This is Zeno's dichotomy paradox [1]. Finitely-defined infinitely-complex systems (e.g. fractals and anything chaos theory) are the escape.
[1] https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Dichotomy_p...
Sure. The point is the gedankenexperiment proves nothing. We don't need to "[record] an infinite amount of information" to encapsulate the infinity between any pair of real numbers.
[0] https://www.physicsforums.com/insights/hand-wavy-discussion-...
A Noether Theorem for discrete Covariant Mechanics, https://arxiv.org/abs/1902.08997
Statements were made that shielding would improve after Ver 1.0 .. it got worse. Statements were made that sats would go low power over quiet zones, they do not.
Returning to your erudite point "and stuff"
The NASA Cosmic Background Explorer (COBE) satellite orbited Earth in 1989–1996 ...
Inspired by the COBE results, a series of ground and balloon-based experiments measured cosmic microwave background anisotropies on smaller angular scales ...
The sensitivity of the new experiments improved dramatically, with a reduction in internal noise by three orders of magnitude.
Hmmm, it appears the ground based results were a dramatic improvement over the sat based data.
Not necessarily so?
https://en.wikipedia.org/wiki/Xuntian
As of 2024, Xuntian is scheduled for launch no earlier than late 2026 on a Long March 5B rocket to co-orbit with the Tiangong space station in slightly different orbital phases, which will allow for periodic docking with the station.
Leaving aside the fact that an optical telescope isn't a microwave array nor is it a Square Kilomtre Array of radio telescopes with each component larger than your example ...
Putting an instrument in orbit has all the costs of development of a ground based instrument, additional costs to space harden and test, additional costs to lift, limited ability to tune, tweak or extend when in orbit, hard constraints on size and weight, and other issues.
Xuntian allows for periodic docking, sure. How will this not be more expensive and limited than (say) walking | driving out daily or weekly to much lager instruments on the ground?
https://en.wikipedia.org/wiki/Xuntian#Instruments <- Terahertz receiver
( https://en.wikipedia.org/wiki/Terahertz_radiation
This band of electromagnetic radiation lies within the transition region between microwave and far infrared, and can be regarded as either. )
> Putting an instrument in orbit has all the costs of development of a ground based instrument, additional costs to space harden and test, additional costs to lift, limited ability to tune, tweak or extend when in orbit, hard constraints on size and weight, and other issues.
Who is to say they won't pull a Space-X, maybe even overtaking it, going fully reusable? Which allegedly lowers the costs, giving more economically access to space and lessening the constraints on payloads, while giving all the advantages of being in space?
With the additional cost of lifting it to orbit, the additional cost of difficulty of in orbit maitainaince, and the additional weight and dimension restraints of going to orbit, the additional costs of over designing to harden for space and limited access.
Yes, there are advantages to being in space. They vary by application.
That aside it's still cheaper to build an instrument or instrument array that's deployed on the ground.
Eg: SKA - definitely cheaper on the ground.
Unconvinced. Because building the whole system, not some isolated dishes somewhere amounted to 1.3 Billion EUR, operating it up to 2030 adds another 0.7 Billion EUR. 2 Billions. Chump change for sure.
Now we can compare that with the JWST and typical cost overruns in american boondoggle style, or look at the latest shining star, EUCLID. Just 1.4 Billion EUR for the latter.
Then there was GAIA at about 740 Million EUR, with the orbiting article at 450 Million EUR alone, plus another 250 Million EUR for the data-processing org.
All of these with more or less conventional rocketry, and not co-orbiting anything for more easy maintenance and upgrading.
My gut feeling tells me we will have cheaper and more reliable access to space, with larger payload capacity, necessitating less 'origamics' for the space parts, and that chinese concept seems sound, too. Very much so, in fact.
How much that will cost I have no clue.
But again, if something like this is becoming reality, no matter by whom, some former assumptions about cost, feasibility (at all, because payload weight and dimension constraints are relaxed, needing less 'origamics') will have to be rethought.
That was my point, in general. Not limited to any special application.
In the interim, and as a general rule for all private entities, it'd be nice to not pollute the commons with unnecessary discharges and sparkles and to carry through on pinky promises to maybe do something about that.
There are a large number of continuous physical quantities, not only length (though all continuous quantities are dependent in one way or another on space or time, which are the primitive continuous quantities), and the reason why you cannot encode an arbitrary amount of information into a specific value of such a quantity is because it is impossible to make an object for which such a quantity would have a perfectly constant value. All the values of such quantities are affected by noise-like variations so you could store information only in the average value of such a quantity, computed over some time and any such average would still be affected by uncertainties that limit the amount of information that can be stored.
