[HN Gopher] Why is Maxwell's theory so hard to understand? (2007...
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Why is Maxwell's theory so hard to understand? (2007) [pdf]
Author : badprobe
Score : 230 points
Date : 2024-01-29 05:00 UTC (18 hours ago)
(HTM) web link (www.damtp.cam.ac.uk)
(TXT) w3m dump (www.damtp.cam.ac.uk)
| bsder wrote:
| Maxwell's theory is not hard to understand--once you have the
| proper tools.
|
| The problem is that because of trying to cram a degree into 4
| years, you wind up having a class on electromagnetics without any
| understanding of _vector fields_.
|
| Electrical engineering is _particularly_ bad about this. You
| never get exposed to the Hamiltonian formulations of classical
| mechanics, and you never get exposed to vector analysis.
| Consequently, you are stuck with the Heaviside-Hertz pedagogy
| with silly things like "displacement current" and stupid, weird-
| ass integration contours (which don't work in motors, LOL)--and
| the attendant difficulty in understanding Maxwell's theory.
|
| However, if you have vector analysis and fields, then you can
| understand formulations like Carver Mead's "Collective
| Electrodynamics": https://www.amazon.com/Collective-
| Electrodynamics-Quantum-Fo...
|
| Suddenly, emag is a _whole_ lot more straightforward to
| understand. It 's not _EASY_ as it 's very math heavy, but it has
| a lot fewer weird things that are just "we say it works."
| MichaelZuo wrote:
| I've always thought the Heaviside notation is a bit bizarre...
| is there any advantage to them at all?
| teleforce wrote:
| Heaviside and his proponents avoided quaternion like a
| plague, and like a classic false messiah he somehow convinced
| people not to use it. If we want to easily and completely
| model the EM waves its entirety including polarization we
| need to embrace quaternion, there is no two ways about it.
| The intuitive understanding of EM can be only developed by
| using quaternion and me personally waiting for someone to
| write a Pozar's book version in quaternion approach.
| mitthrowaway2 wrote:
| I think the essay is not about Maxwell's theory being hard for
| college students to understand, but rather, for other 19th
| century physicists to understand. And that's mainly because
| Maxwell himself didn't do a very good job of communicating his
| theory at the time, so it took other talented physicists to
| rework, explain, and popularize his ideas.
| nimish wrote:
| Admittedly the tools to make the mathematics more compact and
| clearer weren't invented yet (except as quaternions).
|
| We did get stuck with Gibbs' vector calculus formulation as
| the canonical view unfortunately.
| bsder wrote:
| People forget that Maxwell's Theory contradicts the
| prevailing belief of the time that waves _ALWAYS_ require a
| medium for propagation--the so called "aether".
|
| That's a _really_ large conceptual jump and physicists did
| _not_ make that jump lightly or easily.
|
| Maxwell's paper was 1865. The Michelson-Morley experiment was
| 1887. Michelson _himself_ couldn 't make the shift. Quoting
| Wikipedia (https://en.wikipedia.org/wiki/Michelson%E2%80%93Mo
| rley_exper...): "The negative result led Michelson to the
| conclusion that there is no measurable aether drift.[1]
| However, he never accepted this on a personal level, and the
| negative result haunted him for the rest of his life (Source;
| The Mechanical Universe, episode 41[8])."
|
| In an attempt to preserve "aether", Lorentz contraction then
| enters the picture as an ad hoc explanation for the
| Michelson-Morley result. It turns out Lorentz contraction is
| correct, but not because of the existence of "aether" but
| because of the constancy of the speed of light--c (Einstein
| Special Relativity--1905).
|
| Once you finally give up on "aether" after _40 years_ of
| trying otherwise, you can finally just roll with the
| mathematical implications of Maxwell 's equations.
| mitthrowaway2 wrote:
| Maxwell saying that his theory "attributes electric action
| to tensions and pressures in an all-pervading medium... the
| medium being identical with that in which light is supposed
| to be propagated" suggests that Maxwell himself did not
| view his theory to be contradicting the existence of an
| aether (or was being coy about it). Which is especially
| interesting because Maxwell based his theory on Faraday's,
| and Faraday didn't believe in the aether.[1]
|
| [1] Michael Faraday's _Thoughts on Ray Vibrations_ , 1846,
| cited by Maxwell in his paper. Faraday says: "The view
| which I am so bold to put forth considers, therefore,
| radiation as a kind of species of vibration in the lines of
| force which are known to connect particles and also masses
| of matter together. It endeavors to dismiss the aether, but
| not the vibration."
| (https://pwg.gsfc.nasa.gov/Education/wfarad1846.html)
| andyferris wrote:
| I mean, language is hard, and it evolves. What they
| called mediums, we now call (quantized) fields. QED would
| say the photon field _exists_ at all points in space and
| that it is kinda fair to say that it is a "medium" for
| electromagnetic forces and waves to propagate (with the
| additional point that special relativity is required to
| understand how to correctly transform it into a different
| reference frame, rather than imagining it as a strictly
| Newtonian medium).
| zinclozenge wrote:
| Which is interesting considering Freeman Dyson was the guy
| that made the connection between Schwinger's and Feynman's
| QED.
| LeapingLennie wrote:
| Are there any textbooks you would recommend for learning vector
| analysis / vector fields before studying EM?
| bsder wrote:
| I do not. None of the books I know of are very good because
| they are mostly targeted at Mathematics majors rather than
| physicists or engineers.
|
| Gerard 't Hooft used to have a _humongous_ list of textbooks
| for aspiring theoretical physicists. I 'd sure look there for
| starters.
| sidlls wrote:
| EM's vector fields formulation is fairly straightforward:
| it's all curls and divergence. Any 3rd (-ish) semester
| undergrad multivariate calculus course is likely to cover it
| in sufficient depth. "Mathematical Methods for Physicists"
| covers it in sufficient depth, for example, provided you
| already have a thorough understanding of the prerequisite
| material. Most undergrad physics degree curriculums should
| have E&M courses whose texts (e.g. Introduction to
| Electricity and Magnetism, Griffiths) cover enough of the
| details. If you want to pursue it further than an
| undergraduate level study, you'll also want a good text on
| differential equations that has or is supplemented by
| material covering, e.g., spherical harmonics and Bessel
| functions (among other things). I wish I could remember what
| I used, but it was...more years ago than I care to say when I
| was an grad student.
| rramadass wrote:
| I was recommended Nathan Ida's _Engineering Electromagnetics_
| as being comprehensive in that all the necessary Mathematics
| is introduced in place as needed. Lookup the reviews for this
| book on the web.
|
| Perhaps somebody who has read this book can comment in more
| detail.
| jiggawatts wrote:
| 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...
| guardingit wrote:
| 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.
| nyssos wrote:
| > 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.
| guardingit wrote:
| Thanks for your comment.
|
| > They do not do a good job of tracking the physical
| structure of electromagnetism
|
| What do you mean by tracking the physical structure?
|
| By this, do you mean the typical EM formulation of
| Maxwell's laws produces Gauss's law and Faraday's law
| (which are instructive) where as the Geometric Algebra
| formula produces [?]F = J (less instructive?).
|
| > and in fact end up baking in a lot of assumptions about
| the setting that fail to generalize.
|
| Can you explain a bit more what you mean here please?
| nyssos wrote:
| > By this, do you mean the typical EM formulation of
| Maxwell's laws produces Gauss's law and Faraday's law
| (which are instructive) where as the Geometric Algebra
| formula produces [?]F = J (less instructive?).
|
| > Can you explain a bit more what you mean here please?
|
| It's all downstream of geometric algebra leaving the
| metric implicit in its operations, basically
|
| - The metric is an extremely important physical quantity:
| it doesn't necessarily look that way when everything is
| classical and flat, but you have to start caring about it
| in curved spacetimes.
|
| - Even when the metric can be safely neglected, doing so
| makes it very easy to confuse degree (n-k) elements of
| your vector space V with degree k elements of its dual V
| _. V and V_ are isomorphic but not canonically isomorphic
| - you have to choose a basis. And keeping track of where
| you introduce a choice of basis matters, because all
| physical quantities are basis-independent. Nature has no
| preferred coordinate system.
|
| - Geometric algebra doesn't make sense on a general
| manifold: you have to embed it in a sufficiently large
| geometric algebra and inherit structure from the
| embedding. This is better than working in a particular
| coordinate chart but worse than doing differential
| geometry in a purely geometric way.
