Stable 4D atoms revisited

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Stable 4D atoms revisited

Postby quickfur » Wed Aug 15, 2012 5:21 pm

Yesterday I was browsing through the old discussions on this forum about the stability of 4D atoms (specifically, the existence of the 4D analogue of the hydrogen atom), and decided to google for academic research papers on the subject. One very interesting paper that turned up was Is there a stable hydrogen atom in higher dimensions?. It basically confirms what we concluded, that if we assume an inverse cube law, then there is no stable hydrogen atom.

However, what is very interesting is that the paper pointed out (p.627, third page in the linked pdf) that the reason for this is because we assume Maxwell's equations to hold in higher dimensions. If, instead, we start with general physical properties of, in this case, the desired phenomena, then we can deduce a different set of physical laws that would produce those effects. If we were to formulate 4D electromagnetism in such a way that point charges will have a 1/r potential, for example, then it can be shown that stable hydrogen atoms exist, with analogous (but not the same) properties for electron orbitals.

Section V of this paper (p.631, 7th page in the pdf) derives a modified set of Maxwell's equations that describe higher-dimensional electromagnetism in which stable atoms exist. The force between two charged particles obey an inverse square law, and Gauss's law does not hold (but a modified form does). In even dimensions, the field equations are not differential operator equations, but "pseudo-differential" operator equations, which are non-local. In spite of the non-locality, though, macro-causality continues to hold (see The pseudodifferential operator square root of the Klein-Gordon equation).

In any case, it's clear that electromagnetism cannot be directly generalized to 4D, but a different, modified kind of "electromagnetism" can lead to stable atoms.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Stable 4D atoms revisited

Postby Higher_Order » Thu Aug 16, 2012 2:55 pm

Tell me if I'm wrong, but wouldn't this paper confirm not only stable hydrogen in the 4th dimension, but in any dimension? To me, it seems that in each dimension higher than the 4th would have an atom analogous (but not the same) to the previous dimensions' atoms. That would mean that for each higher dimension atoms are increasingly abstract (relative to ours), but still analogous, and (abstractly) possible.
Higher_Order
Dionian
 
Posts: 16
Joined: Tue May 22, 2012 11:44 am

Re: Stable 4D atoms revisited

Postby quickfur » Thu Aug 16, 2012 6:00 pm

This paper claims that stable hydrogen is possible in any dimension provided we modify the rules of electromagnetism such that a charged particle always has a 1/r potential. The resulting electromagnetism may be very foreign, and look nothing like the electromagnetism that we're familiar with. In the even dimension case, as I've noted, the modified electromagnetism violates causality (or rather, exhibits non-local effects) below a certain threshold (but fortunately still exhibits macroscopic locality, so it won't be too strange).
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Stable 4D atoms revisited

Postby Higher_Order » Thu Aug 16, 2012 7:50 pm

Ok, thanks. Just checking that I read the paper right :D
Higher_Order
Dionian
 
Posts: 16
Joined: Tue May 22, 2012 11:44 am

Re: Stable 4D atoms revisited

Postby gonegahgah » Fri Aug 17, 2012 1:04 am

It's my personal belief that all our perceived forces are actually realisations of a single type of force - without needing to go to extremely high temperatures to unify them - so I am very comfortable with the notion that electromagnetism should step up its dimensional aspects, in a higher dimensioned universe, as well as everything else that does.
gonegahgah
Tetronian
 
Posts: 490
Joined: Sat Nov 05, 2011 3:27 pm
Location: Queensland, Australia

Re: Stable 4D atoms revisited

Postby Keiji » Sat Aug 18, 2012 6:51 pm

Hi quickfur.

It's good to know that you can have stable hydrogen, but - this paper at a glance being way over my head - does it help to show what other arrangements of particles we can expect? i.e. other atoms, and their chemical properties.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: Stable 4D atoms revisited

Postby quickfur » Sat Aug 18, 2012 7:08 pm

Keiji wrote:Hi quickfur.

It's good to know that you can have stable hydrogen, but - this paper at a glance being way over my head - does it help to show what other arrangements of particles we can expect? i.e. other atoms, and their chemical properties.

Well I've to admit it's over my head too; I just glanced over it briefly and jumped to the end where it summarizes the main results. The math is a bit beyond me, I'm afraid. But nevertheless I think the calculation was mainly for hydrogen (or its higher-dimensional analogue thereof), but it doesn't really say much about other atoms or other properties of the modified electromagnetism.

In any case, it's a pretty drastic change, since we're assuming the electromagnetic force obeys an inverse square law even in higher dimensions. So magnetic flux would have to behave in odd ways in order for this to happen -- at least that's my intuition, but I may be wrong.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: Stable 4D atoms revisited

Postby dodecahedron » Sun Sep 23, 2012 12:50 pm

[/quote][/quote]In even dimensions, the field equations are not differential operator equations, but "pseudo-differential" operator equations, which are non-local. In spite of the non-locality, though, macro-causality continues to hold (see The pseudodifferential operator square root of the Klein-Gordon equation).[quote][/quote]




We already have quantum non-locality in 3D
dodecahedron
Dionian
 
Posts: 26
Joined: Sat Sep 01, 2012 9:28 pm

Re: Stable 4D atoms revisited

Postby anderscolingustafson » Sun Oct 07, 2012 4:37 am

If it is possible to have stable atoms in 4d with very different sets of physical laws for magnetism or different equations might it also be possible to have stable orbits for planets and moons in 4d provided that the equations for gravity were drastically different? Could there even be life in 4d provided that the laws of physics were drastically different?

Anyway there are different sets of laws of physics such as different values for the forces of nature such as magnetism and gravity that would prevent stable atoms in 3d and stable orbits in 3d as well as ones that would prevent stars from using their fuel so why would anyone assume that just because there could not be any stable orbits in 4d with our physical laws that there could be no stable orbits in 4d at all? Why the assumption that a 4d world would have our set of physical laws?
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
anderscolingustafson
Tetronian
 
Posts: 316
Joined: Mon Mar 22, 2010 6:39 pm

Re: Stable 4D atoms revisited

Postby quickfur » Sun Oct 07, 2012 11:08 pm

The issue is not so much whether 4D "has" our set of physical laws or not, it's a question of what makes it easier for us to understand 4D geometry. It's very easy to invent a set of physical laws that are totally unlike what we're familiar with and apply it to a 4D world. It's also extremely hard to get any insight into 4D this way, because we'd be too busy trying to wrap our brains around the bizarre consequences of these laws than actually understanding how 4D space interacts with it. That's why dimensional analogy is so useful: it takes what is familiar to us, and extrapolates to what happens in 4D -- the familiarity helps us comprehend what's going on when there's an extra spatial dimension available to us.

So the underlying issue here is more of a matter of what set of physical laws in 4D will produce a universe that's similar enough to our own that we can get a good intuitive feel for how 4D works. Deliberately trying to be different, just for the sake of being different, doesn't really help us understand how 4D works. I mean, we already have mathematical analysis and algebra that we can use to work with 4D; anyone can posit some arbitrary set of equations that will produce some kind of universe in 4D. The question is whether the result is of any interest in giving us a better understanding of 4D.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North


Return to Higher Spatial Dimensions

Who is online

Users browsing this forum: Google [Bot] and 4 guests