One of the most constant lengths that have ever characterized an artificial object has been the length of the international prototype meter kept in France and used to define the meter until 1960. To minimize the length variations, that meter bar was made of platinum-iridium alloy and it was measured at a temperature as constant as possible.
Despite the precautions, which included gentle removing of the dust and handling with soft grippers, the length of that meter bar fluctuated continuously. Even if it was attempted to keep a constant temperature, very small fluctuations in temperature still caused thermal expansions and contractions. Every time the bar was touched, a few metal atoms were removed from it, but other atoms from the environment remained stuck to its surface, changing the length.
All these continuous variations have nothing to do with the possibility of the space being discrete, but they limit the amount of information that can be stored in any such value.
For now there exists absolutely no evidence about the space or time being discrete and not continuous. There have been attempts to make theories based on the discreteness of the space and/or time, but until now they have not provided any useful result.
Instead, my way is simpler by generating an absurd result that if you could build and measure a thing to arbitrary precision you can encode infinite information into it. This is enough for me to reject the counter-factual without going through the messiness of thinking through hypothetical realistic experiments.
The one interesting place to consider is at the Schwarzchild radius of a black hole, where presumably information accumulates to an absurd degree, monotonically over time. I don't really know enough about it to comment intelligently, so I won't except to note its existence.
The Schrödinger wave-function is expressed in a unit which is the square root of an inverse cubic meter. This fact alone makes clear that the wave-function is an abstraction, forever hidden from our view. Nobody will ever measure directly the square root of an inverse cubic meter.
Freeman Dyson, Why is Maxwell’s Theory so hard to understand?
https://www.clerkmaxwellfoundation.org/DysonFreemanArticle.p...
>> Just because you can't record something...
>>> Who said anything about recording?
Depends on what you're measuring. To illustrate why that isn't a facetious response, consider the difference between 'measuring' pi, 'measuring' a meter and 'measuring' the mass of a proton. (Or, for that matter, the relative mass of three of something to one of it.)
It's worse than that: you also need an unambiguous way of determining whether the needle is overlapping a stripe.
Pick your method. It’s the ratio of a circle’s circumference to its diameter.
I think it's reasonable to say we can't truly measure pi, though.
And you can neither know nor measure a random real.
> We know pi in the sense of "a unique real number satisfying many useful properties".
We know it a lot better than that. We have efficient programs that output the numerical value of pi for as many digits as you want.
There's a bunch of real numbers we can identify that are far harder to make use of or approximate, and don't have easy exact description of their value.
Sure, but how would you compare those against a measurement?
So, how would result of measuring e.g. length of something to an infinite precision look like? It would look like two particles that are kept at rest relative to each other; the distance between them is the measured distance. Whether this distance has to be commeasurable with the Planck scale or not is an interesting question but it really can go either way.
And how do you do that in the face of Heisenberg uncertainty?
To actually try to answer your question: I don't know. But that's just me; and ain't there some interesting experimental setups with super-cooled crystals? In any case, inability to imagine something is hardly a convincing proof of anything.
Hm.
Momentum space being compact does seem weird..
Of course, if rather than a discrete group for space, you just have a discrete uh, co-compact(? Unsure of term. Meaning, there is a finite radius such that the balls of that radius at each of the sites, covers the entire space [edit: “Delone set” is the term I wanted.]), uh, if you take a Fourier transform of that lattice…
Err… wait, but if the lattice is a subgroup, how does the Fourier transform relate to…
I think the Fourier transform of a Dirac comb is also a Dirac comb (with the spacings being inversely proportional) If you multiply the Dirac comb by something first… Well, if you multiply it pointwise by e^(i x p_0 /hbar) , then the Fourier transform will have whole thing shifted by p_0 , and this is periodic in (width of the spacing of the comb in momentum space)
So, if you consider all the pointwise multiples of a Dirac comb in position space (multiplying it by arbitrary functions), then I guess the image of that space under the Fourier transform, is going to in some way correspond to functions on S^1, I guess it would be functions periodic in the width of the comb in momentum space.
So, if instead of a regular comb, you jostle each of the Dirac deltas in the position space comb by a bit first (a different random amount for each)… I’m not sure quite what one would get…
The operative word being "seems". Position and momentum (and indeed real numbers in general) are mathematical models that predict observations. But the observations themselves are the results of physical interactions that transfer energy, and those can only ever be discrete because energy is quantized.
Maybe one can make the argument that position itself is quantized (thus the position of the mirrors can not be varied continuously), but we do not have experimental reasons to believe space is discrete (and quantum mechanics does not require it to be discrete). And while it is pleasing to imagine it discrete (it is more "mathematically elegant"), we do not have any significant rigorous reasons to believe it is.