|
| - Geometric algebra is not invariant under
| diffeomorphism, which means GR is dead in the water. You
| can build mostly equivalent theories by enforcing the
| equivalence principle in your dynamics instead, but it's
| more complicated and ironically far less geometric.
| jiggawatts wrote:
| > baking in a lot of assumptions about the setting that
| fail to generalize
|
| That's the point! That's the _entire point!_
|
| Mathematicians want the most general, most abstract
| approach. They want to generalise to a wide range of
| problems and not be painted into any one specific
| example.
|
| Physics theories have an opposite goal to this: the ideal
| theory ought to take no parameters, and produce "reality"
| as the one and only possible outcome. The ideal theory
| ought not generalise to un-physical models.
|
| For example, the mathematics of general relativity have
| excess degrees of freedom that must be constrained
| through additional restrictions. Similar issues turn up
| almost anywhere matrices are used: they have too many
| degrees of freedom.
|
| Geometric Algebra is typically a better fit for what
| actually goes on in physics.
|
| For example, rotation matrices have precision issues,
| gimbal lock, and can't be robustly interpolated.
| Rotations implemented using GA have none of these issues.
| kragen wrote:
| i wonder if further progress in physics will require abandoning
| the maxwellian paradigm in the same way that maxwell had to
| abandon the newtonian paradigm? presumably for something even
| further removed from everyday experience. dyson must have thought
| about that possibility, but evidently at least at the time of
| this paper he rejected it. i'd like to read his reasoning
|
| unrelatedly, i feel like i kind of understand maxwell's theory in
| terms of vector analysis, but the clifford algebra formulation is
| still beyond me. it sure looks a lot simpler
| eru wrote:
| Keep in mind that we for all the talk of 'abandoning' the
| Newtonion paradigm, it's predictions are still valid for a
| large space of conditions. Eg NASA uses Newtonian mechanics to
| fly all their spacecraft.
|
| Special relativity includes classical mechanics when speeds are
| low. In the same way, any replacement for Maxwell's paradigm
| must reproduce the predictions of Maxwell's equations under the
| large swathe of conditions where they agree with reality.
| kragen wrote:
| yes, i thought that was too obvious to be worth saying, but
| i'm glad you've said it so the knuckle-dragging contingent
| don't think i'm endorsing their untutored 'theories'
|
| for that matter i use the aristotelian paradigm of physics
| when i expect my bed to stop moving when i stop pushing it
| across the floor; i don't bother with calculating the
| deceleration due to the friction coefficient with the floor
| eru wrote:
| Sorry, I think the first part of your original comment was
| a bit confusing. Upon rereading: the second part already
| made it clear that this is what you meant.
| kragen wrote:
| i don't think there's a non-confusing way to discuss
| questions like this, so plausibly this is not the right
| forum for it
| DiogenesKynikos wrote:
| Maxwell's Equations have already been superseded by Quantum
| Electrodynamics, in the same way that Newtonian gravity has
| been superseded by General Relativity.
|
| Both Newtonian gravity and Maxwell's Equations are still very
| good approximations in their regimes of validity.
| kragen wrote:
| we are commenting on an article which describes in depth how
| quantum mechanics, including qed, falls squarely within the
| paradigm maxwell pioneered. that's why my comment
| specifically talks about the 'maxwellian paradigm' and not
| 'maxwell's equations', which is, by the way, not a brand name
|
| i thought it was too obvious to be worth saying that
| classical physics is still an excellent approximation to
| reality, but hopefully you've enlightened someone reading
| this thread
| denton-scratch wrote:
| > 'maxwell's equations', which is, by the way, not a brand
| name
|
| Are you commenting on the capitalization of "Maxwell"? It
| is a proper name, and it should be capitalized.
| kragen wrote:
| admittedly so, but no, on the capitalization of
| 'equations' (and 'quantum' and 'electrodynamics'), which
| are not
| DiogenesKynikos wrote:
| 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.
| nyssos wrote:
| > quantum mechanics, including qed, falls squarely within
| the paradigm maxwell pioneered
|
| Does it, though? Sure, QED is a field theory, but it's
| perturbative where classical EM is exact, its fields are
| operator-valued distributions where classical fields are
| number-valued functions, its interactions are transition
| probabilities rather than forces - it's not clear to me
| that these are smaller jumps than introducing fields in the
| first place.
| kragen wrote:
| many people have argued that qm is a larger jump than
| maxwell's approach, and i agree that that's a very
| reasonable position, which is why i found it interesting
| to read dyson arguing the opposite
| Onavo wrote:
| > _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, he
| would have made sure that his work was not ignored. Maxwell was
| as great a scientist as Newton and a far more agreeable
| character. But it was unfortunate that he did not begin the
| presidential address in Liverpool with words like those that
| Newton used to introduce the third volume of his Principia
| Mathematica, "It remains that, from the same principles, I now
| demonstrate the frame of the system of the world". Newton did not
| refer to his law of universal gravitation as "another theory of
| gravitation which I prefer"_
|
| It's a good thing Maxwell is not alive and that he did not follow
| Dyson's advice, lest Hacker News accuses him of attempting to
| abuse the attention economy and promote his research like a
| salesman.
|
| https://news.ycombinator.com/item?id=33043945
| https://news.ycombinator.com/item?id=22297855
| https://news.ycombinator.com/item?id=39144845
| staunton wrote:
| 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.
| kragen wrote:
| 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
| dukeofdoom wrote:
| One thing I would like to understand is why in our universe we
| can have two things that combine to nothing. But we can't have 3
| things that combine to nothing. Can someone smarter give an
| explanation.
| DrDeWitt wrote:
| Which are the two things that combine to nothing?
| GolDDranks wrote:
| 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?
| DrDeWitt wrote:
| 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.
| andyferris wrote:
| One line of thought here is that maybe there do exist
| larger, more complex symmetry groups that can broken to
| create more complex "charge" particles (e.g. electric
| charge < color < something else < ...), but that the
| masses of particles involved are so large, their
| lifetimes so short, etc, such that we'd never actually
| observe them (or at least, we can't yet observe them) and
| that the more complex the group, the less relevant the
| physics actually becomes.
|
| (Obviously I can't say whether that is true or not, but
| it might be a possible explanation of current
| observations).
| truckerbill wrote:
| Occam's razor is a heuristic not a law so this isn't
| really a satisfactory explanation
| im3w1l wrote:
| Maybe sound is an illustrative example. The air pressure is
| a positive scalar quantity. We are interested in the
| deviations from the mean. Deviations can be both positive
| and negative. A positive and a negative deviation cancel
| (in absolute terms: a larger than normal pressure and
| lesser than normal pressure will average to something more
| typical)
| andyferris wrote:
| This is a good explanation for e.g. electric charge
| (which is scalar, just like presure), but color in QCD
| really is a bit more multidimensional in behavior where
| you can get things like red + green = -blue and so-on.
| sprash wrote:
| In Quantum Chromodynamics we have 3 things that combine to
| nothng though.
| _Microft wrote:
| What are you thinking about here? If it is "matter/antimatter"
| "particles/antiparticles" then this is not true. There are
| still conserved quantities in annihilation of particles and
| antiparticles which makes other particles and/or energy come
| out of these annihilations.
| DrDeWitt wrote:
| Excellent essay. The thing I do not agree with is how they
| mention that one has to abandon natural language to understand
| quantum mechanics. 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
| eru wrote:
| > 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 spread amongst more people. (Eg Lorentz' work was on the way
| to formulating special relativity.)
|
| So it's hard to say that there was any single 'only reason' for
| any discovery here.
| bigboy12 wrote:
| Everyone seems to think they would have figured out
| Einstein's theory but they didn't until Einstein did.
| eru wrote:
| Huh? Where did I say I would have figured it out?
|
| Albert Einstein was smart, and he definitely was smarter
| than me. But his achievements weren't beyond all the other
| smart people. Especially individually and with more time.
|
| See also how Newton and Leibniz came up with calculus at
| the same time. Or how public key cryptography was invented
| independently multiple times.
| arcanemachiner wrote:
| I don't think you're the "they" here. I think "they"
| means "one of Einstein's contemporaries".
| dang wrote:
| " _Don 't be snarky._"
|
| " _Please respond to the strongest plausible interpretation
| of what someone says, not a weaker one that 's easier to
| criticize. Assume good faith._"
|
| https://news.ycombinator.com/newsguidelines.html
| DrDeWitt wrote:
| 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.
| eru wrote:
| I agree that what Einstein did was beyond most people. But
| not beyond all of the great physicists of the time,
| especially if given more time.
