Advances in number theory inspired by physics | Hacker News Reader

Advances in number theory inspired by physics

quantamagazine.org

214 pointsdigital5569 comments

He's not alone in applying physics math to number theory. One thing that makes mathematicians leery of physics-inspired proofs is that physicists accept mathematical transformations that produce correct experimental results but that have not been proven axiomatically.

For example, here is a proof of the Riemann Hypothesis:

http://aip.scitation.org/doi/10.1063/1.5012170

It uses operations on an analytic continuation of the zeta function. Are they OK? I don't know. If they are, it's Fields Medal material.

But this is nothing new. Oliver Heaviside revolutionized differential equation solving with the Heaviside Operator, which gives correct answers to electromagnetic problems. Nowadays we call it the Laplace Transform, because Laplace had used what turned out equivalent, for other purposes.

Never heard of Oliver Heaveside and his operator despite using Laplace transforms on a regular basis for year. You learn something new every day. Thanks.

My diff equations textbook referred to it as the Heaviside operator. In my EE and control classes, it's called the Laplace transform.

Other reasons Heaviside is often ignored are that he was a commoner, and he published all his papers in a newspaper. He originated the theory of transmission lines (inventing the loading coils you still see hanging between telephone poles) and atmospheric radio propagation, both often credited to others. See Paul Nahin, "Oliver Heaviside: Sage in Solitude"

Now that's very interesting!

According to his wiki page, he reduced Maxwell's equations from 20 equations (in 20 variables) down to the 4 iconic equations we now refer to as Maxwell's equations. Imagine having to learn EM theory with 20 equations... as if current EM theory courses aren't tough enough already!

One of his creations, the step function, is also a key part of many proofs and equations in signal processing.

It should be noted that Maxwell's quaternion formulation of the same relations has just two equations.

Vector notation won out over quaternions not from elegance or power, but from practical details of working them on paper. Now that we don't, anymore, quaternions are probably the better notation. Somebody should rewrite an E-M and an optics textbook.

pls, we use tensor calculus and exterior algebra. More, precisely, the electromagnetic tensor is the exterior derivative of the 4-potential.

Exactly, vector calculus is a useful learning tool, but not used in most higher level (e.g. General Rel, plasma physics, etc) or numerical (super computer) physics calculations.

Raising and lowering Kartoffel symbols and their covariant derivative forms make manipulating the coordinate transforms possible (4-6 dimensional)... otherwise it would just look like pages of equations.

Interestingly, mechanical engineering also deals almost entirely with tensors... and they've developed extremely intuitive visualization tools.

> Interestingly, mechanical engineering also deals almost entirely with tensors... and they've developed extremely intuitive visualization tools.

What visualisation tools are you referring to? That sounds interesting.

Since all 3D stress/strains are tensor quantities (even for purely linear models), MEs project them into principal axes so that you can see their angle and their peak tesnsile/shear stress.

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

Look at Mohr's circle:

https://en.wikipedia.org/wiki/Mohr%27s_circle

Within an object the stresses and strains (and principal components) vary as you move through it. That's part of what makes predicting failures and effects of thermal expansion so complex, etc difficult even ignoring nonlinear effects and failure modes (e.g. crack propagation).

https://en.wikipedia.org/wiki/Christoffel_symbols is probably what you wanted to say

Lol... german potatoes... autocorrect? Thanks, that's actually hilarious.

Isn't a quaternion just a 3D representation of a vector in terms of the coordinate system's unit vectors? From what I recall, using such a representation makes it easier to perform transforms between coordinate systems.

Quaternions are four-dimensional, hence the radix quater, meaning “four”.

Yeah... let's also not forget that quaterion multiplication is noncommutative (AB≠︎B A), which makes theem an absolute pain to work with. This is why it made sense to reformulate them from Maxwell’s two to ythe then-prevailing twenty. Then they got reformulated in terms of 3-vector calculus (by Heavside, whose existence and achievements I was ignorant of until just a few hours ago, courtesy of Hacker News).

Matrix multiplication is noncommutative too. What made quaternions so particularly painful?

They’re not particularly more painful than matrices (indeed you can represent quaterions as matrices, beyond their ”traditional” representation of a tuple of a scalar and a 3-vector), but there’s no point going through the pain since there’s alternative formulations of Maxwell’s laws (in terms of vectors) that are commutative and get you out of the pickle (at the expense of going from two to four equations).

