
Math That Takes Newton into the Quantum World - dnetesn
http://nautil.us/issue/69/patterns/the-math-that-takes-newton-into-the-quantum-world
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m12k
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-...](https://www.quantamagazine.org/physicists-discover-geometry-
underlying-particle-physics-20130917/)

~~~
_Microft
You might also like [https://www.quantamagazine.org/the-octonion-math-that-
could-...](https://www.quantamagazine.org/the-octonion-math-that-could-
underpin-physics-20180720/)

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vankessel
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](https://en.wikipedia.org/wiki/Split-octonion) [2]
[https://en.wikipedia.org/wiki/Minkowski_space](https://en.wikipedia.org/wiki/Minkowski_space)

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danbruc
_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.

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dreamcompiler
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.

~~~
furyofantares
> 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.

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dreamcompiler
> 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.

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Myrmornis
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.

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Koshkin
> _algebraic geometry_

Then there's always geometric algebra!

[https://en.wikipedia.org/wiki/David_Hestenes](https://en.wikipedia.org/wiki/David_Hestenes)

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Koshkin
> _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.)

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

~~~
gus_massa
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.)

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

Nobody does that for Quantum Mechanics.

~~~
gus_massa
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.

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

Good reference books?

But first and foremost any recommended _primers_ into algebraic topology?

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paulpauper
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

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chunky1994
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.

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

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chunky1994
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.

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danharaj
Can you really be that pompous?