Edit: Moreover, if you want to describe (in quantum mechanics) the interaction between a finite system and the open environment around it, the only way to get a mathematical description that matches real-world experiments is to have continously parameterized energy levels for the systems making up the open environment. If you assume that only discrete values are possible, you will simply get the wrong result. Most quantum optics textbooks have reasonably good discussion of this. E.g.:
Quantum Optics by Walls and Milburn
Quantum Optics by Scully and Zubairy
Methods in Theoretical Quantum Optics by Barnett and RadmoreSure, but can you measure those continuously-parameterized energies? I don't see how.
Continuously parameterized energies are no different from continuously parameterized space. They are part of the mathematical model we use to make accurate predictions, but we have no direct access to either, and (AFAICT) we cannot possibly have access to them because that would violate the no-cloning theorem.
The following is a (simplified, abstracted) way to measure an arbitrary energy value.
1. Set the system up so that the carrier of the energy is a photon (e.g. let the two-level system decay[1] or use some form of transduction or whatever).
2. Send that photon to pass by two semi-transparent mirrors at a certain (continuously parameterized) distance between each other.
3. If the photon passes through both mirrors (as detected by a photon detector at the other side), it means its energy is equal to some known constant divided by the distance between the mirrors. If it does not pass it means it has a different energy.
4. Repeat the experiment many times as you slowly vary the distance between mirrors.
I guess in point 4 there is an issue that you need to repeat the experiment with a new realization of your photon each time. Does that have bearing on the initial point being discussed?
You are probably seeing this at this point, but just for completeness: this technique is no different from tuning a musical instrument with a tuning fork.
I papered over some details about whether we want to detect transmission or reflection and exactly what type of transparency the mirrors need to be, etc.
[1] Funnily, the usual way in which someone would prove that decay can happen at all does rely on the existence of a continuous spectrum of energies. This is the same topic I raised above when citing Quantum Optics textbooks.
Yes, I get that. But there are two problems. First, you cannot tune an instrument precisely. Precise tuning of a real musical instrument isn't even a meaningful concept because any wave with finite temporal extent has non-zero bandwidth. It's the same with energy. The exact same uncertainty relationship between frequency and time produces the Heisenberg uncertainty relation between energy and time, so it is not possible to produce an isolated photon at a known time with a known energy. The best you can do is produce a lot of photons so you don't have to wait forever for one of them to arrive at your detector. So the problem with the setup you describe is that in step 2 the concept of "that photon" is not well defined.
Second...
> I guess in point 4 there is an issue that you need to repeat the experiment with a new realization of your photon each time.
Yes, that too. But you need to do more than that: in order to get a meaningful result you'd need to produce photons with the same energy, i.e. you'd need to use a laser or some other kind of tuned cavity. But it is not meaningful to identify individual photons emitted from a laser because they are identical bosons.
You're assuming spacetime behaves like the set of reals (something with cardinal ℵ1, if you accept the continuity hypothesis), an object that even if you stay confined within the bounds of pure mathematics, behaves in very, very weird ways.
It may be that spacetime at small scales maps better to a different kind of mathematical object and not even a grid-like one.
Jorge Borges' way of telling a story as analogy is beautiful and simple.
It takes the resources of the universe to simulate the universe.
The electron might be smaller. Its diameter is known to be smaller than 10^-22m, but could be much smaller than that.
Further below the Planck Length, there are strong indications that the universe isn't continuous -- it's discrete. That there's an absolute limit to precision, something really quite analogous to a pixel. This elementary length could be somewhere around 10^-93m.
The theory that the Planck length has any significance is just a speculation.
Nobody knows how interactions would behave at distances so small and there are no known methods that could compress anything into volumes so small. There is no basis to believe that extrapolating the behavior from normal distances and sizes to the scale of the Planck length is valid.
There are pure speculations that are interesting, but in my opinion any speculation about the Planck length is not interesting, because nobody has been able to formulate any prediction based on such a speculation that can be verified in any way.
Most speculations about the Planck length are made by people who obviously know very little about the meaning of the so-called fundamental constants or of about the significance of the useful natural units for physical quantities, to which the Planck length does not belong.
The Planck length is just one way to express the intensity of the gravitational interaction, i.e. an alternative to Newton's constant of gravitation. Its numeric value does not say anything about any other physical phenomena.
The numeric smallness of Planck's length is just an expression of how weak the gravitational interaction is in comparison with the other interactions. It does not have any other significance.
There are indications discrete space is plausible. It's actively debated.
There are also strong indications space is continous, e.g. Lorentz symmetry. (This was recently the death knell for a branch of LQG.)
For example, I pound the picnic table. Presumably this is somehow transmitted thru the entirety of the Earth, or at least thru a tiny portion of it. But is there a cutoff ? Where is the cutoff ? Where is the effect simply too small to "register" in any conception of reality ?