What kind of newspaper publishes scientific papers?

I guess Nahin called it a newspaper because it came out on newsprint, folded over, weekly. It would be called a trade rag, these days, if we had those anymore.

https://books.google.com/books?id=ehJbAAAAYAAJ&dq=intitle%3A...

He's responsible for the form of Maxwell's equations that we use today.

> One thing that makes mathematicians leery of physics-inspired proofs is that physicists accept mathematical transformations that produce correct experimental results but that have not been proven axiomatically.

I think it's way beyond that. Physicists are know to be fast and loose with their maths.

For a period of time they were basically deleting infinities which appeared in the equations and pretended everything was fine, because after this operation the results agreed with experiments.

To quote Dirac: I must say that I am very dissatisfied with the situation, because this so-called "good theory" does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it turns out to be small—not neglecting it just because it is infinitely great and you do not want it!

Only later mathematicians put this technique on more proper grounds - renormalization.

In other words, physicists were entirely correct to toss the infinite values, and mathematicians were later persuaded it was mathematically sound.

Nowadays, although physicists say they are "renormalizing", they still just toss the infinite values. They're justified by results of experiments. Mathematicians are limited because, typically, they have no handy universe to run experiments in, never mind budget for a supercollider.

> In other words, physicists were entirely correct to toss the infinite values, and mathematicians were later persuaded it was mathematically sound.

Rather: Mathematicians were able to develop an axiomatic framework in which these transformations have a well-defined meaning.

Potato, potahto. If the experiment produced different results, mathematicians would not be asked for an opinion until the physicists came up with some other math. Because the experiment matched, mathematicians were obliged to invent a framework to justify it.

Or to phrase this differently, physicists do not really understand what they are doing. They do some operation because it sometimes happens to produce results that match some physical experiments that they are trying to model. Their derivation rules are due to habit (in the sense of Hume), not due to inference or logic.

Almost certainly, a rigorous formulation will come about later and explain when these derivations should actually hold (rather than just assuming they hold "whenever necessary").

This is actually how a lot of mathematics works: researchers notice some relationship empirically, which gives some intuition and suggests some hypothesis, and then the serious mathematical work is trying to determine exactly what assumptions are necessary and proving it.

>Or to phrase this differently, physicists do not really understand what they are doing.

I think this is very unfair to say. I would like to remind who was it that discovered the renormalization group: Wilson, a physicist.

I think I agree with you, and I also think that, "physicist do not really understand what they are doing" is phrased a bit flippantly.

Physicists understand what they're doing perfectly well, and they're not, for the most part, producing rigorous math. They are identifying phenomena which can be described systematically, as well as the systematic representations of those systems. Rigor, and math as a whole, is meaningful for a physicist only insofar as it facilitates that outcome.

It's interesting that you say that, "almost certainly, a rigorous formulation will come about later..." You seem to agree that experiment is a useful test for the rigor of operations, at which point it comes down to simple division of labor. The physicist has an expectation that an operation is rigorous, and his job isn't to establish its rigor; it's to arrive at the expression. He does his job, and mentions to the mathematicians that this operation 'should' be rigorous, and it'll be interesting to find out exactly how.

I think it's exceedingly wrong to say, "Their derivation rules are due to habit (in the sense of Hume), not due to inference or logic." Derivation rules support only one goal: predict outcomes of systematic phenomena. Physicists are neither nor are they trying to be mathematicians. The property that guides derivation is physical intuition, which I would say mathematicians malign at their peril, since the universe has thus far proved to be a useful machine for checking rigor.

And that's the heart of it. The universe has, for whatever reason, proven to be a detector, or perhaps executor, of mathematical rigor, and in describing the universe with ever-increasing precision, we develop results that feed back in to be explored and made complete. We can say, "it must be mathematically rigorous that this infinity can be discarded, so long as the universe is still functioning as a system that exhibits mathematical rigor," and be satisfied because the universe has shown us it is so. This does amount mathematically to not understanding exactly what it is you're doing.

For interesting reading on this by someone smarter than myself: The Unreasonable Effectiveness of Mathematics in the Natural Sciences (https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...)

