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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]

#181

Post by 1980phipsi »

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…

I'm jealous. Unfortunately, people like me with aphantasia have no visual imagination. The hardest part in physics was converting the problems into equations. Once it is an equation, I could solve it (depending on the problem, with some difficulty or not, I guess).

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

#182

Post by mrguyorama »

lupire wrote: post #137
psychoslave wrote: post #73

Earlier quoted context omitted.

Given that no one, or at least no human, can experiment what reality in its whole, and as far as we want to honestly recognize the effective scope of our knowledge, probably we will never know in absolute terms. What matter is a subjective topic. What we all have in common is logistics constraints. So if some people set as a goal something that requires to settle if reality is more easily handled when modeled in cont…

Continuity just hides the ball. You say you can't comprehend how something can move from 1 to 2 discretely. But the paradoxical notion of infinite continuous change has been known since antiquity. It's faith either way. Discrete doesn't mean state changes are wholly globally arbitrary. Imagine a graph with nodes and edges, a state machine as computer sciences call it. I think it's easy to agree that the universe coul…

Zeno's """Paradox""" was nonsense even in it's own time. Easier now that we understand Newton's laws of motion but his contemporaries were able to sufficiently dispute his idea even without them.

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

#183

Post by mrguyorama »

mr_mitm wrote: post #112
metricspaces wrote: post #110

Earlier quoted context omitted.

> computer science leaking out Planck constant would like to have a word with you. But it is true that CS shines a light on the matter of mapping the infinite into bounded spaces. This matter of ‘cognition’ is the entire matter (npi) of contention. What is the actual relationship between number and perceived phenomena ? What is the deeper meaning of the concordance of mathematics and physics? Where do these magical c…

> Planck constant would like to have a word with you. Do you want to flesh this out? Are you suggesting that because phase space is quantized, position space must be quantized as well?

Most nerds who "understand quantum mechanics" misunderstand (usually do to incorrect explanations) that the Planck units are somehow the fundamental units of reality

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

#184

Post by eigenket »

nyssos wrote: post #141
mr_mitm wrote: post #115

Earlier quoted context omitted.

All observable quantities are eigenvalues of some operator, which are real numbers but discrete. How can they contain infinite amount of information?

There are operators with continuous spectra. The previous commenter was accidentally half-right, in that the usual intro QM picture where everything lives in L2 really isn't fully rigorous, but this is fairly easy to resolve. The correct setting is a rigged Hilbert space: given an algebra of operators A on a Hilbert space H, let S be the maximal subspace of H such that |sa| is finite for any s in S, a in A. These are…

I take minor issue with the phrase "correct" here. Thats one way you can do things but its also works completely fine to not do that. Another way of setting these things up has your states be honest elements of L2, and says observables are just POVMs (i.e. maps from a space of measurable sets to positive operators which obey some natural restrictions like additivity). Then given a measurable subset A of the spectrum of some operator the Born-rule probability is just given by an inner product like where P is the projector you get if you integrate the spectral measure of the operator over A.

This has the advantage of not having any funky "rigged" states suddenly appearing in your calculations and is also exactly how we deal with non-projective measurements in finite dimensional quantum mechanics.

See here, for example

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

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

#185

Post by kergonath »

rini17 wrote: post #61
mjburgess wrote: post #52

Earlier quoted context omitted.

> continuity of spacetime is a convenient approximation I disagree, and there's no evidence for this. This is computer science leaking out; physics has no formulation of spacetime in discrete terms, and indeed, all of physics presumes continuity. In QM, the space of wavefns is infinite-dim continuous, and if wasnt, QM wouldnt be linear. Cognition is discrete, but the world is continuous.

If it really was continuous so that physical quantities were real numbers as defined in mathematics, then it is in contradiction to maximal information density. Because almost all real numbers contain infinite amount of information. Full argument is elaborated here "Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real? by Nicolas Gisin": https://arxiv.org/abs/1803.06824

> If it really was continuous so that physical quantities were real numbers as defined in mathematics, then it is in contradiction to maximal information density. Because almost all real numbers contain infinite amount of information.

Yeah. As they said, it’s computer science leaking out.

It can be misleading to reason about entropy, which is the relevant physical concept, as if it were strictly equivalent to information as computer scientists understand it. Entropy works perfectly fine with continuous densities of states and real numbers.

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

#186

Post by kergonath »

n4r9 wrote: post #179
mjburgess wrote: post #68

Earlier quoted context omitted.

