Complex Geometric Theories and Molecules

[EuclidPhoton]

If QM is a statistical model to approximate something underlying space time we don't quite understand yet, and there is a complex geometry underlying space time, is it possible to find other ways to simplify molecular optimizations and electron interactions in computational chemistry using complex geometric theories outside of euclidean space? Or will wavefunctions always be the only way?

 

[EuclidPhoton]

I don't know if I'm asking this question properly.

If wavefunctions are statistical approximations necessary due to entropy in the experimental universe, (since we must use the universe to measure the universe we always cause an interaction and lose some information about what was happening prior to measurement), but particles seem to be connected in a geometric way outside of our spacetime, it might be possible to simplify molecular modeling calculations or at least find other potentially more efficient approximations using complex geometry rather than wave functions? Since when we are modeling something we don't always have to worry about entropy or modeling things from within the perspective of our spacetime for that matter.

 

[Arsenic&Lace]

I don't really understand your question, but in the case of molecules the symmetries of the molecules are routinely exploited when computing electronic configurations.

 

[EuclidPhoton]

I realize this, I'm just wondering if complex geometric theories like E8 theory or other higher dimensional non-euclidean geometries might lead to better ways to do molecular modeling?

 

[cgk]

EuclidPhoton, if you come up with a good *concrete* idea[1], feel free to try to use it for electronic structure modelling (or to contact someone in the field, e.g., me, to get an impression of whether it is viable). In the electronic structure field you would have the best chances of it being picked up---provided it works. Here anything which works and is useful for making calculations faster, more accurate, or more robust, is adopted quickly, no matter how far out the theory behind it is. And there are some very, very far out things.

So far I am only aware of strange geometric theories being used in wave function space, not in real space, however. All electronic structure theory I know uses electron coordinates with three spatial dimensions and one spin dimension per electron. But this does not mean it has to stay that way if progress could be made by abandoning it.

[1] it must be concrete enough that an expert in the field will know what would in principle have to be done in order to implement a prototype application.

 

[EuclidPhoton]

I'm not sure if it needs to be abandoned in any way. I think that to obtain results comparable to experimental data we may always need these methods. I'm just wondering if it might be worth investigating if any complex geometric theories might provide simplifications for some types of calculations like optimizations of transition state structures, etc. Since all theories of quantum gravity point towards the universe being based on higher dimensions giving rise to our three dimensional euclidean geometry, more advanced geometries might lead to simplifications in electron interactions?
I don't know enough mathematics yet to know whether this is a possibility or not, but I'm learning as fast as I can, until someone tells me this is not possible or beats me to answering this question.

 

[cgk]

Well, in order to get results comparable to experiment, at some point the theory, whatever it is, needs to be converted back into the 3+1 dimensional space. Since this is where the electrons live. However, in principle it could be possible to perform the actual computations in a different manifold, and just obtain the 3+1 dimensional picture via a down-projection at the end, just before physical properties are calculated. Will this help? No idea. But it cannot be excluded.

I would recommend you to not argue with quantum gravity or high-dimensional space-times, however. In electronic structure, the goal is to obtain an accurate approximation of molecular properties arising from the (usually non-relativistic and time-independent) interacting Schrödinger equation. For all practical purposes, this is all the physical reality you need. The main question is: Can strange geometric theories be used as a viable tool for obtaining those approximations numerically? This is what you would have to find an approach for.

Note that geometric optimizations (e.g., transition state searches) use techniques of numerical optimization which are near 100% orthogonal to the techniques used for computing electronic structure (effectively, such geometric optimizations need only energies and gradients from the electronic structure calculations, and are more or less agnostic about the actual wave functions or electronic structure used.) To get into the electronic structure problem, you would probably best start with a method like Hartree-Fock, and see if you can find any way of making it "better" by using non-standard electronic manifolds (e.g., by making the orbitals, which are normally functions of one 3+1 dimensional coordinate (3 space, 1 spin dimension) functions of some other geometric manifold, and defining a way of down-projecting the obtained wave functions into 3+1 space for computing properties)

 

[EuclidPhoton]

Thank you! This is the most helpful answer anyone has been able offer yet. Now at least I have an idea of which direction to proceed my research =)

 

[EuclidPhoton]

Do you know if there is any way to narrow down the possibilities of which complex geometries might lead to simplifications? Or is the only way to pursue one and see if it might be promising or not?

