The spacetime-based analysis of Matsas, Pleitez, Saa, and Vanzella defines the number of fundamental constants as the minimal number of independent standards that must be fixed and communicated so that arbitrarily separated laboratories can express and compare all observables, and shows this number to be two in Galilei spacetime (bona fide clocks and rulers) and one in relativistic spacetimes (bona fide clocks alone), reducing the speed of light cc, Newton's gravitational constant GG, Boltzmann's constant kBk_B, and Coulomb's constant kek_e to conversion factors. The argument presumes ideal bona fide clocks, but quantum mechanics admits no arbitrarily good clocks, so as clock precision approaches the Planck time (∼10−44 s\sim10^{-44}\,\mathrm{s}) the smooth relativistic spacetime underpinning the count fails. The open problem is the quantum-gravitational generalization: determine how many independent dimensional units — zero, one, or more — a quantum-gravity spacetime grants for expressing observables of the emergent laws, within an operational framework that recovers the classical counts in the ideal-device limit and is consistent with quantum limits on clock accuracy and gravitational backreaction on real clocks.

Background

How many fundamental constants does physics have? In a well-known debate, Okun defended the traditional count of three dimensionful constants (cc, ℏ\hbar, GG), Veneziano argued for at most two within superstring theory, and Duff advocated zero, holding that only dimensionless parameters such as the fine-structure constant α\alpha (the dimensionless measure of the strength of electromagnetic coupling) are physically meaningful. Matsas, Pleitez, Saa, and Vanzella (MPSV) proposed to settle the debate operationally by starting from spacetime itself. They define the number of fundamental constants as the minimal number of independent standards that must be fixed and communicated — in their image, the minimal set of apparatuses a cosmic factory must deliver to arbitrarily separated laboratories — so that all observables of the physical laws can be expressed and compared numerically. The construction of spacetime already demands idealized devices: bona fide clocks, pointlike apparatuses that assign reproducible real numbers to infinitesimal proper-time intervals along worldlines, and, only in the nonrelativistic case, bona fide rulers, identical one-dimensional rods at rest with respect to an inertial congruence of observers. MPSV show that the units fixed by the very apparatuses needed to construct the spacetime suffice to express every observable of the theories built over it: in Galilei spacetime, whose foliation into surfaces of absolute simultaneity requires both clocks and rulers, the count is two; in relativistic spacetimes (a smooth manifold M\mathcal{M} endowed with a metric gg), where distances can be reconstructed from proper times registered by inertial clocks alone, the count is one. The constants cc, GG, kBk_B, and kek_e (the speed of light, Newton's gravitational constant, Boltzmann's constant, and Coulomb's constant) are thereby downgraded to conversion factors between units, and the cesium-133 hyperfine transition frequency can be elected as the single communicated standard. The analysis, however, rests on the operational availability of ideal clocks. Quantum mechanics tells us there are no arbitrarily good clocks (Peres, 1980): clock accuracy is quantum-limited, and a clock capable of resolving intervals near the Planck time ∼10−44 s\sim10^{-44}\,\mathrm{s} would gravitationally disturb the very geometry it probes. MPSV note that present atomic clocks, with precisions better than 10−17 s10^{-17}\,\mathrm{s}, are excellent realizations of bona fide clocks for all practical purposes, and state explicitly that at the Planck scale we will most probably be urged to replace the smooth-spacetime concept with something else — adding: How many dimensional units (if any) will be granted by the quantum gravity spacetime (to express observables of the new laws of physics) we do not know. Two further points are noted in the source: whether nature realizes perfect bona fide clocks and rulers is, in the authors' words, a separate issue, which they touch on briefly in connection with the quantum limits on clock accuracy; and the metrological choice of which empirically constant quantity to fix as the single standard — for instance, fixing hh (Planck's constant) rather than GG in the 2019 definition of the kilogram — is discussed as a pragmatic convention. Related work develops the quantities and devices in which the comparison of measurements by distant laboratories can be re-expressed once spacetime itself is quantum: relational observables — defined relative to other physical systems or dynamical fields rather than to a fixed external background — and quantum reference frames — frames treated as quantum systems that may themselves be in superposition or entangled with the systems they measure (Vanzella and Butterfield, arXiv:2406.15838).

Problem Statement

Determine how the minimal-number-of-standards question generalizes once the background spacetime is itself subject to quantum-gravitational modification. Concretely, within an operational framework parallel to the classical analysis — in which observables are expressed through standards fixed and communicated between arbitrarily separated laboratories — establish how many independent dimensional units, if any, a quantum-gravity spacetime grants for expressing the observables of the emergent physical laws: zero (no dimensional unit is granted or needed; all observables of the final theory are ultimately dimensionless and no standard need be communicated), one (a single standard, for instance a fundamental time or discreteness scale, replaces the bona fide clock), or more. Establish this by working in one or more explicitly specified quantum-spacetime settings and making precise, there, what an operational comparison between distant laboratories means, stating the setting(s) explicitly and whether the resulting count is specific to them or holds more generally across quantum-gravity settings, and by recovering the classical counts — two standards in Galilei spacetime and one in relativistic spacetimes — in the appropriate classical, ideal-device limit.

References

  1. George E. A. Matsas, Vicente Pleitez, Alberto Saa, Daniel A. T. Vanzella, 'The number of fundamental constants from a spacetime-based perspective', Scientific Reports 14, 22594 (2024), DOI: 10.1038/s41598-024-71907-0, arXiv:2311.09249, https://arxiv.org/abs/2311.09249
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