Dear Igor,

When we think of the smallest distance of space-time, we have today two physical theories that help us: Gravitation and Quantum Field Theory. Gravity tells us that spacetime is directly associated with the energy momentum through the Einstein's equations. Its Schwarzschild solution provide us with the minimum distance where this theory could be applied at low distances, i.e. the Schwarzschild radius

Rs=2Gm/c2

for a particle of mass m. On the other hand there is another critical dimension for such particle mass if it would be created virtually in QFT, i.e. at half its Compton wavelength

Lc=hbar/2mc

This allow us to define a new length only dependent of universal constants and without taking into account the mass of the particle which called the Planck length

Lp=(Rs.Lc)1/2=(hbar.G/c3 )1/2

which could be taken as the lower distance that we can divide the spacetime using our present physical theories. Below this distance Einstein's gravitation doesn't work and QFT cannot determine also the behaviour of a particle in such volume of the spacetime. This is the physical the real physical behaviour although we could speak about its role in the Hawking entropy definition of a black hole or other interesting physical quantities related with singularities of the spacetime.

This is a great question. In addition to the excellent answers already given, there is a bit more to add.

Planck length lp is defined to be

lp = 1.62 x 10-35 m

Considered in the context of discrete space, John A. Wheeler observed in 1957 that Planck particles have an average spacing equal to Planck length. See p. 27 in the paper by D. Crouse and J. Skufca:

https://arxiv.org/pdf/1803.03126.pdf

For more about this, see

https://arxiv.org/pdf/1805.11212.pdf

Thank you all for the answers, but if I may, I will add from myself. I have my own picture of the world in which the universe is placed in a drop (the shape of a seven-dimensional sphere). However, for simplicity, you can consider a two-dimensional sphere. Then the winding of the torus stretched on a sphere with the poles punched is a plane, pseudo - Riemannian metric which is induced by the ratio of the diameters of the defining circles of the torus and evolves due to the fact that one diameter increases and the other-decreases. In addition, the torus winding is quantized because its latitude on the sphere is a multiple of \pi.

Yes, this this size is really too small. For example, it is almost 1020 time the classical electron radius, i.e. me c2=Ke2/re where K=9.109 (re has a value around 2.8 fm) the electrostatic constant in MKS system of units

lp/re= 10-20

Obviously this distance cannot be compared with any acceleration or Lorentz contraction because there is not associted motion with respect to one observer. This is just a comparition of Compton distance with Schwarzschild radius, where the dependence with the mass disappears.

Igor, are you joking?

Is this a form to say that you don't understand anything of our explantations? My answer can be reduced to say that the Planck length is one unit of distance as can be the meter. But this distance is obtained from physical constants and the physical meaning for reducing it to such constants is comparing the quantum field theory with gravitation theory in their smaller limits. This is all!

Obviously your suggestion of the acceleration is not related with this distance and less the Lorentz contraction because there is not a relative motion to take into account.

BTW the torus and the sphere are very different objects. One has genus 1 while the is 0.

Daniel, I'm not kidding and your explanations I understand well, but things like Planck quantities need to be looked at wider otherwise you will not advance in the unification of gravity and quantum theory not an iota. And then you didn't notice that sphere with the poles poked out.

Dear Igor,

your phrase:

Then the winding of the torus stretched on a sphere with the poles punched is a plane, pseudo - Riemannian metric which is induced by the ratio of the diameters of the defining circles of the torus and evolves due to the fact that one diameter increases and the other-decreases. In addition, the torus winding is quantized because its latitude on the sphere is a multiple of \pi.

Then the winding of the torus stretched on a sphere with the poles punched is a plane, pseudo - Riemannian metric which is induced by the ratio of the diameters of the defining circles of the torus and evolves due to the fact that one diameter increases and the other-decreases. In addition, the torus winding is quantized because its latitude on the sphere is a multiple of \pi.

absolutely nonsense for me. Even if I understand that you try to map a torus on a sphere, you need to say what is the relation with the Planck constant that you have asked about. And on the other hand this just topology, algebraic topology or just geometry depending of how you work with the local charts of the manifolds, but the important is that we need what these geometrical figures mean. In another case, I think that you are joking with your thread or you are asking without understanding anything of gravity, quantum gravity or physics in general. Do you understand me?

