A 1939 Reviews of Modern Physics note by Werner Heisenberg, summarizing his theory of 'explosion showers' in cosmic rays, reported a correspondence-principle result — total cross sections for particle production tend to a finite, non-zero value at extremely high collision energies, so that every interaction acquires a finite effective range — and closed by remarking that this fact 'may be important in the discussion of the question as to whether the measurement of position or of distance can be performed with arbitrary accuracy.' The research question is that question in its original generality: do the established laws of physics (quantum theory, special relativity, and gravitation) impose a fundamental lower bound on the uncertainty of position and distance measurements — a smallest resolvable length — or can measurement accuracy be made arbitrarily high in principle? The specific nuclear-scale length proposed in the 1930s was historically refuted, and measured total cross sections rise roughly logarithmically rather than saturating; but the question itself remains open and is pursued today chiefly at the gravitational (Planck) scale, where gravitational backreaction, black-hole formation, and generalized-uncertainty-principle arguments suggest a minimal measurable length without a universally accepted derivation from established principles, and without an accepted demonstration of unbounded accuracy either. A definitive answer in either direction would reshape the operational meaning of locality and distance in fundamental physics.

Background

In the late 1930s, cosmic rays were the only source of particles far above accelerator energies, and the showers of secondary particles they trigger in the atmosphere were the central testing ground for the young quantum field theories. Two pictures competed: cascade showers, in which a primary particle builds up a large multiplicity through a sequence of two-body processes, and 'explosion showers', in which a single collision produces many particles at once. Yukawa's meson theory of the nuclear force (Yukawa 1935) introduced a new particle, then called the mesotron, whose interaction with nuclear particles leads to cross sections that increase with collision energy, so that ordinary perturbative quantum mechanics was expected to fail at high energies. Heisenberg therefore analyzed high-energy mesotron collisions using the classical field equations supplemented by correspondence-principle arguments — extracting the high-energy behavior from the classical wave equations in the regime where ordinary perturbation theory fails, the same logic as the Bloch–Nordsieck theory of the radiation field of a relativistic electron (Bloch and Nordsieck 1937). The outcome, reported in a one-page review note (Heisenberg 1939) that summarizes his two-part investigation (Heisenberg 1936 and the companion paper in Zeitschrift für Physik 113 of 1939), was that in the limit of extremely high collision energies the total cross sections for the production of other particles tend to a finite value and 'do not decrease to zero': since any collision can create new particles, every interaction acquires a finite effective range. A total cross section measures the effective collision area of a scattering process; its remaining finite and non-zero at arbitrarily high energy means that short-distance probes stay strongly coupled rather than becoming transparent.

The note closes by connecting this result to localization: 'This fact may be important in the discussion of the question as to whether the measurement of position or of distance can be performed with arbitrary accuracy.' The connection runs through the Heisenberg microscope: to resolve a distance Δx\Delta x one needs a probe of momentum of order ℏ/Δx\hbar/\Delta x, hence ever higher collision energies; if at extreme energies every collision turns into a many-particle 'explosion' whose total cross section never vanishes, then no scattering measurement can resolve arbitrarily small distances. This was exactly the program of Heisenberg's 'universal length' paper (Heisenberg 1938), which proposed a fundamental length r0r_0 — expected not far beyond the classical electron radius (2.8×10−152.8\times10^{-15} m, the scale at which the electron's electrostatic self-energy equals its rest mass) — such that 'measurements of positions are not possible to a precision better than r0r_0' (quoted in Hossenfelder 2013). Notably, that paper set gravity aside as irrelevant, because the gravitational (Planck) length, ℓP=ℏG/c3≈1.6×10−35\ell_P=\sqrt{\hbar G/c^{3}}\approx1.6\times10^{-35} m, lies far below r0r_0.

