Brad DeLong, in the course of something completely different, suggests that the theory of relativity really isn’t all that hard. At least, if your standard of comparison is quantum mechanics.
He’s completely right, of course. While relativity has a reputation for being intimidatingly difficult, it’s a peculiar kind of difficulty. Coming at the subject without any preparation, you hear all kinds of crazy things about time dilating and space stretching, and it seems all very recondite and baffling. But anyone who studies the subject appreciates that it’s a series of epiphanies: once you get it, you can’t help but wonder what was supposed to be so all-fired difficult about this stuff. Applications can still be very complicated, of course (just as they are in classical mechanics or electrodynamics or whatever), but the basic pillars of the theory are models of clarity.
Quantum mechanics is not like that. The most on-point Feynman quote is this one, from The Character of Physical Law:
There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time. There might have been a time when only one man did, because he was the only guy who caught on, before he wrote his paper. But after people read the paper a lot of people understood the theory of relativity in some way or other, certainly more than twelve. On the other hand, I think I can safely say that nobody understands quantum mechanics.
“Hardness” is not a property that inheres in a theory itself; it’s a statement about the relationship between the theory and the human beings trying to understand it. Quantum mechanics and relativity both seem hard because they feature phenomena that are outside the everyday understanding we grow up with. But for relativity, it’s really just a matter of re-arranging the concepts we already have. Space and time merge into spacetime; clocks behave a bit differently; a rigid background becomes able to move and breathe. Deep, certainly; inscrutable, no.
In the case of quantum mechanics, the sticky step is the measurement process. Unlike in other theories, in quantum mechanics “what we measure” is not the same as “what exists.” This is the source of all the problems (not that recognizing this makes them go away). Our brains have a very tough time separating what we see from what is real; so we keep on talking about the position of the electron, even though quantum mechanics keeps trying to tell us that there’s no such thing.
@Zwiestein: Schro’s interpretation of the wave function is a lovely interpretation for electrons, but for any multiple particle states you need to live in a Hilbert space equipped with tensor products, and cannot put the distribution over configuration space.
It seems like relics of Schro’s style have survived in the popular press too. An image of the electron as smeared *in space* is better than thinking of it as a misbehaving point particle, but it’s not quite enough.
General Relativity is hard for the same reason that it took Enstein so long to develop it. People try and understand it without knowing the mathematics. I remember my undergraduate quantum class, I had no problem with it. I could do all the calculations, no sweat. But I understood nothing. I look back on it, and part of the problem was the insistence on thinking the wave function “collapsed”. This was encouraged by the text book and the professor, and as you can tell, I remain quite bitter about the experience!
There is an electron, it is not a point particle, it has a position, but when you measure it you bump it out of position. The problem is that once you eggheads learn the current jargon and the math you figure you understand it, stop thinking about what is really going on, and forget what Feynman said (above) and (elsewhere): “No one has found any machinery. . . . We have no ideas about a more basic mechanism. . . .”
When I was a physics undergrad at the University of Massachusetts, I was interviewed by the school newspaper. They asked me what my hardest final was going to be that semester and I said “Quantum Mechanics.” The interviewer asked me why, and my reply was “Because it’s quantum mechanics”. It was published (my first publication?), and the result was my professor (William Gerace, retired now) put a question on the final, “Why does Chris think quantum mechanics is hard?” I think my solution to the problem was similarly circular.
Thanks for that. Now I’m comfortable knowing what I don’t (can’t?) know … {grin}
It’s certainly true that the nature of measurement in quantum mechanics is hard to understand, perhaps so much so that no one really *does* understand it. But I find there are other things that people *think* of as quantum weirdness that really aren’t much different than what one sees in classical wave mechanics. Superposition, for instance. Even the uncertainty principle has an analog in classical wave packets, where one must superpose waves of increasingly many wavelengths to reduce the width of the packet in space (as one can see from the Fourier transform). Of course the connection between wavelength and momentum is from QM.
