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Math That Takes Newton into the Quantum World (nautil.us)
133 points by dnetesn 53 days ago | hide | past | web | favorite | 31 comments



This was an interesting read - reminded me of another article [1] I read in 2013 about using a high-dimensional 'jewel' to describe reality, which then turned out to be an extremely efficient way to do the calculations from a Feynman Diagram. I'm still holding out hope that a unified theory will be found within my lifetime, and that these kinds of math will hold the key.

[1] https://www.quantamagazine.org/physicists-discover-geometry-...


If you are interested in this, there are quite a few lectures by Nima Arkani-Hamed available on YouTube. From some aimed at the general public about the big picture of fundamental physics and the unresolved issues in it to pretty technical discussions of the amplituhedron and the related physics.



I personally think that the split-octonions[1] hold promise for physics. The subset of the split-octonions where the i, j, k, and l components are 0 have a metric that is similar to that of Minkowski space[2].

  (t + xli + ylj + zlk) * (t - xli - ylj - zlk)
  = t^2 - x^2 - y^2 - z^2
[1] https://en.wikipedia.org/wiki/Split-octonion [2] https://en.wikipedia.org/wiki/Minkowski_space


I believe this is somewhat controversial


That was an interesting read! Interestingly, John Baez, author of the originally posted link is also quoted in it.


Why should our universe be so readily governed by mathematical laws? As he put it, “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

This is something that gets repeated again and again but I don't find this puzzling at all. Mathematics is an extremely general tool for modeling all kinds of things, so if something follows some sort of rules, you most certainly can use mathematics to describe, model, and analyze it. While I am not aware of any specific results or hypotheses, I would still guess that mathematics - or even some small subset like set theory - is in some sense universal for modeling things in a way similar to universal models of computation like Turing machines or lambda calculus.

I can not even imagine a universe with laws of physics that could not be tackled with mathematics. Even if the universe was pure chaos, everything happening totally randomly without any recognizable rules, we still could try throwing statistics at it, at least ignoring the fact that such an universe would probably not be a place for life to flourish. Finally mathematics does not exist in a vacuum, we invented large parts of it specifically to deal with real world phenomenons. And then of course »some« more that does not describe anything existing in the physical world.

Somewhere deep down below there are certainly many puzzling issues, from why there is a universe at all and why it is the way it is to human consciousness and why and how we are able to make sense of the universe, but that mathematics can describe the universe does not seem to be one of those, at least to me.


I think of it in terms of Kolmogorov complexity: The number of mathematical symbols you have to write down to describe some phenomenon of the universe. Given the apparent complexity of the universe, it still amazes me that that number of symbols is so small. There doesn't seem to be a reason why that should be true. We know from complexity science that complexity can arise from simple rules, but there's still no reason why those rules have to be simple in the first place.


My layman's perspective on this:

I think there's some hidden complexity at play here also. Yes, the number of symbols required is small. But those symbols are only meaningful to someone with the education to understand them. Namely, usually, a human being with a very complex brain, both in terms of requiring millions of years of evolution and years of training time. Actually, those alone aren't enough, since these people are building on top of hundreds of years of previous work in the field.


> Given the apparent complexity of the universe, it still amazes me that that number of symbols is so small.

Yeah, except, maybe the number of symbols isn't so small. The number of symbols needed to model very simple relationships is small, like the relationship between two gravitational bodies.

Now add a third body, and yes, in some sense this system is still described by the simultaneous relationship between pairs of bodies, but it's not really usable in that form. You've got to explode it, and you've got to do a lot more calculation.

I'm not totally confident here. I do have some intuition that the Navier–Stokes equations, for example, fully describe the motion of a very complex phenomenon in a very small number of symbols. But then if they require so much computation to actually apply, is my intuition wrong that this really counts as a full description of the system?

Is there a distinction between running the equations and running a simulation? And what exactly is the distinction between running a simulation and setting up the system and just seeing what happens?

It's definitely different, but I'm not so sure it's this amazing compression that it's made out to be.


> Now add a third body, and yes, in some sense this system is still described by the simultaneous relationship between pairs of bodies, but it's not really usable in that form.

The third body makes the system nonlinear and non-analytic, so the equations can only be investigated via numerical simulation. But the number of symbols is still small.