If my metaphysical ideas don't correlate with physics, then either I'm not good at explaining, or you don't understand something, or these ideas don't really relate to physics

Sorry, forget my answers which were all within a physical context. My ignorance in metaphysics is absolute and I don't know what can mean Planck size there.

Let me make a small remark: "Physics is a view of the world through the eyes of a local observer, and metaphysics is a view of the world through the eyes of an absolute (global) observer."

Dear Sidney,

The metric is not at all invariant in the Universe, in fact it is the variable which gives the gravitation information. On the other hand QFT can have special topological solutions as magnetic monopeles, instantons, etc but its fundamentals are always local (thanks to that is possible to use the Feynman diagrams among other perturbative techniques). If you want to go to a global quantum field theory you have quantum topological field theory which is being developing nowadays with quite success

https://en.wikipedia.org/wiki/Topological_quantum_field_theory

Dear Sidney,

Quantum Field Theory is well developed subject which almost all the students in Physics needs to know nowadays. There are classical textbooks as:

Itzykson, Zuber: Quantum Field Theory, Dover 1980

Weinberg: Quantum Field Theory I + II, Cambridge 1995

The Topological Quantum Field Theory is really a new subject, but you can consider it as you want, but the aim is clearly different because it is devoted to study the topological invariants as Atiyah-Singer topological index or so on. Nowadays is also evolving quite well in Condensed Matter with subjects as Topological Insulators, Wey semimetals and so on.

But in any case, could you explain me the Igor's sentence that I was speaking on:

Then the winding of the torus stretched on a sphere with the poles punched is a plane, pseudo - Riemannian metric which is induced by the ratio of the diameters of the defining circles of the torus and evolves due to the fact that one diameter increases and the other-decreases. In addition, the torus winding is quantized because its latitude on the sphere is a multiple of \pi.

Dear Sidney,

The projection of the torus, S1xS1 on the sphere S3 (not a S2 sphere as it is written) is not trivial at all and I cannot see the pseudo-Riemann metric in anywhere. On the other hand the torus has not an only winding (number) associated as it is written, it has two.

But what is more strange, and the reason for asking if Igor was really joking with us, is that this is for answering at a question on the Planck's length. What is the realationship of all this with the Planck's length?

Follow this link

https://arxiv.org/pdf/physics/0610127

kind regards

What is the Planck length [and other Planck units, though] – see https://www.researchgate.net/publication/273777630_The_Informational_Conception_and_Basic_Physics DOI 10.5281/zenodo.16494

Cheers

The Planck length is just the result of dimensional analysis: there exists a combination of Newton's constant, Planck's constant and the speed of light, that has the dimensions of length. Whether something physical occurs at such a length scale, whose numerical value in meters is obtained by replacing in the expression the known numerical values of these constants, isn't known. And the reason is that at such a length scale no mathematical description of natural phenomena is currently available.

There's nothing to interpret.

On the one hand you are right - there is physics and there is nothing to interpret. However, if you had provided us with an interpretation of the principle of minimal action within a geometric-dynamic model, no one would have condemned you.

It is absolutely true, there are nothing to interpret, what we need to know is that below this distance Einstein's gravitation doesn't work and QFT cannot determine also the behaviour of a particle in such volume of the spacetime.

No-there's no known physics to be described at the Planck length, that's all. The only thing that's possible to say is that an object of mass greater than the Planck mass, whose size is less than the Planck length, would have collapsed to a black hole-assuming Einstein's equations are valid, which it is known they aren't, since they assume that matter is classical, while it has known quantum properties at scales much greater.

Dear Stam,

The Planck scale is a limit for being explored if theories as quantum gravity want to be developed. Let me to suggest the reading of the very interesting nature:

https://www.nature.com/articles/nphys2262

No, because it's not known whether there's anything particular really happening there and the reason is that it's not known how to count the quantum states at that length scale, so it's not possible to do any calculation in quantum mechanics. As, already, mentioned, it would have been the Schwarzschild radius of a black hole of mass the Planck mass, were Einstein's equations taken to be valid. But these are classical notions.

In the article in Nature, unfortunately, they don't do what they claim to do. They study the quantum fluctuations of a mesoscopic quantum system-i.e. of length scales orders of magnitude greater than the Planck length-in a fixed potential, which describes a classical, non-relativistic, gravitational background. Absolutely nothing to do with quantum gravity. Indeed such experiments have been done in the past with cold neutrons, for instance.