| adrian_b wrote:
| Unlike the special relativity, which was just a new
| interpretation for facts and formulae already established
| by others (like Lorentz and Poincare) in electrodynamics,
| so it can hardly be called as "big leaps", and which was
| fully developed also only by others, like Minkowsky, the
| general relativity, which was developed more than a decade
| later, was the most original work of Einstein, being a
| completely new mathematical model of gravity and inertia.
| Nevertheless, if Einstein had not existed, it is likely
| that Hilbert would have become the author of the general
| relativity.
|
| 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.
| deepnet wrote:
| Natural language can conjure the visual imagination of
| intuition, but it is less specific than images themselves.
|
| Einstein reportedly ran thought experiments in his imagination.
|
| His intuition was visual and dynamic.
|
| That is images that change - animations.
|
| A natural language description of relativity is one step
| removed from the moving images of Einstein's intuition.
|
| Fields are not grounded in everyday experience but most modern
| movie-goers are happy with rapid and grand changes of scale and
| thanks to wizard and superhero movies there exists a
| sophisticated visual grammar of rapidly propagating fields,
| strange paradoxes and simultanous weakly interacting realities.
|
| Many of the concepts of quantum field theory can be grasped by
| a wide audience when presented as animations.
| DrDeWitt wrote:
| You are probably right that natural language is not
| synonymous to a physical intuition. However, I believe that
| the way quantum theories are taught or even understood are in
| a much less intuitive manner than GR. Something that is
| evident by the number of different interpretations of quantum
| mechanics
| kordlessagain wrote:
| I'm still waiting for Weber to win out.
| amerine wrote:
| I think you mean 10^-8 Wb ;-)
| notresidenter wrote:
| The two-layer separation of our world between imperceivable
| objects that define perceivable objects seems quite similar to
| how philosophy is essentially split between metaphysics and
| philosophy that takes things for granted from that "first layer".
|
| > The reason for these arguments is that the various interpreters
| are trying to describe the quantum world in the words of everyday
| language, and the language is inappropriate for the purpose.
| Everyday language describes the world as human beings encounter
| it. Our experience of the world is entirely concerned with
| macroscopic objects which behave according to the rules of
| classical physics. All the concepts that appear in our language
| are classical. [...] The battles between the rival
| interpretations [of quantum dynamics] continue unabated and no
| end is in sight.
|
| Replace 'quantum dynamics' with metaphysics (or post-Kantian
| metaphysics) and the statement seems true as well.
| lchengify wrote:
| 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 field.
|
| This has always confused me and I was very irritated when I first
| learned electromagnetics about how rote all the initial learnings
| are. I wish more work was put earlier into making everything
| relate back to Maxwell's equations to make it make sense.
|
| [1] https://www.youtube.com/watch?v=bHIhgxav9LY
| simonbarker87 wrote:
| 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.
| lupire wrote:
| 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.
| lupire wrote:
| 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."
| hiAndrewQuinn wrote:
| 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 think that's the _modern_ fundamental difficulty in Maxwell 's
| reworked equations - the 4 we all know and love, not the 20 or so
| he originally published. To even begin to get a true intuition
| for them, you have to get really really good at visualizing
| idealized objects with flows running through surfaces, and (if
| you're lucky) symmetries that cancel each other out. You can't be
| afraid of imagining the infinitely small and the continuous to
| really get the most out of it, even if you "know" on some deeper
| level that the continuity of spacetime is a convenient
| approximation.
|
| 14 years later I am still grappling with the beauty of saying
| "yeah yeah, this area of interest is _technically_ discrete, but
| let 's pretend it's continuous and see what kinds of stuff falls
| out." If you have examples of things like this in other areas
| like mathematical finance, I'd love to hear about them.
| 3abiton wrote:
| Can't agree more, mental visualization is such an asset for any
| understanding as it relies on compressing information and
| forces oneself to digest the material.
| mjburgess wrote:
| > 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.
| yard2010 wrote:
| 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? :)
| omnicognate wrote:
| No, position and time are typically continuous variables in
| quantum mechanics. You can have formulations in which they
| are discrete but they are not required and are relatively
| exotic. QM certainly doesn't say they must be discrete.
| hnfong wrote:
| I feel that a lot of people here are confusing the math
| and "reality".
|
| You're definitely correct about the math, i.e. the
| systems that we humans have invented to model reality.
| But I guess most of us don't _really_ care about what
| mathematical model scientists like to use (especially not
| whether they 're "exotic" or not), but rather what
| reality could be like.
|
| And the quantum properties of QM do seem to suggest that
| there's some sort of fundamental discreteness in reality.
| And it seems to run contrary to the resolute claims that
| reality _must_ be continuous as if it were a proven fact.
| What I understand is that the math most commonly used by
| scientists is definitely continuous, but whatever we can
| measure seems to have some kind of planck limitation.
|
| So are we talking about empirical science or science-
| flavored theology here? Have we actually found empirical
| evidence or proven the continuousness of space/time?
| omnicognate wrote:
| I responded to a couple of people who claimed with great
| certainty that QM meant spacetime had to be discrete,
| when it says nothing of the sort. I haven't claimed we
| have proof that it is continuous and I doubt we ever will
| as that seems akin to proving a negative existential.
|
| Your penultimate paragraph suggests some confusion about
| ideas like Planck scale and quantisation.
|
| Firstly, there is nothing special about the Planck length
| itself. It's just a unit of length. Around that sort of
| scale, though, our current theories of physics happen to
| break down because both quantum and gravitational effects
| become significant. That doesn't imply spacetime is
| discrete (or preclude it being discrete) at that scale.
| It's just a realm that our current theories don't work
| in.
|
| Secondly, while describing aspects of nature that are
| quantised was a large part of why quantum mechanics was
| developed (and the source of its name), it in no sense
| says anything like "there's some sort of fundamental
| discreteness in reality". Quantum mechanics deals with
| both discrete and continuous observables in a single
| framework: functional analysis, essentially. The set of
| possible values for an observable is modelled as the
| spectrum of an operator, which can be either continuous
| or discrete. Which sort of observable is appropriate for
| a given physical theory is a choice made in constructing
| that theory. For things like charge and spin we use
| discrete (quantised) values because we have evidence that
| those things are quantised. For things like position we
| use continuous values and have no evidence that using
| discrete observables would better match nature.
|
| Space could in reality be either discrete or continuous,
| or not even exist in any form we'd recognise as "space"
| on those scales. Quantum mechanics doesn't give us any
| hints one way or another.
| diffeomorphism wrote:
| 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.
| eigenket wrote:
| 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.
| saalweachter wrote:
| Wasn't it more the observation the theory was designed to
| explain than the first prediction?
| eigenket wrote:
| I think its a linguistic difference only. At least where
| I studied it was quite normal to call phenomena you can
| derive from a physical theory "predictions" even if they
| have been observed before. I agree the photoelectric
| effect strongly suggested some quantization before
| quantum mechanics was formalised.
| lll-o-lll wrote:
| > 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?
| greysphere wrote:
| 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 of preferred alignment, but as far as we
| can tell there no such preference. Physics in free space is
| rotationally invariant and thus not on a grid thus
| continuous.
| chr1 wrote:
| 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.
| mjburgess wrote:
| The "underlying issue" often at stake in the debate is
| whether reality is a computer, since it would need to be
| discrete if so, and often whether a computer can be made
| to simulate it.
|
| However, what's missed here is that discrete is a
| necessary but not sufficient condition.
|
| Once you give any sort of plausible account of _how_
| reality could be discrete, as you 've done here, you end
| up with non-computable aspects (eg., typically
| randomness). So the metagame is lost regardless: reality
| isnt a computer (/ no complete physical theories of
| reality are computable).
|
| Though the meta-meta-game around "simulation" is probably
| internally incoherent in itself -- whether reality is a
| computer or not would really have nothing to do with
| whether any properties had by it (eg., mass) are
| simulated.
|
| Since either you take reality to have this property and
| hence "simulation" doesn't make sense, or you take it to
| be faked. If it's faked, being computable or not is
| irrelevant. There's an infinite number of conceivable
| ways that, globally, all properties could be faked (eg.,
| by a demon that is dreaming).
| calf wrote:
| 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 proof sketches in
| these classes.
| greysphere wrote:
| 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.
| chr1 wrote:
| In physics you never have measurements differentiating
| between distance 2 and say 2+10^-20, and that gives
| enough space to hide any 'bell curve' you want.
| tzs wrote:
| 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.
