81–90 of 250 posts
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...
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It is true that there is no experimental evidence, but I think there are some convincing arguments that something must happen at the Planck scale (for very short distances) in a full quantum-gravity theory. Here are some quotes from "Covariant Loop Quantum Gravity", Rovelli and Vidotto (slightly redacted). I suggest the whole chapter 1, in particular 1.2 to get an idea of why fundamentally spacetime may be discrete.…
You're talking about minimum lengths, not discrete spacetime. It may be the case that there's a minimum length beyond which "no meaningful laws of physics apply", but it really says nothing about whether real numbers are indispensable in the formulation of physics, or about whether spacetime is continuous. There being a minimum length doesnt mean that everything is a discrete multiple of this length, or that space is…
Also, for what is worth, in QM the space of wavefunctions can also be finite dimensional (for instance the Hilbert space of a spin 1/2 particle).
Years 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...
Earlier quoted context omitted.
You're talking about minimum lengths, not discrete spacetime. It may be the case that there's a minimum length beyond which "no meaningful laws of physics apply", but it really says nothing about whether real numbers are indispensable in the formulation of physics, or about whether spacetime is continuous. There being a minimum length doesnt mean that everything is a discrete multiple of this length, or that space is…
I don't understand your point, I never said that "everything is a discrete multiple of this length, or that space is broken into units of it, or that objects have to be aligned on grid boundaries defined by it", I just wanted to mention that "continuity of spacetime is a convenient approximation" may be a correct sentence in the context of quantum gravity. Also, for what is worth, in QM the space of wavefunctions can…
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
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 original equations of Maxwell, which are also necessary to determine how to handle discontinuities.
The differential form of the equations looks prettier on paper due to a simpler notation, but it is less helpful for understanding and for solving practical problems than the integral form.
In my opinion, it is a serious mistake that almost all manuals show the equations of Maxwell in the Heaviside form, instead of showing them in their original form. This is one of the main reasons why they are hard to understand for many.
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I'm not going to bother attempting a wall of text which ends with an obviously wrong conclusion.
"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 the electric field.
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Particles and antiparticles. But I think he's trying to make a slightly more general point: why are "parities" (2-fold symmetries) so common in nature and mathematics? Why not more 3-fold symmetries?
As the other reply said, in QCD you have "3 things that combine to nothing." If I had to guess why they are not so common I would say that the more variables you add in a theory the more complicated you make it. So by Occam's razor we try to go for the simpler models/structures.
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> The only reason why we made the jump from Newton to Einstein through Maxwell's theory is Einstein's intuition of the physics behind electrodynamics I'm not sure that's true. What kind of model of causality are you using here? Other people were also on the cusp of doing (most of) what Einstein did. Ultimately, without Einstein most likely progress would have been delayed by a few years, and the laurels would be spre…
I would argue that Einstein made big leaps in knowledge to formulate SR and even more to formulate GR. He was able to imagine the motion of a body through a curved space-time as the source of gravity imo something that most people would not be able to do.
However the general relativity has no relationship with the equations of Maxwell discussed here.
While general relativity is the most original work of Einstein and the one best known, the second most original work of Einstein is the one that had the greatest impact on practical technology: the discovery that for computing the properties of electromagnetic radiation one must take into account also the stimulated emission (like in lasers), not only the spontaneous emission and the absorption.
For me, Einstein's paper on stimulated emission is the most important of his work. Even if more than a century has passed, it is not yet clear if Einstein's mathematical model is the best for gravity and inertia or if there exists another model that would be more comprehensive and which could relegate Einstein's model to be an approximation that would be no longer useful.