The fascination of minimum length and minimum time

As shown below, combining three limits that hold across nature implies that twice the Planck length $\sqrt{4G\hslash /c^3}$ ≈ 3⋅10^-35 m is the smallest length in nature. It is the minimum length, the shortest measurable length. The result has been known for many decades. There is also a minimum time, given by twice the Planck time, or $\sqrt{4G\hslash /c^5}$ ≈ 10^-43 s. These minimum values force us to change many habits of thought and have fascinating consequences for the unified theory of nature, as told below.

A Clubhouse podcast in Paul Borril's show "It's about time!" on the topic of minimum length is available here (1-hour talk plus 2 hours of questions).

As shown on a separate page, the limits of nature also imply that 9 lines describe all of physics and all of nature.

See C. Schiller, From maximum force to physics in 9 lines and towards relativistic quantum gravity, Zeitschrift für Naturforschung A (2022).

The minimum length has a particularly fascinating consequence: the unified description of motion is unique.

The minimum length also contradicts an important contemporary philosopher, as illustrated by the added blue figure and as explained below:
 

 
● A simple deduction of minimum length
● Points, continuity, discreteness and sets do not exist
● Time is an approximation
● General relativity and quantum theory are consistent and compatible
● Space is made of constituents
● Elementary particle mass and energy are limited
● No more equations - mathematics is approximate physics
● Research: what the unified theory of physics cannot be
● Research: what the unified theory of physics must be
● Research: celebrities on quantum gravity
● A summary in one statement
 

A simple deduction of minimum length

In nature, there is no system faster than light: $v\le c$, where $v$ is speed.

In nature, there is no action below a quantum of action: $W\ge \hslash$, where $W$ is action.

In nature, there is no force larger than the maximum force or, equivalently, no system is more compact than a black hole of the same mass: $F\le c^4/4G$, where $F$ is force, or, equivalently, $m/d\le c^2/4G$, where $m$ is mass and $d$ is distance.

All observations and all experiments confirm these three limits. They are properties of nature.

The three limits of nature can be combined. Let us insert the mentioned limits into the dimensional definition for action $W$ (the quantity that measures the change of a system and that appears in the principle of least action) given by
            $W=F d t=F d^2/v$,
or, equivalently, into the dimensional definition
            $W=m c^2 t=(m/d) c d^2$.
In either case, inserting the limits for $W$, for $F$, for $v$, and for $m/d$, solving for $d$ we get
            $d\ge \sqrt{4\mathrm{G\hslash }/c^3}\approx \text{3⋅10^-35 m.}$
In other words, the combination of the three observed limits implies that nature has a minimum length. The minimum length is given by twice the Planck length.

A smaller length does not exist, in the same way that nothing faster than light exists, no mass smaller than a black hole of that mass exists, and no change smaller than Planck's quantum of action exists.

A further argument. Black holes have finite entropy that follows from the smallest system entropy k ln 2. (Note: the entropy of a black hole has never been measured. Nevertheless, there is no doubt that its order of magnitude is correct.) Thus they consist of a finite number of constituents. The entropy value, the so-called Bekenstein-Hawking entropy (look it up), implies that there is a smallest area in nature. It is (of the order of) the square of the minimum length, four times the Planck area. No smaller length or area values can be measured. Thus, no smaller length or area values exist.

There are additional ways to deduce the minimum length. Most important are the explorations of measurement errors by Mead or by Garay. The minimum length arises as the minimum measurement error in nature. All arguments, also those based on microscope resolution, are due to the three limits given above when they are applied to the explicit topic of length measurement error. A corresponding minimum measurement error holds for time.

There is not even a way to get near the minimum length. The smallest length that arose in any measurement so far is the limit on the electric dipole moment of the electron, which is about 1000 times larger that the minimum length.

The domain of nature where maximum speed, maximum curvature and the quantum of action play a role at the same time is called the domain of relativistic quantum gravity. It is a fascinating domain of nature.

Note: it is sometimes stated that at lengths smaller than the Planck length, our description of nature breaks down and that we need other theories in that domain. This is wrong. Nothing breaks down. Smaller lengths do not exist, like systems faster than light do not exist. Physics does not break down beyond the speed of light. Instead, such system speeds do not exist.

