What is your favorite deep, elegant, or beautiful explanation?

Science publishing impresario John Brockman’s Edge web-site each year runs a “Question of the year” feature, with short pieces from a wide range of people providing their answer to the question. The past few years I’ve passed on their invitation to submit something, but this year the question was one that I couldn’t resist. It was “What is your favorite deep, elegant, or beautiful explanation?” and you can read people’s answers here.

There are quite a few answers from various physicists, with General Relativity, inflation and the multiverse getting a lot of attention. To me though, the most satisfying answer to the question involves the remarkable role of symmetry principles at the foundations of both our everyday laws of mechanics and our deepest ideas about quantum mechanics. Far more so than in classical mechanics, in quantum mechanics these principles are built into the fundamental structure of the theory. This makes it clear why quantum mechanics works the way it does, and indicates that the structure of quantum mechanics is likely to always be fundamental to our understanding of the physical world, not some approximation like the classical picture. In addition, it links together fundamental physical principles and a fundamental set of ideas that occur throughout modern mathematics, a veritable grand unification of the two subjects.

Here’s what I sent in:

Any first course in physics teaches students that the basic quantities one uses to describe a physical system include energy, momentum, angular momentum and charge. What isn’t explained in such a course is the deep, elegant and beautiful reason why these are important quantities to consider, and why they satisfy conservation laws. It turns out that there’s a general principle at work: for any symmetry of a physical system, you can define an associated observable quantity that comes with a conservation law:

1. The symmetry of time translation gives energy
2. The symmetries of spatial translation give momentum
3. Rotational symmetry gives angular momentum
4. Phase transformation symmetry gives charge

In classical physics, a piece of mathematics known as Noether’s theorem (named after the mathematician Emmy Noether) associates such observable quantities to symmetries. The arguments involved are non-trivial, which is why one doesn’t see them in an elementary physics course. Remarkably, in quantum mechanics the analog of Noether’s theorem follows immediately from the very definition of what a quantum theory is. This definition is subtle and requires some mathematical sophistication, but once one has it in hand, it is obvious that symmetries are behind the basic observables. Here’s an outline of how this works, (maybe best skipped if you haven’t studied linear algebra…) Quantum mechanics describes the possible states of the world by vectors, and observable quantities by operators that act on these vectors (one can explicitly write these as matrices). A transformation on the state vectors coming from a symmetry of the world has the property of “unitarity”: it preserves lengths. Simple linear algebra shows that a matrix with this length-preserving property must come from exponentiating a matrix with the special property of being “self-adjoint” (the complex conjugate of the matrix is the transposed matrix). So, to any symmetry, one gets a self-adjoint operator called the “infinitesimal generator” of the symmetry and taking its exponential gives a symmetry transformation.

One of the most mysterious basic aspects of quantum mechanics is that observable quantities correspond precisely to such self-adjoint operators, so these infinitesimal generators are observables. Energy is the operator that infinitesimally generates time translations (this is one way of stating Schrodinger’s equation), momentum operators generate spatial translations, angular momentum operators generate rotations, and the charge operator generates phase transformations on the states.

The mathematics at work here is known as “representation theory”, which is a subject that shows up as a unifying principle throughout disparate area of mathematics, from geometry to number theory. This mysterious coherence between fundamental physics and mathematics is a fascinating phenomenon of great elegance and beauty, the depth of which we still have yet to sound.

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45 Responses to What is your favorite deep, elegant, or beautiful explanation?

  1. plm says:

    In mathematics we have a classic anthology: Proofs from the Book, by Aigner and Ziegler, http://www.springer.com/mathematics/book/978-3-642-00855-9 .

    I guess we can pick many proofs from that book.

    I may select Cantor’s diagonal argument, or a similar logic or computer science reasoning, like Gödel’s liar paradox proof of incompleteness, or Turing’s proof of undecidability of Halting.

  2. Hendrik says:

    This is a very nice proposal, but as your post on Weinberg’s symmetry views shows, it is a bit idealized. The elegance is a bit less in broken symmetries, supersymmetry, and quantum local gauge symmetry is not even mathematically rigorous yet. There is also plenty of confusion in physics literature on what precisely constitutes a symmetry e.g. what is the difference between physical transformations and symmetries, must symmetries be automorphisms of the observables, or is it enough to be just derivations, and what about superderivations?

  3. Jeff M says:

    Well, as a geometric analyst, I should say the proof of the Selberg trace formula, which relates the lengths of closed geodesics, a geometric quantity, to the eigenvalues of the Laplacian, an analytic quantity. Not like there is any easy way to explain the proof though. Another would be the Hodge decomposition. But for pure elegance I’m going with plm, Turing’s proof of Goedel’s theorem is truly beautiful and shockingly clever.

  4. Peter Woit says:


    I think we’re far from completely understanding many things about representation theory, as well as far from completely understanding fundamental physics. All that’s clear is that the two subjects are closely related. I hope that in the future we’ll understand this relationship better, allowing new ideas to flow both ways between math and physics.

