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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]

#141

Post by nyssos »

mr_mitm wrote: post #115
rini17 wrote: post #61

Earlier 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 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 your states. Operators in A don't necessarily have eigenvectors in H, but they do have eigenvectors in the space S* of all continuous linear functionals on S. So delta_x(psi)`.

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

#142

Post by saalweachter »

eigenket wrote: post #103
yard2010 wrote: post #55

Earlier quoted context omitted.

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? :)

Quantum mechanics does not mean everything is quantized. It got its name because the first predictions of quantum mechanics were quantized energy levels in some example systems, but that does not even mean that all energies are quantized in quantum mechanics. There are many systems you can study where energies are continuous, and many examples where other quantities are continuous in quantum mechanics.

Wasn't it more the observation the theory was designed to explain than the first prediction?

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

#143

Post by kragen »

staunton wrote: post #42
Onavo wrote: post #5

> The moral of this story is that modesty is not always a virtue. Maxwell and Mendel were both excessively modest. Mendel's modesty setback the progress of biology by fifty years. Maxwell's modesty setback the progress of physics by twenty years. It is better for the progress of science if people who make great discoveries are not too modest to blow their own trumpets. If Maxwell had had an ego like Galileo or Newton…

Newton was the first to act on the revolutionary idea that "physics should be formulated using math". Nowadays a theory of gravity that doesn't use math is basically not a theory. Maybe we can forgive him at least some of the extravagance.

galileo and kepler used quite a bit of math to formulate their physics theories too, but in that they were following in the footsteps of ptolemy and archimedes

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

#144

Post by lupire »

adrian_b wrote: post #85
dwenzek wrote: post #62

I'm just a bit surprised that this post says nothing about Heaviside who rewrote Maxwell's equations in the form commonly used today. According to wikipedia [1], Heaviside significantly shaped the way Maxwell's equations are understood and applied in the decades following Maxwell's death . [1] https://en.wikipedia.org/wiki/Oliver_Heaviside

Which was not very useful. The integral equations of Maxwell, which few know today, are much more generally applicable and actually easier to understand. The differential equations of Heaviside are valid only when certain restrictions about continuity are true. Moreover, the meanings of curl and divergence are hard to understand otherwise than by deriving them from the integrals over curves and surfaces used in the o…

IEEE floating point is far more practical for computation than axiomatic arithmetic, but the fundamental axioms are a more intuitively enlightening description of what arithmetic is. Same with Maxwell and Heaviside. Understanding how it all fits together in fewer words makes the rules make sense. Heaviside gives meaning to Maxwells equations.

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

#145

Post by nyssos »

guardingit wrote: post #123
jiggawatts wrote: post #48

Earlier quoted context omitted.

Vector fields are just as incorrect, and ought to be relegated to history books as a mere stepping stone on the way to understanding. Geometric algebra can reduce Maxwell’s equations into a single, hilariously terse equation: ∇F = J Ref: https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...

Thanks for the pointer on Geometric Algebra. This looks to be a promising path to understanding relativity/QM/EM, and goes some way to explaining my unease with cross products and imaginary numbers. Disclaimer: maths degree, so my unease was not a plain lack of understanding.

> maths degree

Then you want differential forms for EM, differential geometry more broadly for GR, and a bit of functional analysis for QM.

The hype around geometric algebras (Clifford algebras over R) just comes from the fact that it's not the plug'n'chug explicit numbers and coordinates approach, which is all most people ever see. They do not do a good job of tracking the physical structure of electromagnetism, and in fact end up baking in a lot of assumptions about the setting that fail to generalize.

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

#146

Post by mjburgess »

lupire wrote: post #140
mjburgess wrote: post #84

Earlier quoted context omitted.

minimum lengths arent relevant to whether things are continuous or not. these arent related.

That's literally the definition of continuity. You have an object at position p, and the behaviors of the system are discretely different between P and P + h, without an intermediary at P+h/2.

And that's not what a "minimum length" in this case means. We're not talking about space having a minimum unit. We're talking, at best, about (presumably massive) objects having a minimum extension in space .