 

[Arsenic&Lace]

One comment which I think is in order is that historically making the mathematics significantly more complex rarely (as in, almost never) results in better algorithms; usually it is insight into the physics, not the mathematics which produces the lion's share of improvement.

My experience with "clever" mathematical methods is that a considerable amount of effort is expended, a toy model is solved, and no progress is made, because while one can use infinitely more complex mathematics than is necessary to attack a problem, there are serious limits on how simple the mathematics can be made.

 

[EuclidPhoton]

Thanks for your reply. I think this is why I'm trying to bring the complex geometric theories of quantum gravity into the picture. These theories might help provide some constraints on which math might lead to simplifications for some calculations.

 

[Arsenic&Lace]

I think that's a pretty weird approach. You're basically taking vaguely related mathematics, putting your gun to your hip, closing your eyes, and firing. Not only are the complex geometric theories of quantum gravity bad at explaining quantum gravity (i.e., they don't at all so far as we know), but we already have good theories for molecular electronic structure.

One approach mathematicians interested in funding and applications have taken is topological data analysis, where one tries to obtain algorithms to find low dimensional embeddings in high dimensional data sets (among other things); I actually don't work in small molecule physics but I'd wager that the data sets are pretty high dimensional. This approach hasn't really born fruit yet but it might be worth exploring if you insist on pre-deciding what mathematics you want to use.

 

[EuclidPhoton]

Yeah I realize that they don't work well yet and that it's a long shot. But some have already shown they can simplify particle interactions using complex geometry like the amplituhedron paper. There's a long ways to go before these theories may be useful, but it is looking like locality and spacetime arise from something more fundamental.
Which suggests that electrons are directly connected in some complex geometry outside of our euclidean space. Which would mean that for modeling purposes we could find a transition state structure much more efficiently if we understood how all the electrons are connected geometrically.
Instead of saying one electron is interacting with another electron which is interacting with another etc in 3 dimensions, we could have a geometry that approximates the system as a whole more efficiently.

 

[Arsenic&Lace]

No, the amplituhedron suggests no such thing, because it only works if you assume things about reality which are false as far as we know, such as that we live in a universe with 4 spatial dimensions, or that supersymmetry is real, which so far it is not.

That being said, an unlikely but not entirely impossible avenue for improving computational atomic/molecular physics is pondering what quantum mechanics really means. I personally vouch for a more physical rethink if one is going to rethink it at all (something like Bohmian mechanics), but I also like to point out that even this seems incredibly unlikely. The amplituhedron might hint at a more geometric rethink after all so I wouldn't completely disregard it even if the history of theoretical physics is not exactly full of major breakthroughs predicated by pure mathematics.

 

[EuclidPhoton]

I agree with everything you've said here, I only mentioned the amplituhedron paper as an example. There are many other theories like E8 theory or whatever that are also not yet successful as you pointed out.
Hopefully the LHC will shed some light on which math we should be focusing on, and hopefully physicists will get funding for a much more powerful collider so we can get an even better idea of what's going on (there are probably many more fields to be discovered). But it looks to me like there is something in these complex geometric theories underlying spacetime. We may never have a theory of everything but we can get better approximations can't we?

 

[Arsenic&Lace]

I guess the question is whether or not we are likely to get better approximations using a technique which has no physical motivation or history of success or using techniques which have physical motivation and/or a history of success. Weighing how fruitful an avenue might be is hard and it's dangerous to write something off too soon, but it is generally considered bad practice to run down a blind alley.