It seems like if shortcuts ‘just work’ in physics, that should be a good sign that there is probably a mathematical justification waiting to be found.

Other times, it is a good sign of edge-cases that are physically impossible or highly unlikely.

> if shortcuts ‘just work’ in physics, that should be a good sign that there is probably a mathematical justification

Or new theory.

Dirac?

My favorite example of bogus-but-effective mathematics in physics is the Dirac Delta "function".

That also has a formal mathematical interpetation as a distribution rather than a function now and the theory of distributions found many applications in PDEs.

An interesting thing is that before Dirac, Green was already informally using distributions in the 1830s to solve differential equations with what we now call Green's functions.

Closely related to Green's functions are the fundamental solutions to an elliptic operator, consider the Laplacian and its fundamental solution in Rⁿ (n≥3), what's the Laplacian of the fundamental solution? A Dirac's delta!

The big trouble with the delta function in quantum physics is that it's in some sense the natural basis for your continuous Hilbert space, but they aren't in the Hilbert space. Which necessitates a whole lot of mathematical explaining-away-formalism.

Physics has an aim: to provide a rigorous description of reality on such a way that predictions about future experiments can be made. Physicists found that by altering certain quantities they got rid of the divergences and obtained a theory of remarkable experimental accuracy. That's all that matters.

Fortunately, of course, for our understanding, more people came along soon after to elucidate this mechanism.

Your comment seems to be: Who cares whether or not QED was developed when it was? They should have stopped for a decade or two while they figured out the theory.

Are you serious?

That's not what he's saying. He's just making an observation that physicists have a tendency to make mistakes with math if it's convenient for them, so it's generally not a good idea to use their proofs when you're wanting something mathematical.

I guess that's more charitable. But to my ears it sounded like a baldly pedantic critique of what is easily one of the greatest theoretical results of the past century.

Heaviside's comment, famously, was "Shall I refuse my dinner because I do not fully understand the process of digestion?"

That's interesting. Any good examples where this turned out badly, in the sense that the formalism turned out to be inconsistent or something?

String theory has no experiments, so has no touchstone to justify dodgy maths. String theorists routinely depend on the infinite sum 1 + 2 + 3 + ... = -1/12, without apology. It's anybody's guess how well that will turn out in the end, but it hasn't interfered with grants, publication, or tenure, so it's OK so far.

>>infinite sum 1 + 2 + 3 + ... = -1/12

Sounds interesting. What is the name of this?

https://youtu.be/w-I6XTVZXww

I learned about it through this Numberphile video: https://www.youtube.com/watch?v=w-I6XTVZXww

There's also a nice write-up from Terry Tao: https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri...

Tl;Dr

It's slightly more accurate to say if

F(s) = 1^s + 2^s + 3^s + ...

Then

F(1) = -1/12.

This is still a lie, but the truth is that F(s) does make sense for a large domain of s, and there is a unique and canonical way of extending F(s) to a larger domain which contains 1. And then that new function evaluated at 1 is equal to -1/12.

It's a rigorous result if you define it as an analytic continuation of the Riemann zeta function defined in the normal way. It doesn't take much to prove, but the way it's used in physics it's not always immediately clear that interpreting infinite and divergent sums as analytic continuations of finite sums is justified.

Yet it seems to work...

The trick is that normal everyday summation is only defined for convergent series, so a slightly different definition of summation (called zeta function regularization) is used to assign a value to this divergent series.

Physicists seem to be suggesting that our universe behaves in accordance with this second definition of summation.

Is there a reason that a separate symbol and term are not used for this different form of summation?

Some people say that they are using "=" to mean something else, and relying on context to distinguish which "=" they mean. But it's less fun to say that the sum of the natural numbers "is associated with" -1/12, especially if you have to say how, exactly.

Mainly that in physics, these sums crop up naturally, using normal summations over real numbers, often sums of energies over different modes.

For instance, if you look at an idealized quantum violin string, each mode of vibration has a minimum energy (zero point energy) proportional to the frequency of vibration - classically they can all be 0, but not quantum mechanically. When you try to ask, then, what the minimum energy of the string is, you end up with a term that is literally 1+2+3+... and no particular reason to intepret those as complex or anything. But in a lot of ways, if you just barrel through and treat it as if you can do the sum, you get real results - the Casimir effect is an example where a real force can be predicted and measured based on calculations that are zeta regularized.