But – the reader might say – there are many authorities saying that the Planck length is not a smallest length. Answer: those authorities are wrong. Ask them for an experimental example. They cannot provide one. They confuse mathematical space and physical space. And they refuse to think about the mindboggling consequences of the minimum length. Which is understandable.

The three limits on speed, action, and force (or mass per length) also imply that no physical observable ever has trans-Planckian values. Nature has no trans-Planckian aspects or effects. For example, being invariant, length contraction cannot be used to produce smaller length values than the Planck length. There is no trans-Planckian physics.

But – the reader might say – there are many authorities saying that trans-Planckian physics exists. Answer: those authorities are wrong. Ask them for an experimental example. They cannot provide one. (Also point them to the section on the Planck mass later on.) They confuse mathematical ideas and nature. And they refuse to think about the mindboggling consequences of the lack of trans-Planckian physics. A few follow.

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Points, continuity, discreteness and sets do not exist

The minimum length implies that there are no points in nature. Points are extrapolations of our imagination into domains of nature where they do not apply. In nature, points do not exist.

The minimum length implies that continuity does not exist. Space is not continuous. Continuity only exists in our imagination. It does not exist in nature. Continuity is an approximation.

Conversely, the assumption of perfectly continuous space would imply a vanishing Planck length. This would imply a vanishing quantum of action, or a vanishing gravitational constant, or an infinite speed of light. In other words, exact continuity contradicts all of modern physics.

The assumption of perfectly continuous space and time would also imply the lack of measurement units of space and time. Almost no measurement unit could be defined.

Nothing in nature is exactly continuous. Space, time, fields, and wave functions are not continuous. They are only approximately continuous.

Euclidean geometry is only approximate. It is not the correct description of nature.

As mentioned, the minimum length is also the smallest possible length measurement error. In other words, everything in nature is slightly blurred or fuzzy. This means that the minimum length cannot be reached or observed in any experiment, independently of your budget. (The present world record is about 1000 times the Planck length.) This is in contrast to the speed of light, to the smallest action, or to the maximum force, which all can be observed. All combinations of any two constants $G$, $c$, and $\hslash$ can be observed and reached. But combinations of all three cannot.

In particular, the minimum measurement error implies that space is not a discrete set of points. The minimum measurement error thus implies that space is not discrete. Space is not a lattice - neither a regular one, nor an irregular one. Space is not made of pixels or voxels. (Note that on a screen, the borders between the pixels are defined with much higher precision than the width of the pixel. In nature, at the Planck length, such a higher precision is impossible. Therefore, nature is not made of pixels or voxels.)

Given that everything in nature is slightly blurred, there are no sharp boundaries in nature. Even the boundary of the universe is blurred.

Given that all boundaries are blurred, there are no sets in nature. At the same time, due to the blurring, all objects we observe in nature can only be approximately be modelled as elements. The minimum length implies that there are no elements in nature.

A remark: the minimum length thus eliminates Zeno's paradoxes.

In short, because $c$ is invariant and a limit ("nothing is faster than light"), and because $c^4\mathrm{/4}G$ or $c^2\mathrm{/4}G$ are invariant and limits ("nothing is denser than a black hole of the same mass or stronger than maximum force"), and because $\hslash$ is invariant and a limit ("no smaller action value can be measured"), it follows that the double Planck length given by $\sqrt{4G\hslash /c^3}$ is also invariant and a lower limit. The double Planck length is also the smallest possible length measurement error. Technical details are explained, for example, in this text.

As a consequence, space is neither a set nor is it made of points. Space is neither continuous nor discrete, but slightly fuzzy. In nature, space is something completely different from a set and something different from a mathematical space. In particular, physical space is not a mathematical space.

On the one hand, conventional continuous mathematical space (whether Euclidean or Riemannian) is an excellent approximation to nature. On the other hand, no deviations from continuous space and time are observable in any experiment. No experiment has ever detected an effect of the fuzziness of space. (Actually, this is not correct, as a deeper investigation shows; but this is too technical to be told here.) Indeed, we need the approximation of continuity to think and to talk about everything in nature. We cannot live without using continuous space when we talk and think about nature.
 