AJKamper @25 – I’m in a similar situation to you, probably even worse with respect to math. But much as I enjoy reading popularized accounts of relativity and QM, I constantly remind myself that it’s ridiculous to say I _understand_ them in any significant way. Being able to work the appropriate mathematics may not be sufficient to understanding physics, but I would bet a large sum that it’s necessary.
all i know is that:
In Hilbert Space no one can hear you scream.
If anyone reading this is interested in learning GR by self-study, I highly recommend this course: http://www.physics.mcgill.ca/~maloney/514/ There are video lectures and problems, and the professor is *excellent*.
In general, I’m finding GR to be challenging but much more satisfying than QM. GR was created in Einstein’s head without experimental evidence, so it is by definition more comprehensible to human minds than QM. No one would have imagined something as bizarre as QM unless they were forced to by strange experimental results. QM is something from an alien Lovecraftian universe, whereas GR are laws of nature for a universe that at least *might* have been created by a rational God!
From Bee >> which is all completely irrelevant decoration.
I could not agree more!! But I also feel that the flip side is true – Slightly more advanced textbooks are to “maths-first” with physics added right at the end (I have an ex-student who did GR in maths and didn’t understand a single bit of physics at the end of it).
So – I made a decision to use Hartle’s book for my GR course (but mix it up a bit with most of the maths concepts) with a physics-first (not decoration first though) approach. I *think* students finish up with a workable knowledge of GR.
@rob: That’s because an unbounded scream with definite momentum is not normalizable. Try a rigged Hilbert space instead.
While I’m sure knowing all about the Poincaré group is truly necessary to grasp the significance of Lorenz invariance and the beauty of Very Important Things like Noether’s Theorem, can we glean much of any of that from good pictorial descriptions of translations in Minkowski Space? I’m afraid if I have to learn group theory I may be permanently out of luck. The Road to Reality (the book) has taught me that life’s too short, I’m afraid.
Some lectures by Leonard Susskind are up on youtube. They cover various subjects and don’t skimp on the math, but present the minimum necessary to see what’s going on. They can provide a nice jumping off point from pop. science coverage for folks with some maths background.
http://www.youtube.com/results?search_query=susskind+modern+physics&aq=f
The other thing of course is that the mathematics of special relativity is far simpler than that of quantum mechanics or general relativity.
>> The other thing of course is that the mathematics of special relativity is far simpler than that of quantum mechanics or general relativity.
This is a curious statement – I think part of the problem is that SE is taught with one sort of maths (a little algebra and calculus) where as GR is another (4-vectors and tensors etc). If we taught SR properly (with 4-vectors and tensors), deal with SR in non-Minkowski coordinates (still flat space-time) so things like Christoffel symbols are non-zero and we start using covariant derivatives early, then the transition to GR is easier.
As Bee said – a lot of SR teaching is window dressing.
@Chris J. post #7.
It’s disappointing to hear that you gave up on learning QM, especially as you’ve shown remarkable tenacity in pursuing GR.
A glance at your reading material might suggest why you found QM difficult to self-learn. Griffiths has a truly excellent E & M book. Truly magnificent, a gem of pedagogy. But I don’t think his QM book is anywhere near that.
As an introductory text, I’d strongly recommend you pick up ‘A Modern Approach to Quantum Mechanics’ by John S. Townsend. It begins quite simply with Stern-Gerlach two-state systems, illustrating all (most) of the nuances of the new mechanics. I can’t recommend it enough. You’ll see at once that the mathematics and the physics illuminate each other.
Unfortunately, I don’t have a particular graduate-level QM text to recommend for self-learning. Le Bellac’s book ‘Quantum Physics’ comes to mind, as well as Gottfried and Yan’s ‘Quantum Mechanics: Fundamentals’; I recommend these because I find books like Shankar’s are somewhat outdated. There have been many recent developments in quantum mechanics, and I find Le Bellac’s and Yan’s books capture these developments much better.