It's also worth noting that the dimensionalities in various string theories tend to hinge on the exact values of these infinite sums. Bosonic string theory being 26 dimensional comes out of, IIRC, a consistency equation that ends up including -1/12 because of a 1+2+3... sum. If memory serves, that result can be rigorously established in other ways, as well.

Thanks for the insight. I'll have to look at this more closely sometime... not sure I can grok it casually.

I've been watching the PBS spacetime videos on youtube a lot lately. The recent ones have been diving into zero-point energy (and related stuff), which has been fascinating.

It’s the Ramanujan Summation of the Euler Zeta Function (not to be confused with the Riemann Zeta Function, which deals with complex values, which is a generalisation thereof) when s = 1.

Wikipedia has an article and derivation

https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B...

They call it "1 + 2 + 3 + 4 + ⋯"

If memory serves, 1+ 2 + 3 + ... = –1/12 also figures in the estimation of the Casimir force between two electrically uncharged plates (which has been experimentally verified).

Has anyone else had the thought that physics is actually more fundamental that mathematics?

Also, read this article made me think of Feynman's advice which was something like 'look for symmetries and look for conservation'

Theoretical physics "shut up and calculate" is experimental mathematics.

Yes, but not in a way that's germane here. Physics in the sense of the workings of the world is fundamental. Physics in the sense of science practiced by humans is infamous for spherical cows, point masses and mathematical funny business. Not the same thing at all. :)

The “look for symmetries and conservation” edict is a direct emanation of Emily Noether's (First) Theorem, which basically places conserved quantities of a system of differential equations with the symmetry groups that system must exhibit. A deep, amazing, beautiful, bi-directional relationship between differential dynamics and abstract group algebra.

That's a discussion that doesn't make much sense.

What do you mean by "more fundamental"? The fact is physics and mathematics treat entirely different subjects. Physics treats the physical, empirical world. Mathematics is the study of formal thought. It isn't meaningful to compare them that way.

Or perhaps at the root they are the same thing.

> “What I started out trying to find” was a least-action principle for the mathematical setting, he wrote in an email. “I still don’t quite have it. But I am pretty confident it’s there.”

CS people should think of Bellman's equation, Dijkstra's algorithm, etc. Minhyong Kim is looking for the dynamic programming solution to "thickets of paths emanating from rational points".

Come to think of it -- calculus wasn't really rigorous until a LONG time after Newton introduced it. Isn't Riemann sometimes credited as the guy who finally straightened it out?

Also, IIRC, many very important things like vector calculus, Fourier and Laplace transforms, and other differential equation stuff was added far later.

Stuff is still being added nowadays: non-integer derivatives and integrals (”fractional calculus”), alternative formulations that rely not on the limit of the addition but on the limit of a multiplier (“L-calculus”), stochastic integration (Itō integrals), automatic differentiation (vital to estimating weights for neural networks in machine learning), derivatives of integers (???!!!!), and other things that might or might not be important going forward are all branches that have been developed since the 1960s.

What's "L-calculus"? And what does "alternative formulations that rely not on the limit of the addition but on the limit of a multiplier" mean? Any resources?

An integral can be thought of as the infinite sum of infinitesimal areas below the graph (look up Riemann sums). I think what he is saying is that instead of summing these, you instead to the product. That is, multiply each infinitesimal quantity together.

A derivative is basically how a small addition to the input value changes the output value; this change in output value is also studied in terms of being something “additional” (though it could also be a subtraction).

In L-calculus this approach is altered: how does multiplying the input value by a tiny value greater than one change the amount by which the output value gets multiplied?

Here’s the relevant Wikipedia page to get you started down the rabbit-hole: https://en.wikipedia.org/wiki/Multiplicative_calculus?wprov=...

Cauchy & Weiesraß reformulated ”infinitesimal calculus” rigorously in terms of limits and called it Real Funcional Analysis.

Curious when Topology will become a core math subject.

tl;dr;

Basically he’s trying to find something analogous to ‘action’ in physics, which when minimized in the space of possible solutions, will let him find rational solutions to Diophantine equations.

Reply on news.ycombinator.com