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Time is an approximation

The minimum length implies that there is a minimum time and a minimum time measurement error.

Thus there are no instants in time. Time is only approximately continuous, for durations that are longer than the minimum time. Between two "instants" (which do not exist) there is not always a third. Time is not discrete.

Also the origin of the universe is thus blurred. There is and was no time zero.

In short, like space, time is neither a set nor is it made of instants. Again, there is no trans-Planckian physics. Time is neither continuous nor discrete. In nature, time is something completely different from a set and from a mathematical space. Time is an approximation. But again, we need the approximation to think and talk, in particular about nature.

Nevertheless, no deviations from continuous space and from continuous time are observable. This is the disappointing result of relativistic quantum gravity.
 

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General relativity and quantum theory are consistent and compatible

The smallest length and the other Planck limits imply that general relativity and quantum theory never contradict each other. First, no experiment has ever shown such a contradiction. Second, contradictions could only exist at distances smaller than the minimum length or at other trans-Planckian values. But these do not exist.

All conceptual difficulties due to the combination of gravity and quantum theory disappear once space is seen as effectively continuous, and only so for all lengths longer than twice the Planck length. Gravity and quantum theory are compatible and consistent from the minimum length upwards. The same holds for time.

In the case of black holes, the minimum length fixes the entropy value. This again shows that gravity and quantum theory are compatible and consistent from the minimum length upwards.

The minimum length implies that singularities do not exist in nature. Singularities are figments of imagination. This has been known since at least 60 years. (But singularities are useful to get attention and sell media.)

In short, the Planck limits imply that general relativity and quantum theory are correct, consistent, and complement each other. You might have heard something different, maybe even for a long time. People can be fooled, nature cannot.
 

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Space is made of constituents

Because black holes have entropy, they are made of constituents behaving randomly.

Because the entropy of black holes is finite, black holes are made of a finite number of constituents. The constituents can be counted. They are discrete.

Because the minimum length fixes the entropy of black holes, black holes are made of constituents that are Planck-sized.

Because black holes have a surface and a horizon, the constituents are Planck-sized in two dimensions.

Black holes are both compressed matter and highly curved space. Therefore, the entropy of black holes implies that space and particles are made of common constituents.

Because space is extended, the common constituents must also be extended. Because the common constituents are Planck-sized in two dimensions, they must be extended in the third dimension.

Space is made of constituents that are discrete and extended.

Particles are made of constituents that are discrete and extended.

Black holes are made of constituents that are discrete and extended.

In short, space is neither continuous nor discrete.

In other words, the combination of general relativity and quantum theory implies that nature is made of common constituents that are extended and discrete, and that build up space, particles and black holes.
 

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Elementary particle mass and energy are limited

A particle is elementary if it is smaller than its Compton wavelength. An object larger than its Compton wavelength is called a composite. Electrons are elementary and pieces of soap are composite. The highest elementary particle mass occurs when its size - more precisely, its largest observable effective size, the (reduced) Compton length - which usually is much larger than its Schwarzschild diameter, becomes equal to the latter: $\hslash /mc\ge 4Gm/c^2$.

This yields the mass limit valid for elementary particles, $m\le \sqrt{\mathrm{c\hslash }/4G}$. This is half the Planck mass. The mass limit for elementary particles is about 10µg. Indeed, all elementary particles are much less massive. Note that the argument does not work for composite masses: pieces of soap are always larger than their Schwarzschild diameter. Indeed, composite masses are not limited by the Planck mass, as you can check in your bathroom.

For the same reason, the energy of an elementary particle is limited by $E\le \sqrt{c^5\hslash /4G}$, about 1 GJ, or half the Planck energy. This includes kinetic energy. The limit agrees with all observations about cosmic rays and all other radiation ever performed. Again, the argument does not work for pieces of soap. The chemical energy stored in 100 kg of soap is larger than the Planck energy. Please do not check this in your bathroom.

In short, the combination of general relativity and quantum theory implies that all elementary particles are less massive than half a Planck mass and have total energies smaller than half the Planck energy, as observed. Again, there is no trans-Planckian observation.
 

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No more equations - mathematics is approximate physics

It is common to say that mathematics is purer than physics. In particular, nothing is supposed to be purer than the integers. But can one count without using time or without using space?