| psychoslave wrote:
| 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 continuous or discrete
| manner for logistical reasons, then it this scope it
| matters. But whatever you settle on, human brain is thus
| built that it can always assume that the perfectly fitting
| model is only valid in its scope which is built on top of
| an other more subtle level of reality which is on it's part
| better modelized with an antithetic approach.
|
| Now, on a very personal out of blue opinion, I fail to see
| how any causal series might happen without an underlying
| continuous flow of event. I mean, supposing causal
| discontinuity is to my mind as relevant as supposing that
| universe as it is right now, actually just appeared,
| without anything we can think about it being relevant, and
| in the next instant could be completely different or
| nonexistent since universe is not bound in any remote way
| to what we might expect on our delusional just created
| sense of causality.
| lupire wrote:
| 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 could be parsed by a regex ;-)
| Heck, imagine an integer on the number line that can go
| up or down.
|
| Worlfram has written a ton about this. Despite all his
| issues, his math is solid. (Which is not to say his
| physics is true.)
| mrguyorama wrote:
| 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.
| Tainnor wrote:
| Not nonsense. The argument goes that if time and space
| are both discrete, then to move from A to B in finite
| time means that you have to perform infinitely many
| actions in finite time.
|
| Zeno didn't believe that the latter was possible. But he
| wasn't stupid, he obviously knew that motion was
| happening all the time in real life. His paradox really
| only makes sense in the context of Eleatic philosophy
| which assumes that reality is an illusion because change
| is fundamentally impossible (how can something come from
| nothing?).
|
| If you want to reframe it in more modern terms, Zeno's
| paradox shows a contradiction in axioms. If you want to
| get rid of the contradiction, you have to change some of
| the axioms.
|
| In real analysis, loosely speaking, we remove the axiom
| that an infinite process cannot result in a finite
| outcome - this way we are allowed to sum (some) infinite
| series, for example. But we don't "know" if reality
| behaves that way.
|
| The atomists found a different solution: they argued that
| reality was fundamentally discrete. This way, Zeno's
| paradox also doesn't arise.
| jerf wrote:
| There are many times in physics where people have thought
| they've had to choose between either one thing or its
| opposite, where both choices had clear deficiencies. The
| ultimate solution ended up being a new hybrid that nobody
| thought of for a long time.
|
| I kind of suspect "is the universe continuous versus
| discrete" will come down to that. I don't know what a
| hybrid of such things looks like. With our current
| conceptions it seems impossible. But it always does, before
| the breakthrough comes and then in hindsight all the people
| of the future will get to look back at us going "How could
| they not see this obvious thing?", to which my only defense
| is that you, dear future reader, only think it's obvious
| because it was handed to you on a silver platter and you'd
| be as confused as we are if you were back here with us.
| programjames wrote:
| My guess is every physical value can be written as a sum
| of rational sines (i.e. sin(tau * a/b)).
| rsecora wrote:
| Its 99 years since Einstein published the paper on the
| photoelectric effect whith had far-reaching consequences. [1]
|
| And 93 years since the first Solvay Conference. [2]
|
| [1]
| https://en.wikipedia.org/wiki/History_of_quantum_mechanics
| [2] https://en.wikipedia.org/wiki/Solvay_Conference
| omnicognate wrote:
| What's your point? Everything they did, including
| Einstein's (and everyone else's at the time) quantum
| mechanics work, was based on continuous space and time
| variables.
| eigenket wrote:
| 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.
| rini17 wrote:
| 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
| mjburgess wrote:
| 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 record an infinite amount of information. No one
| claims this, and the formulation of physics (entirely on
| real numbers) does not require it.
|
| Rather to say, eg., space is continuous, is to say its
| unbroken. There is no physical quantity which is becoming
| infinite.
| mikk14 wrote:
| 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.
| mjburgess wrote:
| It's not entirely clear "Energy" is a physical property
| either. By physical I mean a causal property of a system
| which is a basic constituent of reality.
|
| For example in E = 1/2mv^2, a particle has kinetic energy
| in virtue of being matter in motion -- it is motion and
| matter which are basic. Energy is just a system of
| accounting which tracks motion in the aggregate over time
| (with kinetic/potential just being the future/past in the
| accounting) hence why energy conservation is just a
| temporal invarience.
|
| When making arguments about the physical properties
| reality has (eg., whether aspects are continuous) you
| need to be exceptionally clear what your terms mean, and
| terminology in physics isnt designed for this.
|
| There are no "information saturated volumes", this is a
| series of abstractions piled on top of each other.
|
| All the words in this area have quite complex formal
| definitions that are have quite difficult to unpack
| semantics, you cannot just go around saying "saturated
| volumes" -- it is this sort of language which breeds
| cranks, and pop sci does it with abandon.
|
| This entire discussion is a matter of several PhDs, and
| to be done only well by people with PhDs in the matter
| (philosophy of physics), or equivalent research. It's not
| possible to scrap fragaments of what compusci bloggers
| say and derive much that's likely to be actually correct.
| pishpash wrote:
| Nothing in physics is more basic than something else, as
| there are equivalent formulations in other quantities.
| Energy and momentum are as real as matter and motion.
| Matter is bundles of energy exhibiting inertia and motion
| is just some transformational relationship between
| phenomena in different areas of space-time. There is
| nothing "real" about any of this, only what animals like
| humans have evolved to model directly in their brains.
| calf wrote:
| Aaronson is one of the world's top quantum computing
| scientists, he's a professor at I believe UT Austin.
|
| He's also written papers that are basically philosophy of
| physics. It would be interesting to go over what he has
| actually said on this topic.
| mikhailfranco wrote:
| Yes, the thermodynamic properties of information are well
| established.
|
| Various Hawking-Bekenstein results about black holes
| relate to information density, especially, _shockingly,_
| that information is proportional to surface area, not
| volume. This makes perfect sense because a black hole has
| all its incoming matter and energy sprawled, flattened
| and red-shifted on its horizon (to a distant observer).
| It can never export its internal state to the outside
| world, so you might never expect a volume 's-worth of
| states to be exposed.
|
| The idea was generalized by 't Hooft to the Holographic
| Principle, for 2D screens encoding the state of 3D
| volumes on the other side.
|
| However, the full AdS/CFT Correspondence only applies to
| a certain type of AdS space, not our actual dS space. At
| the moment, it seems half of theoretical physics doctoral
| students are trying to extend AdS/CFT to dS space
| (obviously - _strings_ :) and half of observational
| astrophysics doctoral students are desperately hoping to
| show we live in AdS space - LOL
| n4r9 wrote:
| 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.
| kergonath wrote:
| 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.
| mr_mitm wrote:
| All observable quantities are eigenvalues of some operator,
| which are real numbers but discrete. How can they contain
| infinite amount of information?
| nyssos wrote:
| 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 <x|, for instance, is just the map
| `psi -> delta_x(psi)`.
| eigenket wrote:
| 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 < phi | P phi> 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
| nyssos wrote:
| Sure, I should have been clearer: a rigged Hilbert space
| is the right setting for bras and kets. You can also get
| rid of them entirely. In my experience QM classes
| unfortunately tend to split the difference by slinging
| around suggestive nonsense like \int_{x} |x><x|.
| eigenket wrote:
| Consider the Hilbert L^2([0,1]) associated with a
| physical particle that has position somewhere between 0
| and 1, the corresponding multiplication operator X which
| takes a wavefunction f and maps it to Xf where (Xf)(x) =
| x f(x). Then X is a bounded self-adjoint operator. It
| doesn't have any eigenvalues or eigenvectors but it's
| spectrum is exactly the set of numbers [0,1] as you'd
| expect (prefect measurements of position return real
| numbers in [0,1]).
|
| The spectral theorem, rather than decomposing X in terms
| of a sum of eigenvectors & eigenvalues instead decomposes
| it as an integral over the spectrum with respect to the
| (spectral) projection-valued measure.
|
| Now it is fair to question whether this "observable" is
| really observable, but it certainly works out
| mathematically consistently in the normal way we do
| things in quantum mechanics.
| mikhailfranco wrote:
| I am very sympathetic to Gisin and his cause, but he does
| not propose any sensible resolution. By the way, not a
| fault, and no blame for him. Pointing out logical
| deficiencies always comes before a satisfying solution, and
| he is to be praised for his insight.
|
| There are many interesting ways to probe this problem....
| here's one:
|
| Say I tell you to imagine a circle, an ideal Platonic
| circle in a Cartesian coordinate system (real coordinates,
| first uneasiness). Let's ignore translation, so it is
| centered at (0,0). I tell you the radius. Can you imagine
| the circle with Plato? Model the circle? Reproduce the
| circle? Do you need pi? Does the circle include or encode
| pi? But pi is has infinite information.