All the people who use thinks like the word "information" in this context are confusing thermodynamic, logical, probabilistic, (+ many others) and equivocating. "Information" is not a physical quantity, and there cant be a "volume" of it. Nor does this have anything to do with real numbers. It is impossible for there to be any system extended in space and time to "zoom infinitely" into a continuous range and hence re…

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.

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

#187

Post by scotty79 »

PaulHoule wrote: post #131
toth wrote: post #105

Earlier quoted context omitted.

> The originally published equations were "20 or so" because one equation was written for each scalar component. > Rewriting the equations in vector form reduces the number to the modern number. And if you use the differential form or 4d tensor notation they get reduced to 1 equation. Of course, for a lot of practical problems this is not very useful and it's better to work with the 3d vector form. > The variant with…

The fact that you can write it in one equation shows that the theory is very simple because it is an expression of symmetry. E and B are not these two different things related by an inscrutable cross product but just two aspects of the same thing.

You could write all physics in a single simple equation. deltaW=0 Where deltaW is deviation of the universe from the relevant math.

Writing Maxwell's as 1 equation or 4 or more is just esthetic choice where you decide what to accentuate.

20 might be too much because three dimensions are not really different from each other so the notation that maps over them wholesale is probably a good idea.

4 equations seem perfect if you want to differentiate between classical effects of the electric field and relativistic effects (magnetism).

I don't know if single equation really shows that they really have the same source and the relativity is involved or is it just a matrix mashup of the 4 separate equations that doesn't really provide any insights.

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

#188

Post by tzs »

greysphere wrote: post #139
chr1 wrote: post #94

Earlier quoted context omitted.

You don't have to imagine an ordered grid. If grid unit is small enough (say plank length 1,6 10^-35) and the grid is chaotic, for the distances of ~ 10^-16 that we can measure, everything will look the same in all directions. This happens the same way in which steel demonstrates isotropic behavior although its microscopic structure is anisotropic. So there is no easy way to prove or disprove continuity of space.

A chaotic grid would be macroscopically observable because random + random != 2 random, it's equal to 'bell curve'. Everything would be smeared as a function of distance, which we don't see. This characteristic is observable for metals as well. Steel becomes less flexible as it's worked because it's grains become smaller and more chaotic - A microscopic property with a macroscopic effect.

If we are talking about a grid with a very small spacing, say around the Planck length, I don't see how we would be able to macroscopically observe it.

Everything we can see move on the grid is at least 20 orders of magnitude bigger than the grid spacing. Any macroscopic objects we can experiment with are more like 30+ orders of magnitude bigger than the grid spacing and consist of numerous atoms that will all be moving within the object due to thermal jiggling over distances orders of magnitude bigger than the grid spacing.

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

#189

Post by toth »

wch4999 wrote: post #136
ThePhysicist wrote: post #124

Earlier quoted context omitted.

You're right, sorry I was thinking of a Lorentz transformation that would make either the magnetic or electric field disappear under certain conditions.

"transform into each other" would be more appropriate. The gauge choice you mentioned is not totally wrong. The gauge freedom can be used to set the electric field to zero, but only once at a single point.

Sorry, but gauge transformations do not (by construction) affect the physical fields at all. You cannot set E to 0, even at a point, with a gauge transformation.

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

#190

Post by toth »

scotty79 wrote: post #187
PaulHoule wrote: post #131

Earlier quoted context omitted.

The fact that you can write it in one equation shows that the theory is very simple because it is an expression of symmetry. E and B are not these two different things related by an inscrutable cross product but just two aspects of the same thing.

You could write all physics in a single simple equation. deltaW=0 Where deltaW is deviation of the universe from the relevant math. Writing Maxwell's as 1 equation or 4 or more is just esthetic choice where you decide what to accentuate. 20 might be too much because three dimensions are not really different from each other so the notation that maps over them wholesale is probably a good idea. 4 equations seem perfect…

It's true that you can always define notation to combine all equations you want into one. This means that, by itself, the observation that you can write Maxwell's equations as a single equation doesn't say anything very meaningful.

However, the notation that lets you do this in this specific case is very natural and not specific to Maxwell's equations. Differential forms are very natural objects in differential geometry, mathematicians would have likely introduced them and studied without inspiration from physics. The fact that Maxwell's equations are very simple in this natural geometrical language does say something meaningful about their nature and elegance, I think.

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