Even with a "minimum length" (in this specific sense), you have an object at position p, and can (move/observe) it at any p+dx continuously.

importantly, the question is whether the best theories of physics in a world with a minimum extension-in-space require continuous mathematics, and there's nothing about this plank length to suggest they wouldnt

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

#147

Post by lupire »

lchengify wrote: post #34

Only tangentially related, but I highly recommend watching Veritasium's YouTube video on electricity if you're curious as to how Maxwell's fields create the current / amp abstractions in EE [1]. It's a common misconception that electrons or current transfer energy. In reality it's the electric field that exists between the wires that is doing the heavy lifting, the electrons in the wires are just controlling the fiel…

It's a practical simplification, not a misconception.

It's the same argument as "Einstein corrected the misconception that Newtonian mechanics is how bodies interact, and it's irritating how rote mechanical engineering of a car is."

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

#148

Post by PaulHoule »

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…

In quantum electrodynamics there is this problem: if you imagine an electron is a little sphere like the ball on a Van de Graff generator, there is a certain amount of energy in the electric field around it. As the radius gets tiny the field in the space immediately around it gets stronger so if you integrate it the energy of the EM field becomes infinite as the radius goes to zero…. We’ve got no evidence that the electron is more than a point, however.

We use a trick called renormalization which, in this case, is recognizing that the mass of the electron has a term from the EM field. We’d assume that the EM theory is not completely true but that below some distance the theory breaks down. Working in momentum space there is a certain momentum that corresponds to the cutoff distance so we just don’t integrate beyond that. You can vary the cutoff and also vary the other parameters of the theory (such as the bare mass of the electron) so the theory gives the same answers at macroscopic distances so it doesn’t matter where you put the cutoff.

Thus it does not matter much what the “true” theory is whether space is discrete or the electron really is a little ball or the EM field merges with the other forces at high energy to make some different force that (slowly) eats protons or quantum gravity or whatever.

Discretization is problematic in a relativistic world because it breaks Lorenz invariance. That is, if I am moving quickly I would see the gap between the “pixels” get smaller. Now maybe the pixels can be non-Lorenz invariant but can “fake it” at low energies and large sizes but when the energy gets large you’d expect to see some evidence of the grain. Even if the gap was the Planck length you’d probably see things get weird at much lower energies, such as those of the highest energy cosmic rays. There has been a lot of research on that and there is no clear evidence of relativity being broken but it is still highly mysterious

https://en.wikipedia.org/wiki/Greisen%E2%80%93Zatsepin%E2%80...

for instance Lorenz violation might allow particles to bypass that GZK limit.

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

#149

Post by lupire »

simonbarker87 wrote: post #75
lchengify wrote: post #34

Only tangentially related, but I highly recommend watching Veritasium's YouTube video on electricity if you're curious as to how Maxwell's fields create the current / amp abstractions in EE [1]. It's a common misconception that electrons or current transfer energy. In reality it's the electric field that exists between the wires that is doing the heavy lifting, the electrons in the wires are just controlling the fiel…

I always explain it to people like waves in the sea. The bobbing up and down isn’t the water molecules travelling they are simply going up and down as the wave moves through the water. People seem to accept this analogy as, even if the water thing is new information to them, it’s easier to visualise.

That's not what PP and Derek Veritasium Muller are harping on.

It's not about the misconception about "AC is vibrating so how can electrons be delivering their energy from the power plant to the light bulb far away?"

They are talking about how the electric field is outside the wires almost entirely.

Their argument is that in the water wave analogy, the wave wouldn't be in the water at all, because it's "actually" transmitted via an invisible field in the space above the water, which pushes back on the water farther away.

Most respected electricity/physics YouTubers disagree with Veritasium's emphasis on this perspective, by the way. The think he conflated the first misconception I mentioned with the second idea, which is about how you model electric circuits.

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

#150

Post by jshaqaw »

wholinator2 wrote: post #121
mr_mitm wrote: post #112

Earlier quoted context omitted.

> 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?

As someone who was once a, "I've watched a lot of YouTube videos about physics", person, i think it's very interesting how confident people are in their understanding of what is essentially the edge of physics, something only seen in a masters or doctorate degree. Specifically the whole spacetime and qm thing. There's so many videos on it that you begin to feel like you really understand it after half a dozen or so r…

My favorite phenomenon of which you speak are the guys who haven’t thought about math since 11th grade or physics ever aside from seeing some Joe Rogan clips with Eric Weinstein but will pound their keyboards with fury that “STRING THEORY IS A BIG LIE!!!”
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