|
| Perhaps all you need is the square root function? But
| that's also an infinite Taylor series expansion. You can
| plot and recreate the circle to any precision if you have a
| square root function. The series will only need to run to
| the required precision. The circle will always be granular,
| depending on the number of terms you use in pi, or the
| square root function. Yeah, right, obvious, so why is that
| a problem?
|
| What if I tell you the circle is the physical manifestation
| of equipotentials of a stationary charge (say, nucleus), or
| mass (say Earth), with inverse square law - so basically a
| geometric fall-off with range determined by spatial
| (circular, spherical) considerations. What is the force at
| some distant point? Do you need pi? Do you need square root
| function? Or reciprocals? How does the other charge or mass
| feel the 56,323rd decimal place of the force due to the
| potential?
|
| Maybe it doesn't, because by the time it has felt the
| second decimal place, time has moved on, the charges/masses
| have moved on, and the nuance of what would've/should've
| been felt in a never changing universe are never
| experienced. There is a modified differential equation that
| relates various time derivatives to precision of
| experienced forces (this almost sounds like relativity :)
|
| The discrete explanation with photons goes like this: the
| force is produced by radiating photons. They automatically
| encode the geometric expansion as inverse square law,
| because of their pathways, no need for pi, or sqrt
| functions. But that is statistical, the accuracy is only as
| good as the number of photons that can arrive from the
| source. The circular/spherical nature of the force only
| emerges over time, as photons arrive and act. The accuracy
| of smooth circularity and inverse square only establishes
| itself over time...
|
| Elapsed time affects experienced precision - hmmm,
| interesting.
|
| How would you quantify such a thing, where time changes the
| precision of what you feel? Well, the other obvious example
| is the Heisenberg Uncertainty Principle. This is just a
| simple and obvious example of Fourier Analysis for any
| theory based on a linear wave equation. It almost doesn't
| need stating, and if it must have a name, it is certainly
| _Fourier,_ not _Heisenberg._ Anyway, any math
| /physics/engineering student knows Fourier to their core,
| and it gives a nice solution to the information problem:
| coordinates may be real-valued degrees of freedom, but
| there is no way to mathematically or physically resolve all
| coordinates and their derivatives to infinite precision.
| It's just not possible, even if the underlying
| equations/reality maintain the fiction of real-valuedness.
|
| Fourier combines time, waves, amplitude, velocity
| (momentum, etc.) with a specific expression for possible
| information. A picture is worth a thousand words at this
| point, just look at a wave-packet, it's obvious. Fourier is
| a masterwork, and vastly underappreciated as a fundamental
| limit on knowledge, in a real world sitting on smooth
| continuous waves.
|
| So Fourier sets limits on knowledge, even in the wavy world
| of the smooth continuum. Of course, I do not believe in the
| smooth continuum anyway, but Fourier is my wingman to fight
| the real-infinitists on their own _smooth_ turf.
| eigenket wrote:
| > But pi is has infinite information
|
| It does not, according to any sane way of defining its
| information content. For example the Kolomogrov
| complexity of pi is clearly finite - I can write down a
| program for a Turing machine which will run (forever) and
| keep writing down digits of pi as it does so.
| pishpash wrote:
| Not true either. The original Kolmogorov complexity is
| for finite strings. Plus, the program would need to store
| that for whichever special strings you choose you will
| use this function but for other ones not. That will be a
| giant table that is part of your program.
|
| That's also a different point than the parent's. Seems
| they're saying if you were to specify pi as the limit of
| some expansion that describes the physical process of
| photons arriving in some area, then that specification's
| information increases with more terms added. Pi, being
| almost random by every statistical measure, has as much
| information as a random string, in fact, in any normal
| conception of information. You cannot wave that away by
| machine manipulation tricks or by defining a new
| constant, and this is borne out also by the parent's
| physical argument that in reality there are no low-
| complexity universal constants, but that there may be
| limits to information density (in space and time).
|
| Continuous physics can be a manipulation of limiting
| quantities without being literal.
| eigenket wrote:
| > The original Kolmogorov complexity is for finite
| strings.
|
| I wrote "Kolmogorov complexity" not "original Kolmogorov
| complexity" so this isn't particularly relevant. The
| application of the concept to the infinite string which
| represents pi is essentially trivial.
|
| > Plus, the program would need to store that for
| whichever special strings you choose you will use this
| function but for other ones not. That will be a giant
| table that is part of your program.
|
| I honestly can't parse this.
|
| > Pi, being almost random by every statistical measure,
| has as much information as a random string
|
| This is wrong. You can consider something like a simple
| communication task. Alice and Bob share a phone line and
| she is attempting to tell him a number. Every second the
| line allows her to send a bit to Bob. For a truly random
| number she has to use the line infinitely many times to
| tell him the number. To send pi she can send a finite
| number of bits which amount to a program to compute pi
| and he can do the computation on his end.
| pishpash wrote:
| I urge you to go back and look at how Kolmogorov
| complexity is defined. It includes the notion that a
| program needs to decide whether to output the string
| directly or to generate it from some program.
|
| You're assuming Alice and Bob have already pre-
| synchronized what kind of computing machine is going to
| be used, one in which pi is the output of a relatively
| short program, as opposed to another type of machine
| where some other random-looking number has that property
| (random to you, pseudo-randomly generated via some
| machinery for all you know). You are assuming many things
| away.
|
| Also it is absolutely not trivial to extend Kolmogorov
| complexity to infinite strings. There are multiple
| formulations and they are a lot more difficult than for
| finite strings. Not the computation part but the
| complexity assignment part.
| eigenket wrote:
| I agree there is a bunch of complexity in generalising
| Kolmogorov complexity to general infinite strings.
| However I'm not really trying to do that here, all I want
| is enough to back-up the statement I made before, that
| the complexity of pi is finite. Doing that is much more
| trivial than what you're talking about.
|
| Theres a bunch of fine detail in getting it down to
| defining an actual number measuring complexity which I
| don't care about at all, all I care about (in the context
| of this discussion) is that the number is finite.
| waveBidder wrote:
| But we never know any quantity to full precision, so it's
| not like we get infinite bits about any given quantity.
| kergonath wrote:
| > 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.
| codethief wrote:
| As the sibling hinted at, Gisin's statement is quite sloppy
| and - at the very least - confuses "definite"
| "information"1 (a given real number) with "uncertain"
| "information" (entropy), at least if you follow the
| definition of entropy by the book: The probability
| distribution for an observable that takes on (exactly) the
| value of a given real number with probability 1 has entropy
| 0.
|
| That being said, Gisin's approach is still interesting and
| his results can still be valid. But he _starts with the
| assumption_ that real (irrational) numbers are unphysical,
| i.e. that - in a sense - our observable from above can
| actually only take on certain (rational) values, and then
| he derives certain predictions from that.
|
| 1) Putting "information" in quotation marks here because no
| one really knows what it is.
| r0uv3n wrote:
| > physics has no formulation of spacetime in discrete terms
|
| There are some attempts of working with discrete spacetime
| (e.g. causal set theory), but yeah, all our best descriptions
| so far very much assume smooth spacetime.
| omnicognate wrote:
| > Cognition is discrete
|
| There's little evidence even of this, except in the trivial
| sense that language (minus prosody) is composed of discrete
| units.
| mjburgess wrote:
| It is discrete insofar as we're talking about sequences of
| thoughts, ie., reasoning.
|
| What offends the minds of some people is the world might
| not be like their mind at all. They want always to
| analogise everything to Reason.
|
| Everything should be countable, everything should be
| knowable, etc.
| omnicognate wrote:
| That's the "trivial sense" I'm talking about. If we
| restrict "cognition" to the stuff we know is discrete
| then trivially it's discrete. But cognition is a hell of
| a lot more than that.
| lupire wrote:
| I don't see how what we know is discrete.
|
| A word doesn't even have a discrete meaning, except
| locally in relation to other words.
|
| Saying A = B + C looks discrete, just by hiding any
| potential non-discreteness inside B and C.
| simiones wrote:
| Discrete here means "non-continuous" - i.e. there is no
| smooth transition function between
| ideas/thoughts/rationalizations.
|
| For example, a formal proof is a discrete process: it
| follows step-wise rules that you can assign natural
| numbers to (this is the first step, this is the second
| step, this is the third step). A non-discrete process, a
| continuous one, would have a smooth transition between
| these steps, which is hard to even imagine.
