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This is really good copy pasta, thanks.
Why is Maxwell's theory so hard to understand? (2007) [pdf]
91–100 of 250 posts
Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]
#92OK, 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 that sense. Not sure if it's that what he's getting at, though.
Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]
#93Maxwell didn't have the nice differential geometric notations that we use today, which allow us to write his equations in a very concise and easy to understand form. His original paper is way more convoluted, so at the time it must have been really difficult to understand for everyone except the subject matter experts. And he was of course building on the work of Faraday, Ampere and others.
But like with other theories, people find ways to simplify the notation and formalism and explain it better. Quantum mechanics, special and general relativity are similar in that regard.
Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]
#94Earlier quoted context omitted.
> the world is continuous Is it though? Does it matter one way or the other? Do we think reality is the math in some way, or is the math a really darn good model of the reality?
Imagine there was a grid for space. For simplicity consider a regular grid of size 1unit in one direction and 1unit in a perpendicular direction. If such a grid existed, using one unit of ?something? would move you 1 unit along the axes of the grid, but you'd need 2 units of ?something? to move root2 units 45deg to the grid. Any discrete grid of any shape or size or pattern would have something like this, some sort o…
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.
Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]
#95Years ago I completed a post-graduate degree in physics, and although I had studied Maxwell's equations, I didn't have a good "feel" for them. I recently read "A Student's Guide to Maxwell's Equations", and it was perfect for me - it explained enough of the maths to understand the equations, without having to first learn differential geometry. https://www.cambridge.org/highereducation/books/a-students-g...
If you do want to learn the geometric formulation, part 1 of Gauge Fields, Knots and Gravity is a good resource (and has exercises!).
Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]
#96Really enjoyed the interview with Freeman Dyson here on his life story - “someone left some Real Analysis textbooks in the school library but they were in French. It was probably (G. H.) Hardy.”
https://m.youtube.com/playlist?list=PLVV0r6CmEsFzDA6mtmKQEgW...
Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]
#97Earlier 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.
But isn't the whole point of QM is that this assumption doesn't hold in some scale? I mean it's literally in the name. Care to explain? :)
A butterfly also literally has butter in the name. The point of QM is that certain energy levels are quantized. Or more generally that lots of operators/observables on continuous Hilbert spaces have discrete spectra.
Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]
#98[deleted]
Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]
#99My 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…
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.
Moreover, the original equations are the complete system.
The variant with 4 equations is the simplified variant for vacuum, which is mostly useless, except for the purpose of studying the propagation of electromagnetic radiation in vacuum.
The complete system of equations has around 6 or 7 or even more equations, depending on whether one chooses to have distinct notations for various physical quantities, such as electric polarization, magnetization, current of free electricity carriers etc., or not.
The variants with less equations are simplifications that are valid only in linear media, because only there you have proportionality relationships between quantities like electric field and electric polarization.
Instead of learning a large number of simplified variants of the Maxwell equations with limited applicability, it would have been much better if a manual would present since the beginning the only complete variant that is always true, which must be in integral form, as initially published by Maxwell.
The many simplified variants, for media without discontinuities where differential forms are valid, for stationary media, for linear media, for vacuum and so on, can be easily derived from the general form, while the reverse is not true.
Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]
#100Earlier quoted context omitted.
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
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…
Some people would disagree with dismissing information as non physical. For instance: https://scottaaronson.blog/?p=3327 The argument there would be that stuffing an extra bit of information in an information saturated volume would make it collapse into a black hole.