The fascination of minimum length and minimum time
As shown on the dedicated page, 9 lines describe all of physics and all of nature. Combining the 9 lines that describe all of physics implies that twice the Planck length (4G ℏ/c3)1/2≈ 3⋅10-35 m is the smallest length in nature. One often calls it the minimum length. The result has been known for many decades. There is also a minimum time, given by twice the Planck time, or (4G ℏ/c5)1/2≈ 10-43 s. The minimum values force us to change many habits of thought and have important consequences for the unified theory of nature.
For example, the Planck length
contradicts an important
● Time is an approximation
● General relativity and quantum theory are consistent and compatible
● Space is made of discrete constituents
● Mathematics is approximate physics
● Research: What the unified theory of physics cannot be
● Research: What the unified theory of physics must be
● Research: Millionaires on quantum gravity
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. Then combine the three limits.
As imple way is to use the expression for action W
W = F d t = F d2 / v
where F is force, d is distance, and v is speed. Equivalently, action is defined as
W = E t = m c2 t = (m/d) c d2,
where E is energy and m is mass. The expressions can be used to insert the maximum speed c, the minimum action ℏ and, for general relativity, either maximum force c4/4G or, if preferred, maximum mass per length ratio c2/4G. The last two limits are only achieved by black holes. The result in both cases is
d ≥ (4G ℏ/c3)1/2.
In other words, the domain of nature where maximum speed, maximum curvature and the quantum of action play a role at the same time - the domain of quantum gravity - is characterized by a minimum length, given by twice the Planck length.
A second argument. Black holes have finite entropy. Thus they consist of a finite number of constituents. Their entropy value (look it up) implies that there is a smallest length in nature. It is, again, twice the Planck length.
No smaller length value can be measured. No smaller length value exists. There are many other ways to deduce minimum length, for example by exploring measurement errors, as done by Mead and by Garay.
There is not even a way to get near that minimum length. (The smallest length that arose in a measurement so far is the limit on the electric dipole moment of the electron, which is over 1000 times larger that the minimum length.)
(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 latter statement is wrong. Smaller lengths do not exist. At all.)
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 would also imply the lack of most physical measurement units.
Nothing in nature is exactly continuous. Space, time, fields, and wave functions are not continuous, only approximately continuous.
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 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.
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 in order to think and talk, in particular about
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 discrete 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.
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.
Space is made of discrete constituents.
Particles are made of discrete constituents.
Black holes are made of discrete constituents.
In short, the combination of general relativity and quantum theory
implies that nature is made of a finite number
of common constituents that build up space, particles and black holes.
Mathematics is approximate physics
Minimum length implies that discreteness does not exist in nature. The same is valid for 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 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.)
Thus, mathematics only arises when the minimum length and quantum gravity are ignored.
Mathematics is not useful for exploring the consequences of minimum length or for exploring quantum gravity. Mathematics is not useful at the Planck scale.
Mathematics is only useful at low energy. If you explore quantum gravity using mathematics, you cannot reach the Planck scale.
Because equations of motion for quantum gravity are impossible, any equation describing nature neglects quantum gravity.
Mathematics – including sets, axioms, and numbers – arises only once physics is approximated by neglecting minimum length, i.e., by neglecting quantum gravity.
Fractals, continuity, dimensions of space, and real numbers are examples of approximations.
Every equation that contains discrete or continuous quantities - thus every equation - is an approximation.
Mathematics is built on physics.
Mathematics arises once physics is approximated.
Mathematics is imprecise physics.
Research: What the unified theory of nature cannot be
The following are topics of research.
The 9 lines and the minimum length imply that the future unified theory has no evolution equation and no Lagrangian. This lack of fundamental mathematics in the unified theory is shown in C. Schiller, "On the lack of a Lagrangian for the unified theory of relativistic quantum gravity" at https://www.researchgate.net/publication/364261495.
The 9 lines imply that the unified theory of relativistic quantum gravity – the “theory of everything” or “final theory” – is fully defined and determined by the minimum length. There cannot be complicated math in the unified theory.
For the unified theory, the 9 lines and the minimum length eliminate all continuous structures for space. For example, there cannot be additional spatial dimensions. There cannot be additional structures that extend continuous space, such as fermionic coordinates.
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 3d space, because both discreteness and continuity are only approximations.
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 - and black holes - 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 most decisive 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
predictions that are still possible in fundamental physics. All other
constants have already been measured.
Research: Millionaires on quantum gravity
Regularly, rich people make statements about quantum gravity and unification. But there is no royal road to geometry. And there is no shortcut for millionaires to quantum gravity.
Given the minimum length and the minimum time, nature is not computable exactly, but only approximately, at energies lower than the Planck energy. The Planck limits also limit computation.
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 - only rare strand crossing switches.
In contrast, black hole horizons can meaningfully be divided into surface elements of Planck area. (One can thus speak of black hole pixels.) Such surface elements describe their entropy. However, in practice, 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 are made of randomly occurring strand crossing switches. Strands produce tight distributions of such surface elements on black holes.
Again in contrast, not even in 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.
Instead, space, nature and physical systems are made of randomly occurring strand crossing switches.