What they claim is that deviations of the commutation relations from their canonical form would provide evidence of quantum gravitational effects. But they haven't established that such deviations must result from quantum gravitational effects in the first place. The Planck mass just provides a unit of mass, nothing more. And it is of mesoscopic size, in fact: 1019 GeV is, about that many protons, so that's about 10-4 moles, since 1023 (Avogadro's number up to a factor of 6.023) atoms are in one mole. That doesn't imply that such deviations probe quantum gravity, because the length scales are way larger than the Planck length.

Dear Stam,

One of the main problems is that quantum gravity effects must appear at Planck scale, which is very far to be reached with the scale of energies of our technology. Therefore you are right that it is very difficult to do physics for distinguish effects of such quantum gravity. But if we want to understand things as the Big Bang or other singularities it seems difficult to avoid such difficulties. Superstring theory was perhaps the best theory developed for facing such problem, but experiments are still waiting ...even for their fundamental state.

No, quantum gravity need not appear at the Planck length at all-just like diffraction effects don't appear at the wavelength used to probe matter using waves (whether electromagnetic or other).

Quantum gravitational effects are relevant, indeed, for describing the primordial Universe. Claiming they can be probed in tabletop experiments isn't as obvious as the Nature article claims-and this doesn't have anything to do with superstring theory. There doesn't exist any theoretical framework, for the moment, that can describe how to sum over spacetime geometries, in general, at any scale where quantum effects of gravity might be significant.

No Stam, no. I don't agree that quantum gravity can exit at a different scale than Planck's. It is very easy to proof it if you take into account the weaknees of this interaction. On the other hand it is a well accepted statement.

Dear Stam,

Although it is not a deep reference, perhaps can show you how the quantum gravity must be working at Planck's scale only

https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/

Once more: Planck scale doesn't mean anything by itself-as mentioned, the Planck mass is 10-4 moles, nothing special. The arguments about Planck scales (plural) is what I mentioned: stuff, whose mass is greater than the Planck mass-which is easy-when occupying a region of size less than the Planck length, i.e. about 10-35 m-that's the hard part!-would have collapsed to form a black hole, were Einstein's equations, that define what spacetime is, valid.

Since known quantum effects of matter, that have been neglected in this analysis, are known to become relevant at length scales much greater than 10-35 m, there's no reason that would imply that anything special happens there, beyond the fact that the known ways for describing natural phenomena have, already, nothing to say. But it could be at 10-44 m as well as at 10-30 m that hitherto unknown effects appear, as far as is now known. As mentioned in the linked article, it represents an order of magnitude (indeed several) for a scale describing current ignorance.

Dear Stam,

Don't escape of the discussion!. Quantum gravitation needs to be defined at this scale or not? It is clear that a distance by itself is without physical meaning, another thing is that this distance is given by Compton length and Schwarzschild radius. Then there are two important concepts in Physics which play an important role.

The main Planck units, first of all Planck length, lP, and Planck time, tP, are indeed “derived” by Planck from “fundamental” physical constants, i.e. the Planck constant, ћ, the speed of light, “c”, and Newton gravity constant, G.

However with a large probability the situation is opposite – these fundamental constants [the Planck constant is the “Planck unit” directly, that is the elementary action/angular momentum] are “macro revelations” of that Matter is rather simple logical system, where on fundamental depth all material objects, i.e. particles, bodies, etc. are built by using a few fundamental constants; and so just the constants “c” and G can be easily derived from lP, tP and ћ.

Corresponding informational physical model [see the link in the SS post above] follows, first of all, from the ruinously proven in the Shevchenko and Tokarevsky’s “The Information as Absolute» conception absolutely fundamental fact that there exist nothing else then some informational pattern/systems of the patterns and is based on a few in this case natural postulates:

The absolute Matter’s [5]4D Euclidian spacetime is filled by dense 4D lattice of the fundamental 4D binary [in accordance with the von Weizsäcker’s hypothesis 1950-th years] logical elements (FLE),

All 4 FLE sizes are equal to the Planck length.

In accordance with that FLE has 4 independent degrees of freedom at “flipping”, the spacetime is 4D.The 5-th, i.e. “true time” dimension is absolutely fundamental and so obligatorily acts in every dynamical object/system in the “Information” Set, including in the sub-Set “Matter”.

