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

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

#241

Post by lumost »

steamer25 wrote: post #168

Prior to computer-generated 3D animation, I can imagine it was very difficult to float and spin vector-arrows in mid-air with enough accuracy to show what goes on without having to resort to reams of explanatory paragraphs. Eugene Khutoryansky is something of a lesser-known 3b1b that's more focused on physics than math. I found his animations very helpful for building intuition around Maxwell's equations: https://www…

I suspect something similar happens with manifolds for GR. Riemanian manifolds aren't a big deal when you contrast them to what happens inside of a DNN, but physical analogs for these structures start to break down.

e.g.

Imagine a 4-dimensional hyperbolic surface defined by the lightcone of a particular point in space-time, now imagine that this surface is stretched/compressed by the distortions of gravity. Now let's talk about equations which are only loosely tied to this surface.

vs.

Consider the metric tensor defined by this 4x4 matrix g_xz. Distance is computed as a^x b^z g_xz, now consider all possible walks from point a to c to b. Now let's show the relation between these walks and a quantity we'll call the stress-energy tensor which represents the energy/momentum density at any particular point in space, and it's flux towards any other direction in space.

The latter is a very algebraic description, which does not rely on the audience having to visualize the constructs involved. Practically, even if you get a feel for what a Riemanian manifold looks like in 4-dimensions - you'll struggle to visualize the Riemann tensor, or Christophel symbols.

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

#242

Post by kragen »

DiogenesKynikos wrote: post #210
kragen wrote: post #128

Earlier quoted context omitted.

admittedly so, but no, on the capitalization of 'equations' (and 'quantum' and 'electrodynamics'), which are not

It is common to capitalize the names of famous equations or theories. Maxwell wrote tons of equations during his life, but there's only one set of "Maxwell's Equations." Maxwell himself never even wrote down Maxwell's Equations in the form we now know them.

coherent and reasonable, but as a guide to current usage, wrong

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

#244

Post by gosub100 »

s1artibartfast wrote: post #191
steamer25 wrote: post #168

Prior to computer-generated 3D animation, I can imagine it was very difficult to float and spin vector-arrows in mid-air with enough accuracy to show what goes on without having to resort to reams of explanatory paragraphs. Eugene Khutoryansky is something of a lesser-known 3b1b that's more focused on physics than math. I found his animations very helpful for building intuition around Maxwell's equations: https://www…

I wish most explanations wouldn't skip over the fact that field lines arent real, and just a tool to graphically depict what is going on. Statements like the following gets the causality entirely backwards. >the strength of an electric field depends on the number of electric field lines.

isn't that a special case of Plato's argument that "triangles aren't real. show me a perfect triangle...you can't, you can only show representations of a perfect triangle"

Your comment could be reduced to "lines arent real. show me a perfect line"

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

#245

Post by DiogenesKynikos »

kragen wrote: post #242
DiogenesKynikos wrote: post #210

Earlier quoted context omitted.

It is common to capitalize the names of famous equations or theories. Maxwell wrote tons of equations during his life, but there's only one set of "Maxwell's Equations." Maxwell himself never even wrote down Maxwell's Equations in the form we now know them.

coherent and reasonable, but as a guide to current usage, wrong

Capitalization of "Equations" is clearly style-dependent, and thus neither right nor wrong, unlike, say, capitalizing the first letter in a sentence.

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

#246

Post by Mikhail_K »

wch4999 wrote: post #133
Mikhail_K wrote: post #87

OK, I do not understand prof.Dyson's argument at all. "This does not mean that an electric field-strength can be measured with the square-root of a calorimeter. It means that an electric field-strength is an abstract quantity, incommensurable with any quantities that we can measure directly." Electric field-strength is measurable no less directly than energy, it is a force experienced by a unit charge placed within t…

You need to read the paragraph prior to the quote. He is talking about which one of field and mechanical stress is more "fundamental" or less "direct". If one measure the force exerted by electric field using a unit charge, one measures the field by measuring the mechanical stress first. Of course, the context matters. Often if one compares potential and field, field would be the one directly measured. It is just sem…

I have read the prior paragraph. It does not clarify how measurement of energy is more direct than measuring the field.

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

#247

Post by Mikhail_K »

ThePhysicist wrote: post #92
Mikhail_K wrote: post #87

OK, I do not understand prof.Dyson's argument at all. "This does not mean that an electric field-strength can be measured with the square-root of a calorimeter. It means that an electric field-strength is an abstract quantity, incommensurable with any quantities that we can measure directly." Electric field-strength is measurable no less directly than energy, it is a force experienced by a unit charge placed within t…

You can make the electric field disappear by choosing the right gauge. Same goes for the magnetic field (can't make both disappear together though). The vector potential, in that sense, can be regarded as a more fundamental description of the electromagnetic field. It can't be observed directly though, but electric and magnetic field strengths are manifestations of the vector potential, they are not fundamental in th…

> You can make the electric field disappear by choosing the right gauge.

