Can someone clear up the implications of Planck units once and for all?

[SeventhSigma]

Time and time again I always hear people say that Planck Lengths are the smallest units of meaningful distance.

But I don't think this implies nothing can be smaller than this or that spacetime is discrete, correct? It just means talking about anything smaller than this would be like talking about, say, half-pixels. It's meaningless, but it doesn't prevent some deeper framework from existing.

Would this be a correct way to phrase it?

 

[Vanadium 50]

There is nothing special about Planck units. Nothing. They don't represent any sort of "pixel" size. They are just units.

 

[JeffKoch]

Well, it's potentially more than that, which is why someone (I think other than Planck) gave it the name "Planck length". It's a distance unit that can be made up of G, c and h, and it's an extremely small distance, so there is some speculation (and, I think in some cases, more than speculation) that it may represent a quantum unit of space, defining say the dimensional scale of a string in string theory.

 

[rbj]

i wouldn't say that there is nothing special about Planck units.

i wouldn't say that we have a working theory for why the Planck length must be the smallest length nor the Planck time is the smallest time. for sure, the Planck mass is neither the smallest nor largest mass.

but the Planck length is so small, and the Planck time is so small, that it seems like they could be close to a differential length or time for all practical purposes.

expressing equations of physical law in terms of Planck units (or better yet, rationalized Planck units where $4 \pi G = 1$) ends up removing scaling factors between quantities that are identical except for the scale factor. like equating E-field and flux density or (more popularly) mass and energy or time and length.

anyway, if one were to set up a system of cellular automata with cells as big as the (rationalized) Planck length and time discretized to the Planck time (the sampling frequency of reality would be up there around 10⁴³ Hz), then the discrete equations of physical law (derived from the continuous-time equations using Euler's forward differences) have no contrived scaling factors that Nature has to pull out of her butt to convert one quantity (like flux density) to another (like E field).

 

[granpa]

http://en.wikipedia.org/wiki/Talk:P..._.2F_Rest_mass_.2F_Gravitational_field_energy

particle with a Bohr radius of 1 Planck length...Amazingly, at this size, the energy in the gravitational field is 137 (1/α) times the energy in the rest mass

 

[Vanadium 50]

No. The Planck length is not some sort of pixel size. I don't know why people keep saying this.

It is true that around the Planck length, quantum gravity effects become important. But around might be tens or hundreds of times larger.

 

[TrickyDicky]

There is nothing special about Planck units. Nothing. They don't represent any sort of "pixel" size. They are just units.

Would you say the Planck cosntant is just a units conversion constant?

 

[Vanadium 50]

Yes; they're just units. There's nothing more (or less) special than using feet or inches.

 

[SeventhSigma]

That's what I was suspecting -- I just see all too often people trying to argue that irrational numbers don't exist because Planck lengths imply a discrete spacetime

 

[JeffKoch]

That's what I was suspecting -- I just see all too often people trying to argue that irrational numbers don't exist because Planck lengths imply a discrete spacetime

Irrational numbers exist because we define them to exist within the realm of mathematics, regardless of whether or not spacetime is discrete. It seems probable that spacetime is discrete, with a quantum length scale on the order of the Planck length and a time scale on the order of the Planck time, so ultimately it would not make sense (for example) to carry around significant figures of ∏ beyond what is required to measure or calculate distances to a precision better than a Planck length.

 

[SeventhSigma]

See, that's exactly why I made this thread, though.

What's the deal with this "discrete spacetime" thing? I don't understand how that even makes sense (in my mind, nature only makes sense if it's continuous even if there are aspects to it that are discrete such as quantization). Even if we say that within quantum mechanics, it doesn't make much sense for us to discuss anything sub-Planck scale, that doesn't mean there isn't something underneath the quantum mechanics or that things aren't still continuous.

 

[JeffKoch]

What's the deal with this "discrete spacetime" thing? I don't understand how that even makes sense (in my mind, nature only makes sense if it's continuous even if there are aspects to it that are discrete such as quantization).

Well, it doesn't really make sense to me that light appears to propagate at the same speed regardless of my own speed, or that it would seem to me to take an infinitesimal amount of time to travel to a star in the Andromeda galaxy, but that's how reality appears to be. Space and time may well be (probably are) "foamy" at extremely small scales that we cannot directly access.

 

[Naty1]

Jacob Beckenstein discovered that there is a "pixel" size (not that I like that term) associated with Planck units which clearly implies a discreteness:

adding one bit of information will increase the horizon of any black hole by one
Planck unit of area, or one square Planmck unit. Somehow, hidden in the principles of quantum mechanics and the General Theory of Relativity there is a mysterious connection
between individible bits of information and Planck sized bits of area.

so says Leonard Susskind, THE BLACK HOLE WAR, page 154

(In fact, he has the essential mathematics and steps Beckenstein used outlined from p 150-154...)