Counting is impossible without either space or time. Ergo: physics is purer than mathematics. And minimum length provides the reason.

Minimum length implies that points do not exist in nature. The same is valid for boundaries, elements, sets, axioms, continuity, etc.

The minimum length implies that natural numbers do not exist in nature. In nature, natural numbers are approximations. Natural numbers are low-energy approximations because counting is only possible at low energy, i.e., at energies much lower than the Planck energy.

Strictly speaking, the natural numbers are figments of imagination. They arise when our fantasy extrapolates observations by ignoring minimum length and quantum gravity.

But mathematics is built on natural numbers – and on axioms and sets.

Kronecker stated: "Die ganze Zahl schuf der liebe Gott, alles Übrige ist Menschenwerk." Instead, quantum gravity implies that mankind also made the integers. (Though to be honest, all observers can be said to have made them, including animals and machines.)

Furthermore, real numbers and all continuous quantities only arise if the minimum length is ignored.

Thus, most mathematics only arises when the minimum length and quantum gravity are ignored and neglected.

Most mathematics, especially mathematical analysis, is not useful for exploring the consequences of minimum length or quantum gravity. Most mathematics is not useful at the Planck scale.

Mathematics is only useful at low energy. If you explore quantum gravity using mathematical analysis, you cannot reach the Planck scale.

Because of the lack of continuity and of real numbers, there are no equations in the domain of quantum gravity and of unification.

Conversely, because equations of motion for quantum gravity are impossible, any equation describing nature intrinsically neglects quantum gravity.

Most of mathematics – including sets, elements, axioms, points, and numbers – arises only once physics is approximated by neglecting minimum length, i.e., by neglecting quantum gravity.

Continuity, real numbers, three-dimensional space, time, dimensions, Hilbert spaces, fibre bundles, topological spaces and fractals are examples of approximations that arise by neglecting quantum gravity.

Every equation of physics that contains discrete or continuous quantities - thus every equation of physics - is an approximation.

Mathematics is built on physics.

Mathematics arises once physics is approximated - by neglecting quantum gravity.

The part of mathematics using real numbers and continuity is imprecise physics. This does not apply to discrete topology, in particular to knot theory, as told here.

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Research: what the unified theory of nature cannot be

The following statements are topics of research.

The 9 lines summarizing physics and the minimum length imply that the future unified theory has no evolution equation and no Lagrangian. There are several reasons. First, if the unified theory had a Lagrangian, it would assume continuous space and time and thus would contradict the minimum length. Second, minimum length implies minimum measurement errors for all observables. A unified evolution equation thus cannot exist in principle. Third, black hole entropy shows that space and matter are made of common constituents of Planck size. But single Planck-sized constituents cannot be observed. Thus, there is no way to define motion, a Lagrangian or an evolution for single constituents. This is only possible for many of them. Finally, in a unified Lagrangian, the quantities appearing in it would not be explained. But then the theory would not be unified. Lagrangians are intrinsically approximate. However, the unified theory cannot be an approximate description by definition.

The limits of nature imply that the mathematics of the unified theory of relativistic quantum gravity – what journalists call the “theory of everything” or “final theory” – is fully defined and determined by the minimum length and by the properties of the common constituents. There cannot be complicated, analytical mathematics in the unified theory.

The detailed arguments leading to the lack of analytic mathematics in the unified theory are given in C. Schiller, "Absence of a Lagrangian for the unified theory of relativistic quantum gravity" at https://www.researchgate.net/publication/364261495.

For the unified theory, the minimum length eliminates points and thus eliminates all continuous structures for space. For example, there cannot be quantum foam; there cannot be additional spatial dimensions; there cannot be additional structures that extend continuous space, such as fermionic coordinates; supersymmetry is eliminated; and there cannot be dualities or conformal field theory.

Measurement errors eliminate points and thus also eliminate all discrete structures for space. For example, space cannot be a graph or be made of invariant space atoms. Measurement errors also eliminate all discrete structures for space that extend usual three-dimensional space, because both discreteness and continuity are only approximations. This eliminates all dualities between the large and the small.

The lack of sets in nature, the lack of points, and the lack of sharp boundaries, taken together, eliminate all axioms from physics.