|
| While I am not convinced it is correct to say that "human
| reasoning is discrete", human language is definitely
| discrete. Words don't blend smoothly into each other. If
| you don't believe me, try to define a function f:[0,1] ->
| Words, such that f(0) = "red" and f(1) = "blue" and tell
| me what is f(sqrt(2)/2), or what is df/dx.
| omnicognate wrote:
| I entirely agree. I wasn't making a statement that "what
| we know is discrete". I was referring to a particular
| subset of cognition as "what [i.e. the things that] we
| know are discrete".
|
| There are aspects of cognition that are discrete: a
| language contains a finite set of phonemes and words, a
| human mind is capable of (painfully slowly) carrying out
| purely symbolic algorithms like those a computer
| performs, etc. My point was that these things are a small
| subset of cognition, and most of cognition we have no
| particular reason to think depends on discreteness, which
| I think is the same point you're making.
|
| Personally I strongly suspect that the "discrete" aspects
| of cognition are things that have evolved on top of /
| within a system that is fundamentally continuous
| (analogue) in nature.
| hnfong wrote:
| > My point was that these things are a small subset of
| cognition, and most of cognition we have no particular
| reason to think depends on discreteness, which I think is
| the same point you're making.
|
| How do you convince yourself that you have thoughts that
| cannot be accurately written down no matter how many
| words you use?
|
| > Personally I strongly suspect that the "discrete"
| aspects of cognition are things that have evolved on top
| of / within a system that is fundamentally continuous
| (analogue) in nature.
|
| How do you tell whether things are really fundamentally
| continuous, or a really high definition pixel art?
| omnicognate wrote:
| I can't. That's why said "I personally strongly suspect"
| rather than presenting my partially-informed intuitions
| as facts.
| Almondsetat wrote:
| what's a sequence of thoughts?
| Sardtok wrote:
| What are thoughts, anyway?
| lupire wrote:
| And everything should be... "quantizable"?
|
| https://www.energy.gov/science/doe-explainsquantum-
| mechanics
| hnfong wrote:
| You're technically correct.
|
| That said, our Turing Machine model of computation is
| discrete, and the Church-Turing thesis implies human
| thought is Turing Complete.
|
| It's not empirical evidence, but it's something. (I really
| doubt an empirical test is possible at all, so it seems
| philosophizing is all we have, unfortunately.) I'm not
| aware of any (communicable) model of thought that actually
| can't be reduced to the Turing model (in fact, that AFAIK
| precisely the reason he proposed the model).
|
| Analog signals can be approximated to arbitrary precision,
| so while we conventionally think of it as continuous, it
| doesn't imply our cognition really has infinite precision
| floats internally...
|
| I think it's really unfair to only focus on half of the
| picture (saying there's no evidence for "Cognition is
| discrete") where in fact we actually have no evidence at
| all whether _anything_ is fundamentally continuous or
| merely approximated as such with high precision.
|
| Traditionally the math in physics is continuous, and the
| math in computing is mostly discrete. If people point to
| Hilbert space as some kind of justification for believing
| physics is continuous, then it seems equally valid (or
| invalid) to use the Turing model as justification to
| believe cognition is discrete. I think both approaches are
| misguided, but as I said, it's really unfair to point out
| only the convenient half of these invalid arguments.
| discoinverno wrote:
| 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.
|
| "In general relativity, any form of energy E acts as a
| gravitational mass and distorts spacetime around itself. The
| distortion increases when energy is concentrated, to the
| point that a black hole forms when a mass M is concentrated
| in a sphere of radius R ~ GM/c^2, where G is the Newton
| constant. If we take L arbitrary small, to get a sharper
| localization, the concentrated energy will grow to the point
| where R becomes larger than L. But in this case the region of
| size L that we wanted to mark will be hidden beyond a black
| hole horizon, and we lose localization. Therefore we can
| decrease L only up to a minimum value, which clearly is
| reached when the horizon radius reaches L, that is when R =
| L. Combining the relations above, [..] we find that it is not
| possible to localize anything with a precision better than
| the Planck length (~10^-35 m). Well above this length scale,
| we can treat spacetime as a smooth space. Below, it makes no
| sense to talk about distance. What happens at this scale is
| that the quantum fluctuations of the gravitational field,
| namely the metric, become wide, and spacetime can no longer
| be viewed as a smooth manifold: anything smaller than the
| Planck length is "hidden inside its own mini-black hole"."
|
| "The existence of a minimal length scale gives quantum
| gravity universal character, analogous to special relativity
| and quantum mechanics: Special relativity can be seen as the
| discovery of the existence of a maximal local physical
| velocity, the speed of light c. Quantum mechanics can be
| interpreted as the discovery [..] that a compact region of
| phase space contains only a finite number of distinguishable
| quantum states, and therefore there is a minimal amount of
| information in the state of a system. Quantum gravity yields
| the discovery that there is a minimal length lo at the Planck
| scale. This leads to a fundamental finiteness and
| discreteness of the world."
| mjburgess wrote:
| 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 broken
| into units of it, or that objects have to be aligned on
| grid boundaries defined by it.
|
| Whenever people try to do philosophy of physics the
| inevitable place everyone lands at is a series of false
| equivocations, often caused by the language of physics
| being ambiguous and polysemous. But "minimum length" here
| does not mean a sort of grid length.
| discoinverno wrote:
| 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 also be finite dimensional (for instance the Hilbert
| space of a spin 1/2 particle).
| mjburgess wrote:
| minimum lengths arent relevant to whether things are
| continuous or not. these arent related.
| lupire wrote:
| 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.
| mjburgess wrote:
| 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
| simiones wrote:
| Discreteness would mean that there exists some base
| distance p such that the distance between any two objects
| is Np, with N being a natural number (and any surface is
| some Mp^2 and any volume is Qp^3 and so on). Continuity
| is simply the opposite of that. It could be that objects
| can be at arbitrary real-valued distance d from each
| other, but that d > p is a precondition for any other law
| of physics.
|
| By contrast, discretness has various unintuitive
| mathematical properties that mean it's not easy to fit
| into some other theories (particularly those relying on
| differential equations).
| hnfong wrote:
| I agree with all your claims here in the literal sense,
| but suppose there's a minimum length, then it would seem
| to be at least theoretically possible to use discrete
| mathematics to formulate an approximation to the "real
| number formulation of physics"?
|
| The fact that we don't have already a full system using
| discrete maths doesn't mean it is impossible, because our
| current system is based on a long tradition of belief in
| real numbers, and assuming physical space is continuous.
|
| I'd argue (admittedly unhelpfully) that unless we have
| actually tried to formulate physics using discrete
| mathematics and found a barrier that we prove
| unequivocally that it is impossible to overcome, we can't
| claim that physics must be formulated using real
| numbers/continuous math. There's a difference between "we
| don't know how to do this" vs "we know we can't do this".
| Keyframe wrote:
| maybe we should take another look at analogue computing
| metricspaces wrote:
| > 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 constants come from and what does it all mean?
|
| It seems we bring the 'world' into being by _partitioning_.
| See Genesis 1 for details.
|
| > the world is continuous
|
| _Reality_ is actually a unified undivided unity _without
| form_ and _timeless & eternal_ - that is all we can say with
| certainty. The "world" is our perception of this reality. Our
| cognitive machinary is discreet, and a mapping of this
| reality into metric & temporal spaces of the mind.
| mr_mitm wrote:
| > 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?
| metricspaces wrote:
| A discreet unit of measure exists in dynamics & its
| relationship to position space is informed by
| Heisenberg's theorem.
|
| (My actual point is that _Reality_ is neither continuous
| nor discreet - it is an infinitesimal point and it is our
| mind -- that likes to name and number things and relies
| on duality to make 'distinctions' -- that creates the
| universe, the _subjective reality_ that we perceive as
| inhabiting.)
| lupire wrote:
| What is an infinitesimal point?
| metricspaces wrote:
| _The tao that can be told is not the eternal Tao The name
| that can be named is not the eternal Name.
|
| The unnamable is the eternally real. Naming is the origin
| of all particular things.Free from desire, you realize
| the mystery. Caught in desire, you see only the
| manifestations.
|
| Yet mystery and manifestations arise from the same
| source. This source is called darkness.
|
| Darkness within darkness. The gateway to all
| understanding._
| wholinator2 wrote:
| 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 repeat
| the same words at you. But the thing you don't realize is
| that those are literally the only words they could
| possibly communicate, and that's an infinitesimal
| fraction of the nuance of the real thing. Plus, there's a
| selection bias in that the videos that make you feel good
| get more views, so you're more likely to stumble upon
| videos that make you _think_ you understand it that
| videos that actually do, partly because the only videos
| that could make you understand are a graduate degrees
| worth of 80-100 hour lecture courses that you're gonna
| have to take notes on.
