# Fundamental Length

## Fundamental Length

the hypothetical universal constant of length. The fundamental length establishes the limits of applicability of such fundamental physical concepts as the theory of relativity, quantum theory, and the causality principle in physics. The scales of space-time and energy-momentum regions where new phenomena may be expected that fall outside the scope of present-day concepts may be expressed in terms of the fundamental length *I*. Such regions are characterized by dimensions *x < I*, time intervals; < *l/c*, and energies *E > ℏell*, where *c* is the speed of light and *ℏ* is Planck’s constant. If the expectation of new phenomena is warranted, as is indicated by the difficulties and inconsistencies of present-day theory, then yet another radical transformation of physics that will be comparable in its consequences to the development of the theory of relativity or the development of quantum theory is imminent. Accordingly, the fundamental length will be an essential element of a future consistent theory of elementary particles, playing the role of the third fundamental dimensional physical constant, in addition to *c* and *ℏ*. In that role, it will establish the limits of applicability of old concepts.

At various times, the following lengths have been considered as possible candidates for the fundamental length: the Compton wavelength of the electron λ_{e} ≈ 10^{–11} cm (the length at which quantum processes become significant for the electromagnetic interaction), the Compton wavelength of the pion λ_{N} ≈ 10^{–13} cm, the Compton wavelength of a nucleón λ_{N} ≈ 10^{–14} cm (the length at which quantum processes become significant for the strong interaction), the characteristic length of the weak interaction (approximately 10”^{16} cm), and the Planck length (of the order of 10”^{33} cm; the length at which quantum processes become significant for the gravitational interaction). The identification of the fundamental length with one of the above quantities would be of tremendous importance, as it would indicate the type of interaction with which the development of new physical concepts would be associated.

As of 1977, experiments had shown that the fundamental length does not exceed 10”^{15} cm; arguments based on measurements using the Môssbauer effect favor an even smaller upper limit to the fundamental length, that is, an upper limit of the order of 10^{–20} cm. Therefore, quantities associated with the electromagnetic interaction, the strong interaction, and possibly the weak interaction may no longer be regarded as candidates for the fundamental length. The Planck length is very likely to turn out to be the true fundamental length in physics. An argument in favor of the Planck length is the universal nature of gravity, which—in contrast to the other interactions—affects all structural units of matter without exception. In this case, the theory of elementary particles should be constructed on the basis of the general theory of relativity.

An experimental means of determining the fundamental length is to compare experimental results with theoretical results for various physical effects obtained in accordance with existing theory. In all cases where such comparisons can be made, no discrepancies of any kind have yet been demonstrated. Therefore, experiments thus far have yielded only an upper limit to the fundamental length. High-energy experiments carried out in particle accelerators and characterized by a relatively low accuracy are the primary means for making such comparisons. They include experiments aimed at verifying the dispersion relations (*see*STRONG INTERACTION) for pion scattering by nucléons and other particles and experiments for testing certain predictions of quantum electrodynamics, such as pair production and electron scattering by electrons. Other means for making the comparisons consist in high-precision static experiments, such as the measurement of the anomalous magnetic moment of the electron or the muon and the measurement of the Lamb shift. As noted above, the Môssbauer effect yields specific information on the fundamental length. Proposals have been considered for using information obtained from celestial objects, such as ultrahigh-energy (> 10^{19} electron volts) cosmic rays, pulsars, quasars, and black holes. If the fundamental length exists, the radiation from some of these objects would have properties that would be exceptional from the viewpoint of present-day concepts.

Models of a theory that incorporates the fundamental length are being developed. Such models include modifications of nonlocal quantum field theory and modifications of the theory of quantized space-time. In addition to being of value in their own right, such theoretical schemes are of use in designing experiments aimed at determining the fundamental length and in analyzing the results of such experiments.

*See also*MICROCAUSALITY CONDITION; NONLOCAL QUANTUM FIELD THEORY; CAUSALITY PRINCIPLE; and QUANTIZATION, SPACE-TIME.

### REFERENCES

Tamm, I. E.*Sobr. nauchnykh trudov*, vol. 2. Moscow, 1975.

Markov, M. A.

*Giperony i K-mezony*. Moscow, 1958.

Markov, M. A. “O modeli protiazhennoi chastitsy v obshchei teorii otnositel’nosti.” In the collection

*Nelokal’nye, nelineinye i nenormiruemye teorii polia: Materialy 2 soveshchaniia po nelokal’nym teoriiam polia*. Dubna, 1970.

Kirzhnits, D. A. “Problema fundamental’noi dliny.”

*Priroda*, 1973, no. 1.

Kirzhnits, D. A. “The Quest for a Fundamental Length.”

*Soviet Science Review*, September 1971, p. 297.

D. A. KIRZHNITS