### 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. Details below.

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

For example, the Planck length
contradicts an important
contemporary philosopher:

● 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

● 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

### Points, continuity and discreteness do not exist

There is nothing faster than light; there is no action below a quantum of action; there is nothing more compact than a black hole. All observations confirm these three limits. They are without doubt. The 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

$W=Fdt=F{d}^{2}/v,$

where $F$ is force, $d$ is
distance, $t$ is
time, and $v$ is speed. Equivalently, action is also

$W=Et=m{c}^{2}t=(m/d)c{d}^{2},$

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

$d\ge \sqrt{4\mathrm{G\hslash}/{c}^{3}}\approx \text{3\u22c510-35m.}$

In other words, the combination of the three observed 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. (Note: as is well known, the entropy value 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 the square of the minimum length, four times the Planck area. No smaller length or area values can be measured. No smaller length or area values exist!

There are additional ways to deduce minimum length. Most important are the explorations of measurement errors by Mead or by Garay. The minimum length measurement error in nature is again given by the minimum length. The arguments, also those based on microscope resolution, are due to the three limits given above, applied to the explicit topic of measurement errors. A corresponding error limit 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.)

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

As mentioned, the minimum length is also the smallest possible
measurement error. In other words, everything in nature is *slightly
blurred or fuzzy.*

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, nor is it a random set.

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 force (or mass per length)
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,** because c is invariant and a limit ("nothing is faster
than light"), and because c^4/4G or c^2/4G are invariant and limits
("nothing is denser than a black hole of the same mass"), and because h-bar
is invariant and a limit ("no smaller action value can be measured"), it
follows that the double Planck length given by (4 G h-bar/c^3)^(1/2) is
also invariant and a limit.

Thus, 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 or a mathematical space. Physical space is not a mathematical space.

On the one hand, conventional continuous mathematical space (whether
Euclidean or Riemannian) is an approximation to nature. On the other hand,
we *need* this approximation to think and to talk, in particular
about nature. We cannot live without using space.

### 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 and the other Planck limits imply that general
relativity and quantum theory *never contradict* each other.
Indeed, they would do so only at even smaller distances or at
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 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.

### 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 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*
$m\le \sqrt{\mathrm{c\hslash}/4G}$ for elementary particles, 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 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

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.

Minimum length adds the details.

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 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 (mostly) 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 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. There cannot be dualities or
conformal field theory.

Measurement errors eliminate points and this 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 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.

### 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.

* *