Flipping FLEs can cause of flipping of neighbor FlEs, since all 4 FLE’s sizes are equal, any disturbance in the lattice, i.e. a “flipping point” moves with equal footing in all 4 dimensions of the 4D sub-spacetime, since the flipping time intervals for all 4 sizes are equal , all disturbances move in the sub-spacetime with 4D speeds V that have identical absolute values, V=. lP/ tP = the speed of light, c. “c” is fundamental constant on macro level just since it derived from, indeed fundamental units.

All fundamental particles, and essentially all other particles, because of they are stable and, at that, are some constantly running chains of FLEs, are some closed-loop algorithms.

From that in Matter the energy conservation law acts follows, in accordance with the fundamental fact that was discovered by Fredkin and Toffoli, that these algorithms are built on some reversive logic.

Besides 4 universal degrees of freedom in Matter at least 4 fundamental Nature forces act, i.e. the FLE have also additional at least 4 “marked faces” that correspond to 4 forces’ charges, which form on the particles algorithms chains the charges’ sections. Different particles have different charges, but obligatorily have “gravity section”, from the experimental data about that the gravity is very weak force seems follows that unlike other forces the gravity section consists of only one FLE; at least if the particle is at the absolute rest.

From these postulates a number of fundamental points in physics become be essentially clarified.

It is understandable why the particles of the one type are totally identical.

It is understandable what is “quantum quantity spin” and why all fermions, including neutrinos, have “rest masses”, when some bosons, e.g. photons, are “restmassless”.

“A meta law of Nature”, as that wrote de Brogile [see the link in the Issam Mohanna post in https://www.researchgate.net/post/Does_energy_have_mass/1 4 days ago], when “to each portion of energy [a particle] with a proper mass m0,one may associate a periodic phenomenon of frequency hν0” and so hν0 =m0c2, becomes be understandable and correct.

The “Feynman–Stueckelberg interpretation” interpretation in QED, when it is ad hoc postulated that antiparticles move in the spacetime in reverse temporal direction, what is in mainstream physics some mathematical trick, since it contradicts with the SR where everything move only in one temporal dimension, turns out be adequate to the reality.

From correct definitions in the “The information as Absolute” conception of the absolutely fundamental notions/phenomena “Space” and “Time” follows that the Matter’s spacetime is absolute, thus the main postulates in the SR/GR that there is no of the absolute spacetime and that all inertial reference frames are totally and completely equivalent and legitimate, are incorrect. Besides it follows that all temporal and spatial dimensions in the spacetime are independent on each other and cannot be principally “contracted”, “dilated”, “bend”, etc. as that is postulated in the SR/GR also. Further from the postulates above follows what are the Lorentz transformations and real physical effects such as the real contraction of lengths and slowing down of internal processes of/in moving bodies; follows when the relativity principle and so the transformations are adequate to the reality completely and when that isn’t so; thus the mechanical model is developed, which includes the Poincaré-Lorentz-Einstein theory, however from which, in contrast to the SR, no meaningless logical and physical consequences follow.

It is shown that the absolute motion can be observed yet now experimentally, see https://www.researchgate.net/publication/259463954_Measurement_of_the_absolute_speed_is_possible DOI 10.5281/zenodo.48709

In the informational model it is shown that the GR postulate that photons don’t change energy when move in the gravity field is with a large probability wrong, and it is suggested how that can be tested experimentally, see https://www.researchgate.net/publication/277710038_The_informational_model_-_gravity_a_next_experiment

Besides at suggesting that the gravity force is caused by only one FLE in the particles’ algorithms chains, which has Planck length size, it is possible to obtain the model of the gravity, which is correct at least at the statics, i.e. to obtain the Newton’s gravity law, and, if the suggestion above is correct, that allows to observe the quantum nature of the gravity, which can be rather easily made also yet now, see - https://www.researchgate.net/publication/215526868_The_informational_model_-_possible_tests ; at least the section 2.1.2. “Monochromatic photon beam distortion”.

From which, besides, if the experiment‘s outcome will be in accordance with the model above, will follow that the Planck unities, including the Planck length, are indeed ultimately fundamental in Matter.

Cheers

planck length should be accepted to be THE UNIT; case it were not the unit, operations like squaring would make it less then itself, which is in contradiction to the presumption that planck length is the smallest one.

The Planck length, time and mass follow the Lorentz contraction. The speed of light is different in higher dimensions. The Gravitational Constant not Really Constant.

Preprint Do the Planck length, time and mass follow the Lorentz contraction?