Relativists may not like to think about spacetime as discrete because it appears to conflict
with Einsteins relativity...however that apparent conflict might be an illusion:


http://arxiv.org/abs/1010.4354

“The equivalence of continuous and discrete information, which is of key importance in information theory, is established by Shannon sampling theory: of any bandlimited signal it suffices to record discrete samples to be able to perfectly reconstruct it everywhere, if the samples are taken at a rate of at least twice the bandlimit. It is known that physical fields on generic curved spaces obey a sampling theorem if they possesses an ultraviolet cutoff.”

and
http://arxiv.org/abs/0708.0062
On Information Theory, Spectral Geometry and Quantum Gravity
Achim Kempf, Robert Martin
4 pages
(Submitted on 1 Aug 2007)

We show that there exists a deep link between the two disciplines of information theory and spectral geometry.

And I kept this for my own notes from another thread here:


http://pirsa.org/09090005/

Spacetime can be simultaneously discrete and continuous, in the same way that information can.

In this thread

https://www.physicsforums.com/showthread.php?t=391989

"argument for the discreteness of spacetime",

Ben Crowell posted this question...

The following is a paraphrase of an argument for the discreteness of spacetime, made by Smolin in his popular-level book Three Roads to Quantum Gravity. The Bekenstein bound says there's a limit on how much information can be stored within a given region of space. If spacetime could be described by continuous classical fields with infinitely many degrees of freedom, then there would be no such limit. Therefore spacetime is discrete.

(This is very similar to Susskind's discussion which I referenced above.)


and the subsequent long discussion is very good.

Several years ago I posted something like "Are we analog or digital?" and got some good discussion, but at that time I posted maybe a dozen or so reasons suggesting spacetime is discrete..we are DIGITAL..and was leaning that way.

Maybe "continuous" and "discrete" are two sides of the same coin, analogous to wave-particle duality.

PS: there are many other discussions on "discreteness" in these forums.

 

[unusualname]

Nice Post Naty1, we do not have a good understanding of this yet.

So, sorry OP (Original Poster) but no one can clear it up (just yet).

For an interesting essay on Planck's own views see:

http://books.google.com/books?id=S3FOuMYHcqIC&lpg=PP1&pg=PA21#v=onepage&q&f=false

Dimensionless universal constants could now be constructed that necessarily would have the same value for all physicists, human or not, irrespective of their systems of measurement.

 

[rbj]

No. The Planck length is not some sort of pixel size. I don't know why people keep saying this.

See, that's exactly why I made this thread, though.

What's the deal with this "discrete spacetime" thing? I don't understand how that even makes sense (in my mind, nature only makes sense if it's continuous even if there are aspects to it that are discrete such as quantization). Even if we say that within quantum mechanics, it doesn't make much sense for us to discuss anything sub-Planck scale, that doesn't mean there isn't something underneath the quantum mechanics or that things aren't still continuous.

it just makes for elegant mathematical relationships (when these continuous-time, continuous-space differential equations are discretized using Euler's forward method). and the Planck length and Planck time are so small (so far beyond any precision of measurement) that no one would know the difference.

 

[Naty1]

Here is another paper of possible interest:
(This is the same Lee "Smolin" Ben Crowell referenced above. )

Holography in a quantum spacetime
Fotini Markopoulou∗and Lee Smolin†
Center for Gravitational Physics and Geometry
Department of Physics
The Pennsylvania State University

http://arxiv.org/PS_cache/hep-th/pdf/9910/9910146v1.pdf

In this paper, we present a framework for a Planck-scale, cosmological, background independent theory which is holographic in a sense appropriate to a quantum spacetime.
This is motivated by the fact that the formulations of the holographic principle
given to date[1]-[6] are confined to the semiclassical regime. At the same time, results of several approaches to quantum gravity indicate that the description of spacetime
based on smooth manifolds can provide only an approximate description[7]-[12]. If it is true, the holographic principle ought then to be more than a conjecture about the classical and semiclassical theory. Rather, it should be an important part of the
framework of a Planck-scale, background independent quantum theory.

 

[jetwaterluffy]

Of course irrational numbers exist, even if there is "pixels" of space. The very definition of the plank length implies this.

If you look in the Planck length equation above, in there is pi (in h-bar) and a square root. Both of which produce irrational numbers. Therefore, irrational numbers must really exist if the plank length has any meaning.