same atom becomes a simple physical-sheet pole of the time-ordered two-point function, with residue $Z$ in the chosen normalization. This connection underlies the stable external-leg condition in the original LSZ formulation Lehmann, Symanzik, and Zimmermann 1955, pp. 205–225.

The particle subspace is field-independent, but $Z$ is not: a field orthogonal to the species can have $Z=0$ even though another local operator couples to it. Rescaling the field changes $Z$ without changing $\mathcal H_1$. Thus a pole in one correlator is evidence for a spectral particle only after positivity and the physical-sheet spectral representation are established.

Massive scalar application

For the massive scalar with $\rho=Z\delta(\mu^2-m^2)+\rho_c$ and $\operatorname{supp}\rho_c\subset[M^2,\infty)$, $M>m$, the joint spectral projection onto $H_m^+$ gives $\mathcal H_1$. Its vectors have norm from the Lorentz-invariant measure and transform in the spin-zero Wigner representation. The pole residue measures the overlap of the interpolating field with this subspace. This is the exact spectral input used by LSZ reduction: poles, residues, and stable external states.

As an independent check, the shell is invariant and $p^0=\sqrt{\mathbf p^2+m^2}$ has group velocity $\mathbf v=\mathbf p/p^0$ with $|\mathbf v|<1$. The continuum threshold $M>m$ prevents the one-particle atom from being approached by continuum mass values.

Counterexample: resonances and thresholds are not projections

A resonance is commonly represented by a pole reached only after analytic continuation through a cut to a nonphysical sheet. It is not in the spectrum of the self-adjoint $P^\mu$, whose spectral measure lives on real momentum. A threshold singularity lies at the edge of continuous spectrum and likewise need not carry an atom. Treating either as $E(H_m^+)$ fails the spectral theorem and supplies no normalizable stable state. The strongest surviving statements concern a resonance lifetime or continuum enhancement, not an exact one-particle subspace.

In massless gauge theories the charged spectral density can begin continuously at the nominal mass, with no delta function. That is the infraparticle case developed later; assigning $Z$ before locating an atom reverses the logic.

Exercises

Show that a simple rescaling $\phi\mapsto c\phi$ changes the pole residue but not the one-particle subspace.

Solution

Every two-point function is multiplied by $|c|^2$, so $Z\mapsto|c|^2Z$. The translation representation and its spectral measure $E$ are properties of the Hilbert-space theory, not of the normalization of one interpolating field. Hence $E(H_m^+)\mathcal H$ is unchanged.

References

• Lehmann, Harry, Kurt Symanzik, and Wolfhart Zimmermann. 1955. “On the Formulation of Quantized Field Theories.” Il Nuovo Cimento 1: 205–225. DOI.
• Wigner, Eugene P. 1939. “On Unitary Representations of the Inhomogeneous Lorentz Group.” Annals of Mathematics 40: 149–204. DOI.