That, had it been true, would have made the gauge observable.

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

#248

Post by blharr »

lupire wrote: post #130
hiAndrewQuinn wrote: post #43

My proudest moment in high school was getting a 5/5 on the calculus based AP Physics C exams at 15 with no calculus and only rudimentary algebra knowledge at the time. That experience permanently colored my thinking, and made me much more open to practicing thorough visual imagination as a way to solve problems. I found that practice useful all the way through my EE degree's vector fields courses a decade later. I th…

You may have taken a slightly wrong lesson from that exam. AP exams are extremely curved (scoring about 70% or less is a 5), and are half multiple-choice, so a few clues can get the answer), and it's partly a conceptual test that doesn't rely on math. But of course you're absolutely correct that continuous and discrete systems are approximately equal.

AP Physics C is also one of the more heavily curved exams, where a 55-60% is a 5.

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

#249

Post by calf »

nyssos wrote: post #237
calf wrote: post #218

Earlier quoted context omitted.

Why is randomness non-computable? In computer science, the theorem is that the set of all Deterministic Finite Automata is equivalent to the set of all Nondeterministic Finite Automata. It is a non-obvious theorem that is a one page proof taught in every junior level theory of computation course. This theorem is what lets deterministic and nondeterministic Turing machines to be used interchangeably in many subsequent…

> Nondeterministic Finite Automata. The "Nondeterministic" in NFA means its transition function goes from states to sets of states, instead of from states to states. Informally, it can explore multiple paths in parallel for the cost of one. They're not probabilistic.

The computational semantics of the NFA simply requires that the next state be one of the allowable next states in the transition function d: Q x ∑ --> PowerSet(Q).

Thus, this semantics implicitly encodes the notion that the machine is nondeterministically choosing the next state in each execution.

The decision problem of whether an NFA accepts a string w is what allows for the informal parallel interpretation, that it accepts iff you imagine the computation is forking off a new thread at each nondeterministic branch. But to say that this not nondeterministic or not probabilistic is like saying the Many-Worlds Interpretation means there is no real superposition, or something like that. It's like saying a throw of a dice does not really involve probability because of a symmetry argument that a dice has six equal sides. Mainly, I don't understand that, because I see probability as a way to implement nondeterminism: a system is probabilistic only because it is making nondeterministic choices according to some probability distribution. And checking Sipser 2nd ed. p.368: "A probabilistic Turing machine is a type of nondeterministic Turing machine in which each step is a coin-flip step".

Anyhow, my main issue was that the original commenter casually claimed that probability makes things (physics) uncomputable. But Turing computability has nothing to do with probability, since as I recall the closest concept is the Non-deterministic Turing Machine (NDTM) and with that it is a basic proof to show that NFAs vs. DFAs, as well as NDTMs vs. DTMs, are computationally equivalent and there are theorems for that.

Meaning either they are using an idiosyncratic definition of computability or are ignorant of an introductory course on theory of computing which explains formally what Turing/Church's theories were about when clarifying the concept of computability. Okay or maybe they have a deeper philosophical disagreement with computability and complexity theorists - maybe they reject Sipser's definition above - but these are standard undergraduate curricula in CS by now and it could be argued that perhaps it is the non-CS experts who haven't thought deeply enough about what computability really is and would benefit from actually learning from these subdisciplines. I don't know, as they did not reply.

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

#250

Post by n4r9 »

kergonath wrote: post #186
n4r9 wrote: post #179

Earlier quoted context omitted.

I suspect that Gisin has a very clear idea of what he means by "information" in this context, having worked for over 40 years at the forefront of theoretical physics with a specialisation in quantum information theory.

I can link to people who’ve worked their whole life on various fields of Physics who still talk about perpetual motion. I am not saying he is wrong in this specific case, but an appeal to authority is not very convincing.

I've given your comment quite a lot of thought over the last few days. Maybe too much.

At first I was inclined to agree with you that this is an appeal to authority, with the caveat that such appeals do not always constitute a fallacy. For example, if we both agreed that such a Gisin is an expert whose opinion on this topic can be trusted, then his statements are valid evidence for one way or another.

But then I realised that the very claim being challenged is whether Gisin knows what he's talking about. Floating his credentials and experience feels like a valid contribution. For what it's worth, back in my PhD days I read several of his papers and saw a couple of his talks at conferences, and can confirm he's one of the leading researchers in the field and is particularly thoughtful and careful in his work.

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