Both pillars of the 1939 argument were later reshaped. First, the empirical premise: the Froissart bound (Froissart 1961), derived from analyticity and unitarity of scattering amplitudes, limits total cross sections to grow at most like Cln⁡2sC\ln^{2}s (with ss the square of the center-of-mass collision energy) — growth without bound is allowed, and saturation to a constant is not required. Measurements agree with slow logarithmic growth rather than saturation: the proton-proton total cross section rises from tens of millibarns (1 mb = 10−3110^{-31} m2^2, the standard unit of scattering area) at 1970s collider energies to σtot=(110.6±3.4)\sigma_{\rm tot}=(110.6\pm3.4) mb at s=13\sqrt{s}=13 TeV (Antchev et al. 2019), while remaining finite and non-vanishing, as the note's weaker claim required. Second, the interpretation: the high-multiplicity cosmic-ray events were eventually explained by strong-interaction multiparticle production, the Standard Model proved renormalizable — its short-distance divergences absorbable into a finite number of parameters, with no new cutoff length — and experiments probed distances many orders of magnitude below r0r_0 without encountering any fundamental resolution limit; Pauli's 1947 warning that the 'universal' length might be 'a mere figment of imagination' (quoted in Hossenfelder 2013) proved prescient for that scale. Kragh (1995) gives the standard history of this episode. In parallel, rigorous results sharpened what relativistic quantum theory says about localization itself: Newton and Wigner (1949) constructed localized particle states (which behave badly under Lorentz transformations), and Hegerfeldt (1974) showed that a strictly localized state immediately develops tails spreading at superluminal speed — localizability of particles is structurally obstructed in relativistic quantum field theory, though these theorems concern states rather than operational measurement accuracy.

The source's question survived its first wrong answer and migrated to the gravitational scale. Mead (1964) argued, through measurement thought experiments that include the gravitational backreaction of the probe, that no distance can be measured to a precision better than the Planck length; Doplicher, Fredenhagen and Roberts (1995) built a rigorous algebraic model of a quantum spacetime at the Planck scale in which localization below a fundamental cell is impossible; Garay's review (1995) and Calmet (2008) conclude from related arguments that quantum mechanics, general relativity and causality forbid probing — and hence measuring — lengths below the Planck scale. The modern literature implements such limits via the generalized uncertainty principle (GUP), a modification Δx≳ℏ/Δp+α G Δp/c3\Delta x\gtrsim\hbar/\Delta p+\alpha\,G\,\Delta p/c^{3} of the uncertainty relation that possesses a nonzero minimum in Δx\Delta x; Hossenfelder's review (2013) surveys the thought experiments, the model implementations, the phenomenology, and the proposed ways to circumvent a minimal length, and poses the question — whether the fundamental laws of nature limit our ability to probe arbitrarily short distances — as unresolved.

Problem Statement

Whether the measurement of position or of distance can be performed with arbitrary accuracy — that is, whether the established laws of physics (quantum theory, special relativity, and gravitation) impose a fundamental lower bound on the uncertainty of any operational position or distance measurement, a smallest resolvable length, or whether measurement protocols of unbounded accuracy are possible in principle. Here a 'measurement' means any physical procedure by which a position or a distance is determined, and its 'uncertainty' is the position/distance uncertainty Δx\Delta x in the sense of the quantum-mechanical uncertainty relation, as in the generalized-uncertainty-principle expression Δx≳ℏ/Δp+α G Δp/c3\Delta x\gtrsim\hbar/\Delta p+\alpha\,G\,\Delta p/c^{3} discussed in the Background.

The question originates in the source, in its original generality. Heisenberg posed it on the basis of a specific physical claim: that the finiteness and non-vanishing of total cross sections at extremely high collision energies — equivalently, the finite effective range that virtual particle production gives to every interaction — implies that arbitrarily short distances cannot be probed, so that position and distance measurements have an irreducible resolution limit (Heisenberg 1939; Heisenberg 1938). That specific premise was superseded: the nuclear-scale length originally proposed was historically refuted, and the asymptotic behavior of measured total cross sections is slowly rising rather than saturating. None of this settles the underlying question itself, which remains open and is now pursued chiefly at the gravitational (Planck) scale; the target here is that general question, not the fate of the 1939 cross-section argument.

At the level of physical principle rather than of any particular instrument, the question is whether a universal lower bound on the uncertainty of any operational position or distance measurement follows from established principles alone (quantum theory, special relativity, and gravitation) — valid for all measurement procedures, not a restricted class — or whether operational measurement protocols of unbounded position/distance accuracy are possible in principle under the established laws, that is, whether for any prescribed target accuracy a position or distance measurement attaining that accuracy can in principle be performed. A result derived only within a particular speculative framework (a specific quantum-gravity model, string theory, a specific noncommutative geometry) resolves the question only relative to that framework.

References

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