|
| It makes me wonder if this is true for literally every
| subject. I fancy myself are least politically versed in
| modern events but is my entire understanding based on an
| entertainment-first version of an actual education? How
| much do i walk around using jargon that i don't really
| understand to make points whose true depth I'm completely
| unaware of?
| jshaqaw wrote:
| 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!!!"
| jasonwatkinspdx wrote:
| Well, unfortunately some otherwise great physics
| educators intentionally stoke that fire, portraying the
| current particle physics agenda as if its some conspiracy
| to waste funding rather than the consensus of thousands
| of the best minds in physics. Selling people a
| superficial sense of contrarian insight ends up being a
| very successful marketing tactic.
| mrguyorama wrote:
| Most nerds who "understand quantum mechanics"
| misunderstand (usually do to incorrect explanations) that
| the Planck units are somehow the fundamental units of
| reality
| lupire wrote:
| Is wave function collapse continuous? Is photon absorption
| and emission continuous?
|
| No one knows what the universe of truly made of.
|
| Reality is measured up to certain error tolerances.
|
| Don't confuse the map (math and physics) with the territory
| (reality).
|
| Also, Stephen Wolfram would like a word with you.
| mikhailfranco wrote:
| Collapse is not continuous, by definition. Copenhagen is
| discontinuous.
|
| If you want continuity, then shun collapse, and believe in
| Many Worlds. You know you should.
|
| Wolfram is another conversation, and one that does not fit
| in this margin.
| TheOtherHobbes wrote:
| This seems very unlikely.
|
| If you're from Copenhagen every measurement is a lossy
| discontinuity that resets the wavefunction.
|
| This is not an abstraction, it's directly observable.
|
| As for discrete formulations:
|
| https://en.wikipedia.org/wiki/Causal_dynamical_triangulation
| simiones wrote:
| Interpretations of QM and measurement have very little to
| do with whether space-time is discrete or continuous. The
| simple fact is that no common QM formulation uses discrete
| mathematics for space-time, and it's unclear if any that
| does would even work.
|
| Also, your link is not a formulation of QM, it is a
| different theory which makes different predictions (it is a
| quantum gravity theory). And, per the sounds of the
| Wikipedia article at least, it is not actually proven
| equivalent to QM in the regimes where it needs to be
| ("There is evidence [1] that, at large scales, CDT
| approximates the familiar 4-dimensional spacetime", or in
| other words, it is not fully worked out if this is the
| case).
| adrian_b wrote:
| 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.
| toth wrote:
| > 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 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.
|
| > 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.
|
| Here I have to strongly disagree. The version of Maxwell's
| equations that is fundamental and exactly correct [1] is the
| vacuum version. The ones with magnetization and displacement
| vectors are only approximations where you assume continuous
| materials that respond to fields in simple way. In truth,
| materials are made of atoms and are mostly vacuum: there is
| no actual displacement vector if you look close enough.
|
| Also the vacuum Maxwell equations are useful in many
| scenarios. For instance, that's how you compute the energy
| levels of Hydrogen atom or how you derive QED. Also, you have
| to start from them to derive the macroscopic versions with
| magnetization and displacement that you seem to like.
|
| [1] Well, up to non-linear quantum mechanic effects.
| PaulHoule wrote:
| 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.
| scotty79 wrote:
| 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.
| toth wrote:
| 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.
| adrian_b wrote:
| Even the vacuum version is incomplete without adding an
| equation for force or energy, because no meaning can be
| assigned to the electromagnetic field or potential
| otherwise than by its relationship with the force or
| energy.
|
| Even today, there exists no consensus about which is the
| correct expression for the electromagnetic force. Most
| people are happy to use approximate expressions that are
| known to be valid only in restricted circumstances (like
| when the forces are caused by interactions with closed
| currents, or the forces are between stationary charges).
|
| Moreover, when the vacuum equations are written in the
| simplified form present in most manuals, it is impossible
| to deduce how they should be applied to systems in motion,
| without adding extra assumptions, which usually are not
| listed together with the simple form of the equations (e.g.
| the curl and the divergence are written as depending on a
| system of coordinates, so it is not obvious how these
| coordinates can be defined, i.e. to which bodies they are
| attached).
|
| While the vacuum equations are fundamental, they may be
| used as such only in few applications like quantum
| mechanics, where much more is needed beyond them.
|
| In all practical applications of the Maxwell equations you
| must use the approximation of continuous media that can be
| characterized by averaged physical quantities that describe
| the free and bound carriers of electric charge. The useful
| form of the Maxwell equations is that complete with
| electric polarization, magnetization, electric current of
| the free carriers and electric charge of the free carriers.
| It is trivial to set all those quantities to zero, to
| retrieve the vacuum form of the equations.
| toth wrote:
| I agree that to fully specify electromagnetism you also
| need to include how the fields affect charged matter. So
| EM = Maxwell's equations + Lorentz force equation (not
| sure why you say there is no consensus about what this
| is, that is new to me).
|
| This is just a matter of taste, but OTOH I would not
| include descriptions of how some materials respond to the
| fields in the continuous limit as part of a definition of
| EM.
|
| It is true that for most terrestrial applications you do
| need those to do anything useful with EM. But if you want
| to study plasmas you need to add Navier-Stokes to EM,
| doesn't mean hydrodynamics is part of EM. To study
| charged black holes you need EM + GR, but it still makes
| sense to treat them as mostly separate theories.
| neutronicus wrote:
| You also need to include how charged matter affects the
| forcing fields in Maxwell's equations (i.e. moving
| charges depositing a current field).
|
| I actually basically agree with your viewpoint, I studied
| Plasma Physics in graduate school in a regime where we
| did _not_ use Navier-Stokes or constitutive relations and
| everything was in fact just little smeared-out packets of
| charge moving according to the Lorentz Force Law and
| radiating.
| farseer wrote:
| I agree however as a commentator pointed out in a similar
| thread a few years ago, mental visualization only works for
| relatively small problems with limited variables. Most
| reasoning after that is done via equations when the problem
| complexity surpasses the n'th dimension (where n is maybe 3 or
| 4).
| tim333 wrote:
| Mathematical finance is basically all people saying look I know
| in the real world markets are driven by things like idiots
| buying Dogecoin because number go up but let's assume it's made
| of well informed participants who price everything correctly.
| Assuming this we can show...
| lupire wrote:
| 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.
| PaulHoule wrote:
| 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.
| 1980phipsi wrote:
| 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).
| seanhunter wrote:
| > If you have examples of things like this in other areas like
| mathematical finance, I'd love to hear about them
|
| There are tons of examples in mathematical finance but the
| obvious one is the Black/Scholes [1] paper where one of the key
| assumptions they make (which they know to be untrue but
| helpful) is that you can replicate a portfolio in continuous
| time. This allows them to use a constructed portfolio of a
| risk-free intrest-bearing instrument and the underlying to
| replicate the price of an option, and the process is a Brownian
| motion. Everyone knows that actual trading (and thus price
| processes) are discrete in real markets, but continuous time is
| much easier to model. Much later on people like Heston and
| Matytsyn(?sp) came up with stochastic vol models with jumps to
| replicate discrete price discontinuities, but they're a lot
| harder to work with in many ways.
|
| [1]
| https://www.cs.princeton.edu/courses/archive/fall09/cos323/p...
| xhstephen wrote:
| i have tried to transform the pdf into a presentation with AI,
| may help you read faster
| namaria wrote:
| I don't know what's more concerning, the fact that you find a
| six page paper written by one of the greatest communicators in
| science hard to digest or that you think an automatically
| chopped up version with colorful shapes is equivalent to the
| original.
| lupire wrote:
| Reformatting makes a document more digestible, especially
| when not reading on a _printed sheet of paper_ , which the OP
| is exclusively designed for.
|
| What's concerning (aside from your callous disregard for
| people who have small screens) is that the PP created a new
| document, and didn't show it, suggested that we might find it
| more digestible.
| namaria wrote:
| I see hundreds of students consuming PDF files on
| smartphones everyday for their classes. I read this paper
| on my phone just now. I am callously disregarding people
| who cannot bring themselves to read six pages and have to
| make it small and cute first.
| niemandhier wrote:
| William Burk published a little book on understanding
| differential geometry on an intuitive level using visualisations.
|
| One chapters deals with Maxwell. After this it was easy to
| understand
|
| https://www.cambridge.org/core/books/applied-differential-ge...