The recent paper ‘Graduate Quantum Mechanics Reform’ , Carr L.D. , McKagan, S.B., American Journal of Physics, v. 77, p. 308 (2009) does a good job of chronicling the various ‘epochs’ of quantum mechanics and recommends what a graduate syllabus should cover. You’ll find that books like Shankar’s are a bit light on the modern stuff. Just saying.
@Cusp #40
There is an interesting book out by Joel Franklin: “Advanced Mechanics and General Relativity”; his approach is interesting, but he uses variational principles from the jump. His viewpoint is pretty much what you describe, starting with the Minkowski metric.
He then moves on to discuss scalar and tensor fields. Some may find it dry, but I think it’s elegant.
Special relativity is all about moving objects and running clocks. So where, in any textbook, do you actually see moving objects and running clocks? Most students can go through an entire course and never see them. It’s like studying music theory and reading scores but never listening to any music.
(I also agree that SR teaching is too full of personalization and frills, not to speak of the meter stick lattices and clocks crashing through each other! )
Some of this can be overcome by making videos and animations using Java and other tools, but these are usually stand alone items not tightly integrated with the rest of the material.
To seque slightly to a different topic, technical communication and pedagogy is today in something of a state of chaos.
I am somewhat an advocate of Mathematica because of its active and dynamic properties. I consider it to be a revolutionary new technical communication medium. It would be possible, in my opinion, to write far superior tutorials if we can:
1) Get the physical principles and mathematics correct.
2) Devise presentations that illustrate and teach the material.
3) Write clear textual explanations.
4) Supply documented tools for readers to use on their own.
I know that many of the readers will not like Mathematica because of its commercial nature and expense. Furthermore WRI has not quite turned it into a communication medium such that anyone could have a free Mathematica reader to read and operate the controls of any notebook or application (but not write or edit them). Basically the Adobe Acrobat model.
But something like this is going to come and learning how to best use such a medium is not necessarily easy.
And I do like the Bondi k factor and calculus as one tool in learning SR.
David Park
djmpark@comcast.net
http://home.comcast.net/~djmpark/
Try understanding applied vector analysis! Whihc is basically calculus, in electrical engineering.
I’m on a train right now, I can force myself to see my worldline (Poincaré-)tilted against that of the trees outside. (I usually don’t, because I have a hard enough time not bumping into stuff in the Galileian world).
GR is a bit harder because I just can’t really keep all the features and twists of more complex spacetimes in mind at once. But locally it’s just Lorentz anyway.
I also at some point started to imagine (classical particle-)QM as series of waves rippling outwards and interacting. I also tend to favour the many-worlds-interpretation (the waves just keep on rippling) over the conventional wave function collapse. The latter is a useful tool to make sense of practical (apparent?) measurements though. I think the Hilbert space is not really the place to “grasp” QM, although taking it, ignoring infty – 3 dimensions and thinking of states as vectors helps in making sense of the mathematics of QM.
I’m not sure about QFT though. Waves with different modes of vibration rippling through fields, maybe?
Of course, in practice it’s always “shut up and calculate”.
It’s been a looong time since college (BS, physics, 1987)… but to the best of my recollection, it went like this: special relativity – pretty easy. Mindblowing, but not hard. General relativity – harder, still mindblowing, but not too bad. QM – uhh… pretty hard. Finally got it. But the real killer: statistical mechanics. Holy crap. I was at best an indifferent student, and failed a few courses along the way – but my general experience in retaking them was that if I had just buckled down the first time around I would have been fine. But stat mech nearly unmade me – failed it abjectly the first time, and barely managed to pull a C the second… and that was a huge struggle. I felt like I got an acceptable grasp on the topic by the time I was done, but… yeah, hard.
I read “From Eternity to Here” last year and had to struggle with Vietnam-style flashbacks to stat mech… canonical ensembles… Bose-Einstein condensates… GAAAAHHH!
Lol
>> Of course, in practice it’s always “shut up and calculate”.
Not in my course it isn’t.
@Chris J: not sure the level you are looking for in QM but a good soph. level QM book is Morrison’s isbn 0137479085
harris – nonclassical physics. Best soph level QM book ever.