Why is Maxwell's theory so hard to understand? (2007) [pdf] • Hacker News Board

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#61

Post by jules » Sun, Jan 06, 2019, 10:29 PM UTC

wmnwmn wrote: post #14

E&M is far from "simple". It contains special relativity, for starters. Also it is incompatible with thermodynamics: solving this problem is why Planck invented quantum mechanics. Also point charges have infinite energy: this problem leads to renormalization theory. Also it introduces gauge invariance, an essential but complex part of all modern theories. And lastly, the mathematics of E&M is a big step up from Newto…

E&M is simple in the sense that the equations that govern the relationship between the charge distribution and the electromagnetic field are quite simple. And the equations governing how charges move in an electromagnetic field are quite simple. I think this is what Dyson has in mind. Understanding the implications and limitations of those equations is not simple at all.

An even more blatant difficulty with E&M, related to your third point, is that it is schizophrenic. Given the trajectories of charges it will tell you what the electromagnetic field will be, and given the electromagnetic field it will tell you how charges will move. Unfortunately, these two parts of the theory seem to be incompatible, and the theory will not tell you how fields + charges will evolve in time.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#62

Post by ducktective » Sun, Jan 06, 2019, 10:32 PM UTC

ncmncm wrote: post #21

It is unfortunate that Dyson neglects the role of Oliver Heaviside, again. This is in a long tradition of English neglect, ultimately traceable to Heaviside's status as a commoner. Heaviside invented the mathematical tools we still use to understand and teach Maxwell, and most of the important consequences of the theory, but Pupin, Hertz, Marconi, and deForest used his methods and took got the credit. Today Heaviside…

In university in eng.math, diff.eq, electromagnetism and signals and systems courses, we had a "Heaviside Method" (with this exact name) in deriving the numerator coefficients in partial fraction expansion process of yielding time-domain equivalents of s-domain functions. I specifically remember my professor saying that this method is ascribed to "Heaviside" so much so that I later googled him and read his Wikipedia entry.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#63

Post by adrianratnapala » Sun, Jan 06, 2019, 10:34 PM UTC

ncmncm wrote: post #21

It is unfortunate that Dyson neglects the role of Oliver Heaviside, again. This is in a long tradition of English neglect, ultimately traceable to Heaviside's status as a commoner. Heaviside invented the mathematical tools we still use to understand and teach Maxwell, and most of the important consequences of the theory, but Pupin, Hertz, Marconi, and deForest used his methods and took got the credit. Today Heaviside…

> This is in a long tradition of English neglect, ultimately traceable to Heaviside's status as a commoner

This is unfair on the English. Glancing at their wikipedia entries, Heavyside doesn't seem any more common than [Dirac][1] or [Faraday][2]. If I had to guess at a sociological reason why Heavyside got ignored, it's that he got caught in the transition when people started learning engineering through universities rather than apprenticeships.

[1]: https://en.wikipedia.org/wiki/Paul_Dirac#Early_years [2]: https://en.wikipedia.org/wiki/Michael_Faraday#Early_life

Heavyside's uncle Charles Wheatstone was a tradie who never went to unversity, but got knighted for services to electrical engineering. Faraday was another tradie who joined the Royal Society eight years before he got an (honorary) university degree. But in a later age Dirac became a world famous scientist, but would probably have gone nowhere if he hadn't got a scholarship to study electrical engineering at a university.

We can't wag our finger at Victorian English society for a the sin of credentialism when our own society does it much more vigorously. It's especially for our own profession, as programming is much more a craft than a thing you can learn at university.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#64

Post by amatthew » Sun, Jan 06, 2019, 10:38 PM UTC

killjoywashere wrote: post #57

It's not until now, over 20 years later, that I realize what a magician my E&M prof was. After months of careful development of basic E&M, Maxwell's equations simply fell out. As I recall, it was capacitance that was the impetus. It was as though we had struggled in a dark tunnel for months and suddenly came around a turn and there was light. Not a little, but sheets of daylight and all the equations just fell togeth…

In what ways were they that important later in life?

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#65

Post by adrianratnapala » Sun, Jan 06, 2019, 10:44 PM UTC

tntn wrote: post #17

wmnwmn wrote: post #14

E&M is far from "simple". It contains special relativity, for starters. Also it is incompatible with thermodynamics: solving this problem is why Planck invented quantum mechanics. Also point charges have infinite energy: this problem leads to renormalization theory. Also it introduces gauge invariance, an essential but complex part of all modern theories. And lastly, the mathematics of E&M is a big step up from Newto…

Dyson's statement was about Maxwell's theory, not all of E&M and its interactions with every other physical theory. There are no point charges in Maxwell's theory, for example.