| wly_cdgr wrote:
| The moral of this story absolutely is not that "modesty is not
| always a virtue", lol
| dwenzek wrote:
| 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
| adrian_b wrote:
| 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 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.
| lupire wrote:
| 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.
| Helmut10001 wrote:
| > It is better for the progress of science if people who make
| great discoveries are not too modest to blow their own trumpets.
|
| This is a nice remark, but very difficult to implement in
| practice. In reality, many non-modest people will overrate their
| contributions, while the few modest people will have a hard time
| to act non-modest in certain situations. We are in a world where
| modesty is even rarer than 100 years ago. I am sure many
| important discoveries are hidden in the myriad peer reviewed
| publications published just for quantitative reasons.
| frozenport wrote:
| I taught this subject at the graduate level.
|
| With an emphasis on basis functions and computing we could do it
| in a single semester.
|
| The more interesting thing was that most students had practically
| no useful previous knowledge despite formally having some years
| of eduction on the subject
|
| My insight was the biggest issue was the eduction system failing
| the students and not anything specific to Maxwell's equations.
| laiudm wrote:
| 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...
| rramadass wrote:
| Related: _Maxwell on the Electromagnetic Field: A Guided Study_
| by Thomas Simpson.
| computerfriend wrote:
| If you do want to learn the geometric formulation, part 1 of
| _Gauge Fields, Knots and Gravity_ is a good resource (and has
| exercises!).
| Mikhail_K wrote:
| 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
| the electric field.
| ThePhysicist wrote:
| 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.
| toth wrote:
| >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).
|
| What? No you can't. The fields are invariant under gauge
| transformations.
| ThePhysicist wrote:
| You're right, sorry I was thinking of a Lorentz
| transformation that would make either the magnetic or
| electric field disappear under certain conditions.
| wch4999 wrote:
| "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.
| toth wrote:
| 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.
| wch4999 wrote:
| 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
| semantics really.
| ash wrote:
| In the article Dyson retells the story from Pupin's
| autobiography. This 1923 Pulitzer-winning book is now out of
| copyright and freely available:
| https://www.gutenberg.org/ebooks/66886
| ThePhysicist wrote:
| Maxwell 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.
| kurthr wrote:
| True, but he did use quaternions (by 1873), which allow the
| field properties to be written as a single equation. It's kind
| of sad that more physicists don't use or teach quaternions,
| while math and CS have fully adopted them.
|
| I really liked Kathy Joseph's historical reviews of vector
| physics and the people who developed it, which explain some of
| the reason's for how it's taught. Most texts don't even develop
| electrodynamics from relativistic electrostatics as a
| demonstration.
|
| https://youtu.be/CdwxpSInhvU
|
| I think I fell down the rabbit hole from Freya Holmer's "why
| you can't multiply vectors".
|
| https://youtu.be/htYh-Tq7ZBI
|
| The key being that all of the Hamiltonian fields can be found
| in a single quaternion equation, which is just what happens
| when you start multiplying vectors together.
| aap_ wrote:
| Unfortunately he didn't use quaternions in his initial
| formulation (that was all split into xyz coordinates) and in
| his later revision he took apart the quaternions into scalar
| and vectors parts. It could have been so much prettier....but
| luckily we have geometric algebra for that today.
|
| On the other hand he did derive the electric and magnetic
| fields from a scalar and potential field. In that sense
| Heaviside made a step backwards.
| duped wrote:
| Faraday didn't even know trigonometry, allegedly (he never
| studied mathematics). It's interesting that his student
| (Maxwell) who _did_ have the mathematical background would
| extend his theories and figure out the math to explain it all
| vmilner wrote:
| Really 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...
| elashri wrote:
| I enjoyed reading that. Maxwell's equations in differential form
| is the most elegant equations that I saw in my life. I remember
| in my freshman year I had 8 questions in my EM final and the last
| question was name four equations that you can use to solve all
| the other 7 questions. It was a straight question for free grade
| obviously. But I was puzzled that I could not actually derive all
| of what I used. I went on and submitted the exam and returned to
| my seat. I went on with deriving all of them and spent a couple
| of hours. I left the room being a physicist from that moment
| until now.
| sylware wrote:
| All that was a long time ago. The Maxwell magnetic component is a
| result of special relativity, and I am wondering what would
| result from using general relativity instead of special
| relativity to get approximation equations from QED at the same
| scale than those very Maxwell equations.
| lupire wrote:
| > Instead of thinking of mechanical objects as primary and
| electromagnetic stresses as secondary consequences, you must
| think of the electromagnetic field as primary and mechanical
| forces as secondary.
|
| Feynman explained this nicely. He said essentially, you ask me to
| explain what is electmagnetism. Is it like two hands pushing on
| other? Well, if it is, then what is "pushing"? Pushing is just
| the result of electomagnetism in your hands! _It is impossible
| explain electromagnetism to you in terms of anything simpler that
| you already understand._ It is a fundamental force.
| steamer25 wrote:
| 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.youtube.com/watch?v=9Tm2c6NJH4Y
| devmor wrote:
| Wow this video is actually great at imparting the concepts with
| animation.
|
| It's a little distracting how it looks like an ad for an adult
| themed video game, but it's very well thought out.
| Terr_ wrote:
| > It's a little distracting how it looks like an ad for an
| adult themed video game
|
| Well-put: Does that demon-squid in the intro look like a
| Chihuly glass installation to anybody else?
|
| Also I swear I've heard that song long ago on OCRemix.
| s1artibartfast wrote:
| 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.
| timeagain wrote:
| It's like saying that rain falls where there are blue regions
| on the weather map :)
| rolph wrote:
| we can fix that.
|
| the number of electric field lines, depends on the strength
| of an electric field. ,
| psuedobrain wrote:
| I think the question of whether field lines are real is more
| of a philosophical (of physics) question so it usually falls
| outside the scope of introductory material on E&M. However,
| some texts like Purcell and Morin do kinda take a stance on
| whether fields are real: "since it works, it doesn't make any
| difference."
| s1artibartfast wrote:
| it works for a physics test, but is also part of the
| problem, as it misleads and prevents conceptualization,
| even for simple problems.
| azalemeth wrote:
| Very much this. The (standard model's) "answer" is that the
| four vector potential probably is the "most real" and we're
| all just excitons along for the ride.
|
| At some point the definitions become almost circular and
| opinions about what it fundamental have shifted a bit over
| the centuries. The cgs system of units -- which differs
| profoundly from SI in the treatment of electromagnetism --
| was associated with those who viewed D and H rather than E
| and B the most fundamental. I'm quite happy with the level
| of theory used being appropriate to solve the problem at
| hand. There's always a bit of wiggle room around exactly
| what that problem is, however ;-)
| JohnFen wrote:
| So, a bit like how the conventional depiction of electric
| flow is in the opposite direction of the actual electron
| travel?
|
| It doesn't matter in terms of the math (in the vast
| majority of situations), so while the conventional idea of
| electric flow is incorrect, we keep it anyway.
| s1artibartfast wrote:
| I think it is closer to the conventional view of current
| as the travel of electrons down a wire.
|
| Current moves far faster than electrons. it is more
| similar to a wave in the ocean with the electrons being
| the water molecule.
|
| As a result, and counterintuitively for most, the speed
| of electrons will give you a completely wrong answer for
| when a light will turn on after you flip a switch.
| jereees wrote:
| This video is worth watching for the soundtrack alone. I came
| across Eugene's channel before but somehow I missed this gem!
| Ringz wrote:
| > ... for the soundtrack alone
|
| Did I sense mild sarcasm here?
| seanhunter wrote:
| Just today I was watching a cool video by Angela Collier[1] about
| how Faraday's experimental work really laid the groundwork for
| Maxwell by proving that light polarity could be affected by an
| electromagnetic field.
|
| [1] https://www.youtube.com/watch?v=Fbi-_8zOuR8
| Scubabear68 wrote:
| As a lay person, this was beautifully written, and I feel like I
| understand the issues to a degree better than I have before from
| casual reading.
|
| Very important to people like me, because I really struggle with
| advanced math. I dropped an EE degree because while I could do
| the math, it was incredibly hard and in no way intuitive to me.
| lr1970 wrote:
| The Maxwell equations are conventionally being taught and written
| as 4 equations for two 3 dimensional vectors instead of a single
| equation for a single anti-symmetric 4-dimensional tensor. Also,
| the tensor exaction is explicitly relativistic covariant while in
| the vector equations formulation this fact is well hidden and
| requires quite a long proof to see it.
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