It may be the case that most of the universe is nonlinear and we only pay attention to the linear parts because they're easy. It might also be the case that a lot of the universe requires a lot of symbols to describe but we only pay attention to the other part.


But that is of course a different question, the mystery is not that mathematics can describe the universe but that relatively simple mathematics does the job. The complexety in our universe is in the enormous number of particles in it. Could it have been different, few particles, complex rules? Maybe, at least I can not immediately come up with a good reason why not.


> relatively simple mathematics does the job

Well, the key word being relatively. Relative to what? Mathematics of QFT is not simple by any measure (and neither is that of fluid dynamics, for example), nor is algebraic geometry that is being pushed as a new tool of theoretical physics.


Is this perhaps just an error in confusing relative and absolute terms? That is to say, being that we know that simple rules can generate complex results, it is thus possible (likely?) that a series of complex results will result from simpler rules, making them seem "simple". In other words, given any set of rules X, the results Y will often naturally be more complex, such that C(X) < C(Y), leading to the after-the-fact mis-interpretation that the rules themselves are simple, vs. the more accurate, but certainly less exciting, statement that they are mere simpler.


Complex behavior doesn't just come from simple rules. That only happens when the rules are nonlinear. Complex behavior could also come from very complicated rules.


Sure, and my comment in no way disallows that. My point is that there seems to be an assumption that complex behavior comes from complex rules, vs. the complexity of the behavior being assumed to have no necessary relation to the complexity of the rule-set. As such, we go in with an inherent bias of expecting complex rules, which leads the rules to often appear simpler than they are.


You think it's not puzzling because you take it for granted, having grown up with a scientific culture that now assumes that it is true (perhaps to too great a degree).


In addition to giving some idea about algebraic geometry, I appreciated the insight and honesty it contained regarding how a (high profile) researcher decides to try to learn more about an area, and about how the author was intimidated by it. I appreciated it, but, it is also pretty depressing, in that a professional mathematician of Baez's calibre clearly found it very tough-going just trying to survey the existing knowledge, let alone adding to it.


> algebraic geometry

Then there's always geometric algebra!

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


> Newton into the Quantum

Simply put, "the first quantization." (I am amazed how difficult popularizers of science often make it to understand what the hell they are talking about.)


If you don't already know what quantization is, that's just jargon though.


The problem in the title is that "Newton" just means "Classic Mechanics". Newton has a lot of results, like in gravity, light, mechanics, ... Replacing "Classic Mechanic" with "Newton" looks like the weird replacement tables in some xkcd comics. (Inside the article, the idea is clear, the problem is in the title.)


Similar to personalizing all General Relativity advances and issues as being about "Einstein".

Nobody does that for Quantum Mechanics.


For what it's worth, I hate the articles with title "Einstein is wrong/right". (Extra bonus hate point for "Scientist say Einstein is wrong/right")

The case of Quantum Mechanics is strange, there is no clear name to choose. I guess Heisenberg an Schrödinger are the best candidates, but their names are associated in popular culture with some specific effect instead of the whole theory.


Can anyone recommend good course notes, books, ... on algebraic topology?

Good reference books?

But first and foremost any recommended primers into algebraic topology?


to say that Cartesian geometry is in any way like algebraic geometry is like saying that a skateboard is like a Ferrari

>One reason is that quantum physics is based on algebra, while general relativity involves a lot of topology. But that suggests an avenue of attack: If we can figure out how to reduce topology to algebra, it might help us formulate a theory of quantum gravity.

huh? everything boils down to big matrices and hard differential equations in the end

>My physics colleagues will let out a howl here, and complain that I am oversimplifying. Yes, I’m oversimplifying:

no, you're not making any sense. i stopped reading here


I'm in complete agreement. The fact that he entirely glosses over there existing a field like algebraic topology with tremendous application to physics shows the author has little understanding of what he's writing about when it comes to physics.


The author is https://en.m.wikipedia.org/wiki/John_C._Baez who:

> He is known for his work on spin foams in loop quantum gravity.[3][4] For some time, his research had focused on applications of higher categories to physics and other things.

So he probably has _some_ understanding of physics, no?


John Baez is one of the most qualified people in the world to talk about mathematical physics. Who are you?


That's an appeal to authority if I ever saw one. He may be qualified to talk about it, but he certainly isn't showing that here. The parent comment shows what we're taking issue with.


Can you really be that pompous?




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