### 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. The minimum values force us to change many habits of thought and have fascinating consequences for the unified theory of nature. Details are provided below.

For example, the Planck length contradicts an important contemporary philosopher: ### Points, continuity and discreteness do not exist

There is nothing more compact than a black hole; there is nothing faster than light; there is no action below a quantum of action. These three limits can be combined.

Take the maximum speed $v\le c$,
the minimum action $W\ge \hslash$,
and a black hole limit of general relativity: either the maximum force $F\le {c}^{4}/4G$
or, if you prefer, the maximum mass per length ratio $m/d\le {c}^{2}/4G.$

Insert the limits into the definition for action $W$ (the quantity that measures the change of a system and that appears in the principle of least action) given by
$W =Fdt =Fd2/v,$
where $F$ is force, $d$ is distance, $t$ is time, and $v$ is speed. Equivalently, action is

where $E$ is energy and $m$ is mass. After inserting the limits, in both cases the result is

$d ≥ 4Gℏ / c3 ≈ 3⋅10-35m.$

In other words, the combination of the three limits implies that nature has a minimum length. It is given by twice the Planck length.

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

A second argument. Black holes have finite entropy. Thus they consist of a finite number of constituents. Their entropy value, the so-called Bekenstein-Hawking entropy (look it up), implies that there is a smallest area in nature. It is the square the minimum length, four times the Planck area.

No smaller length value can be measured. No smaller area. No smaller length or area values exist! There are many other ways to deduce minimum length, for example by exploring measurement errors, as done by Mead or by Garay. Also the minimum length measurement error in nature is given by the minimum length. And the same for time.

There is not even a way to get near that 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 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.)

The minimum length implies there are no points in nature. Points are an extrapolation of our imagination. 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 either 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.

The minimum length is also the smallest possible measurement error. In other words, everything in nature is slightly blurred.

The minimum measurement error also implies that space is not discrete. Space is not a lattice - neither a regular one, nor an irregular one, nor is it a random set. The minimum measurement error implies that space is not a discrete set of points.

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.

The minimum length eliminates Zeno's paradoxes.

The three limits on speed, action and mass per length (or force) imply that no physical observable ever has trans-Planckian values. (See the section about mass and energy below for details.) There is no trans-Planckian physics.

In short, space is neither a set nor is it made of points. Space is neither continuous nor discrete. In nature, space is something completely different from a set and from a mathematical space.

On the one hand, conventional space (whether Euclidean or Riemannian) is an approximation. On the other hand, we need this approximation to think and talk, in particular about nature.

### 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. 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.

### General relativity and quantum theory are consistent and compatible

The smallest length implies that general relativity and quantum theory never contradict each other. Indeed, they would do so only at even smaller distances. 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 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 is known since at least 60 years.

In short, general relativity and quantum theory are correct, consistent, and complement each other.

### 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 short, 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.

### 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, pieces of soap are composite. The highest elementary particle mass occurs when its largest size measure, the (reduced) Compton length, usually much larger than its Schwarzschild diameter, becomes equal to it: $ℏ/mc ≥ 4G m/c2$.

This yields the limit case $m ≤ cℏ / 4G$ for the mass of an elementary particle. The mass limit is about 10µg and 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, their mass is 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 $E ≤ c5ℏ / 4G$, about 1 GJ. 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 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.

### Mathematics is approximate physics

Minimum length implies that points do not exist in nature. The same is valid for 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." Quantum gravity implies: mankind also made the integers. (Though to be honest, this holds for all observers, including animals and machines.)

Also real numbers and all continuous quantities only exist when the minimum length is ignored.

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

Most mathematics, especially mathematical analysis, is not useful for exploring the consequences of minimum length or for exploring 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 equations of motion for quantum gravity are impossible, any equation describing nature simply 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.

Fractals, continuity, topological spaces, dimensions of space, and real numbers 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.

Mathematics is imprecise physics.

### Research: what the unified theory of nature cannot be

The following statements are topics of research.

The 9 lines 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 contradict the minimum length. Second, minimum length implies minimum measurement errors for all observables. An 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 can be deduced - only 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. But the unified theory cannot be approximate.

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 shown 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 all continuous structures for space. For example, there cannot be quantum foam or additional spatial dimensions. There cannot be additional structures that extend continuous space, such as fermionic coordinates. Supersymmetry is eliminated. There cannot be dualities or conformal field theory.

Measurement errors 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 the 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.

### Research: what the unified theory of nature must be

The minimum length - as black holes show - implies that the common constituents of space and time have a Planck area cross-section and also 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 have to be eliminated; this is done 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, also 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 randomness 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 not correct.

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.

### 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, and only at scales far from the Planck scale. The Planck limits also limit 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 hold for spheres of Planck length, or 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.

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

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