In short, the unified theory cannot have equations, cannot have complicated mathematics, cannot be continuous, cannot be discrete, and cannot be axiomatic.
 

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Research: what the unified theory of nature must be

The minimum length implies - as black holes show - that the common constituents of space and time have a Planck area cross-section and reach, like space itself, the boundary of the universe.

More precisely, the 9 lines imply that the common constituents are filiform, of Planck radius, and fluctuating. Details and consequences are explored in "From maximum force to physics in 9 lines and towards relativistic quantum gravity" at https://www.researchgate.net/publication/365750175.

The minimum length thus suggests that nature consists of strands. (Possible alternatives are excluded, as told in the paper.) A proposal that realizes all the mentioned requirements is found here.

The minimum length implies that space is emergent; it arises from large numbers of strands. Likewise, wave functions, particles, and interactions emerge from strands.

If particles and fields are made of strands, particles and fields must be due to configurations of strands. Wave functions and field quanta must also be due to configurations of strands.

If space, curvature, matter and radiation are made of strands, also physical action and Lagrangians must be due to strands.

If space, curvature, matter and radiation are made of strands, also all events and all motion in nature, with all its random and all its deterministic outcomes, must be due to strands.

The 9 lines imply that the unified theory has only two experimental effects: (1) general relativity and (2) the standard model of elementary particle physics, extended with massive Dirac neutrinos and PMNS mixing. Proposals for unified theories that require additional forces, particles, effects or concepts are unlikely to agree with observations.

Therefore, the 9 lines imply that the first test of any proposed unified theory of relativistic quantum gravity is the explanation of the forces and elementary particles found in nature.

If particles, interactions and space are made of strands, the particle masses and coupling constants must be due to statistics of strands. Fundamental constants must be unique and calculable.

Therefore, the 9 lines imply that the definite test of any proposed unified theory of relativistic quantum gravity is the calculation of the values of the elementary particle masses, of the mixing matrices, and of the coupling constants. Proposals for unified theories that do not allow such calculations are wrong.

The prediction of the neutrino masses and their mixings might well be the only predictions that are still possible in fundamental physics. All other constants have already been measured.
 

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Research: celebrities on quantum gravity

Regularly, celebrities make statements about quantum gravity and unification. But there is no royal road to geometry. And there is no shortcut for celebrities to quantum gravity.

Nature is not a computer. Given the minimum length and the minimum time, nature and what happens in it is not computable exactly, but only approximately, statistically, and only at scales far from the Planck scale. The Planck limits limit the possibility of computation. If you believe that nature is a computer, you are in contrast with the laws of nature.

Nature is not a computer nor a display. Nature cannot be divided meaningfully into cubes of Planck length. The arguments above imply that such a division does not describe any physical system or any physical variable. The same holds for spheres of Planck length, or for any other shape with Planck size in all three dimensions. Neither nature nor any physical system is made of Planck-sized voxels (volume elements). Such voxels would contrast with the difference between matter and empty space, with the observation of particles, and with every other observation. (Instead, nature, space and physical systems are made of randomly occurring strand crossing switches. Strands do not produce tight distributions of volume elements - but only rare and distant strand crossing switches.)

In contrast, black hole horizons can meaningfully be divided into surface elements of Planck area. But their arrangement fluctuates continuously. Such fluctuating surface elements yield black hole entropy. However, even for black holes, the Planck area is not observable. All other physical systems cannot meaningfully be divided into surface elements of Planck area. (Instead, nature, space, and physical systems consist of strands with randomly occurring strand crossing switches. On black hole horizons, the crossings are as tight as possible.)

Not even in everyday physical space can length values be meaningfully divided into elements of Planck length. In no experiment about physical space can Planck lengths be counted. The Planck length cannot be reached or observed.

Another way to put it: if nature were made of Planck-scale pixels or voxels, the boundaries between them would need to be defined more precisely than the Planck limits themselves. This is impossible.

So how can space, nature and physical systems be made of randomly occurring strand crossing switches? They can because only strands comply with the Planck limits and explain the particles, the forces and all their properties observed in nature.

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A summary in one statement

Because of minimum length, we can say: nature is pointless.

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