I think the point is that electromagnetism really is a very complex and knotty phenomenon, and we moderns are spoiled by having

(1) Heavyside's deceptively elegant formulas for the theory (which we now call the Maxwell Equations) and,

(2) A well oiled pedagogy about how those equations the capture specific phenomena named after the likes of Ampere and Faraday, and

(3) Powerful formalisms for studying circuits, optics, transmission lines etc that can mostly be used without going back to the fundamentals.

Most of that was not available to Maxwell when he started out, so it's not surprising that he presented his theory in a way that reflected the real complexity of it all.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#66

Post by jules » Sun, Jan 06, 2019, 10:48 PM UTC

amelius wrote: post #23

Some people hold the view that a better way of teaching electromagnetism is through geometric algebra. See: https://arxiv.org/abs/1010.4947

Yes, but the formulation in that paper is really not the best one. Electromagnetism is best formulated in geometric algebra of 4-space (3 space dimensions + 1 time dimension), not 3-space.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#67

Post by jules » Sun, Jan 06, 2019, 10:56 PM UTC

19f191ty wrote: post #58

Koshkin wrote: post #55

Earlier quoted context omitted.

Also, the approach based on differential forms is very appealing; for a list of various formulations of electromagnetism, see https://en.m.wikipedia.org/wiki/Mathematical_descriptions_of...

Aren't the two very closely related, if not equivalent? Differential forms and Geometric Algebra I mean.

Yes, they operate on the same objects. The main operations on differential forms are the exterior derivative d and the wedge product f /\ g. These are independent of the metric. If you add a metric you get one extra operation called the Hodge star ⋆, which is a prefix operator, so it operates as ⋆f.

Alternatively, if you have a metric on your space you can define the geometric algebra derivative D and the geometric product. These are closely related to the differential forms operations, e.g. Df = df + ⋆d⋆f. Using it, Maxwell's equations become one equation, namely DF = J where F is the electromagnetic field tensor and J is the four current.

So in my opinion it's indeed best to not put this as differential forms vs geometric algebra, but rather as a single theory in which some operations don't depend on the metric (d, /\) and some do (⋆, D, geometric product).

Unfortunately, geometric algebra has been overhyped by its proponents, and the papers they write are less than rigorous, so some people are under the impression that it's crackpottery, so it's safer to call it Clifford algebra :)

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#68

Post by tim333 » Sun, Jan 06, 2019, 10:57 PM UTC

ncmncm wrote: post #50

scabarott wrote: post #40

Earlier quoted context omitted.

You're really caught up on this commoner/non-commoner dynamic (from your other comments). Was it really that important? - many of the well known scientists of that time were 'commoners'.

I'm not, but they were. But it was not discussed openly in print at the time, and so must be inferred.

I just read his Wikipedia and he was sponsored by his uncle Sir Charles Wheatstone of Wheatstone bridge fame and ended up a Fellow of the Royal Society so was not so badly situated.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#69

Post by earthicus » Mon, Jan 07, 2019, 12:03 AM UTC

jules wrote: post #67

19f191ty wrote: post #58

Earlier quoted context omitted.

Aren't the two very closely related, if not equivalent? Differential forms and Geometric Algebra I mean.

Yes, they operate on the same objects. The main operations on differential forms are the exterior derivative d and the wedge product f /\ g. These are independent of the metric. If you add a metric you get one extra operation called the Hodge star ⋆, which is a prefix operator, so it operates as ⋆f. Alternatively, if you have a metric on your space you can define the geometric algebra derivative D and the geometric p…

Is it possible to work with geometric algebra effectively without a metric using some more complicated construction?

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#70

Post by ckocagil » Mon, Jan 07, 2019, 12:11 AM UTC

aj7 wrote: post #48

nonbel wrote: post #16

Earlier quoted context omitted.

>"E&M is far from "simple". It contains special relativity, for starters." A theory (which is just a set of assumed first principles and rules of logic) can be simple but allow you to deduce vast complexity from it. In fact, it is ideal for a theory to be as simple as possible. The "game of life" is not really a theory, but it demonstrates that simple rules can lead to surprising complexity: https://en.wikipedia.org/…

I reflexively downvote the game of life. I have never heard of a single useful application or analogy of “finite automata.” Yes, there are speculations about extremely small Planck lengths, by no new physics has panned out. Finally I am sad that one of the greatest computer scientists and polymaths, Ed Fredkin, got sucked into this. We all have weaknesses.

Surely you mean cellular automata, not finite automata.