  5. AcademicLurker says:

    Interesting coincidence with the timing of this post. I just started a book, “Emmy Noether’s Wonderful Theorem” (Dwight Neuenschwander, 2011). It’s not a popularization. Instead, it works through the proof and then explores some of it’s consequences. It seems to be pitched at the level of an undergraduate physics major.

  6. JB says:

    Peter, can you recommend a good textbook on Representation Theory?

  7. Symmetry is of course the underlying elegant reason for pretty much anything, say the tautological truth of many world modal descriptions (which you seem to actually not like). To reduce it almost mechanistically to something that is indeed more “representation” rather than explanation, is not so good. Especially not if it involves time/energy (as that very symmetry is not given in a general relativistic context) and phase transition/charges. A fundamental explanation kind of symmetry be better fundamental, and not these apparent and emergent symmetries you are talking about. With those, a mathematical modeling/representation/mapping into another language is not an explanation.

  8. King Ray says:

    Peter, you mean U = exp(iH), not exp(H), where H is self-adjoint aka Hermitian. iH is skew-adjoint.

  9. Peter Woit says:

    King Ray,

    I was trying to be concise and avoid equations, so at the level of precision of that text, you should take h-bar=i=-1=1.

  10. Bob Levine says:

    For my money, the prize has to go to Maxwell’s incorporation in the expression for the magnetic field circulation of a term expressing the time rate of change in the electrical field flux, purely on the basis of mathematical parallelism with the relationship between the electrical field circulation and the time derivative of the magnetic field flux. There was absolutely no ‘phenomenological’ motivation for this modification in the original Ampere law, not one bit of experimental evidence for a ‘displacement current’ at the time; it was a matter of pure formal symmetry, yielding the wave equation for the propagation of orthogonal E and B fields that established the wave nature of light on the basis of the coefficient 1/c^2 in a wave equation that could *only* have been derived on the basis of that modification.

    We’re used to this kind of thinking in the era of modern physics. But what Maxwell did was, so far as I can tell, virtually unprecedented in the context of 19th century physics—-and probably the greatest breakthrough in physics since Newton’s original work. The key point is that Maxwell was guided purely on the basis of the principle that, as Dirac was famous for saying, ‘it is more important to have beauty in one’s equations that to have them fit experiment’—but followed that principle just short of a century before Dirac wrote those words. In context, his innovation ranks with the very best that Newton and Einstein achieved.

  11. Abbyyorker says:

    In the answers I liked:

    Natural selection
    Boltzmann explaining entropy
    Geometrization of the quantum

  12. Proudmemberofthecult says:

    @ Peter Woit

    I’ve heard that Seiberg refers to this as “string theory units” 🙂

    (for computational simplicity include 2=\pi=1)

  13. Frank Meulenaar says:

    Why call it “mysterious’? It might be ‘deep’ or ‘unexpected’ but I would never call it ‘mysterious’.

  14. Chris Oakley says:


    I do not agree that Maxwell’s displacement current was just plucked from thin air. It is, after all, the simplest extension of Ampere’s law that makes the equation consistent with charge/current conservation.

  15. plm says:

    Peter, I find it interesting that you chose to underline how fundamental physical theories are selected by stability/symmetry among possible theories, while you are very critical of string theory.

    It seems to me that the main argument in favor of string theory is that it adds degrees of freedom to quantum field theories (or so it seemed), which are then pruned by stability/symmetry conditions (conformal invariance) with mysterious (more or less attractive) consequences. Like for instance special relativity takes a point (or field or wave-function) in 4-space and asks for hamiltonians invariant under the transformations you mention.

    I think this is also what makes string theory a little more attractive than canonically quantizing general relativity. (This may also suggest that some form of canonical GR may be recovered by either a generalization of string theory or even some existing formulation.)

    So we have failed alot to find experimental evidence in favor of string theory, but it is mysterious and embeds this symmetry principle workhorse of theoretical physicists that you promote…
    Then how do we weigh those things, what is the right bayesian framework? How does this mix up with the social structure of scholarship -e.g. personal interests?
    I try to understand your attitude, and others’.

    And thank you for the post, it was very interesting.

  16. plm says:

    I should add that quantizing GR is also based on the symmetry workhorse insofar as quantization is. We add probabilistic fuzziness to our objects and reduce by requiring (biological) observers. This results in projections onto preferred (superselected) states, like eigenstates of conserved observables, or the next best thing, coherent states. And decoherence is a mysterious mathematical ingredient, which affects how much we need to project.

    (My perspective on the interpretation of quantum mechanics has been that experiment supports quantum physics and classical biology so I start with quantum theories for the quantum experiment and try mentally to model a simple quantum neural network coupled with the observed system, and then I ask that what happens is an observation, a neuron either firing or not, which technically can only be made projecting on some subspace of states, and this with some mathematical unraveling, explaining decoherence mostly, should be equivalent to projecting on just eigenspaces. So I usually tell myself “Why should we humans be all of physical reality and not just part of it, projections of states?”. And I think a bayesian view of physics supports this, our experiments give ever-increasing support to QM and we just continually observe consciousness which we can formalize in models of neural networks -from single neurons, to perceptrons with feedback, to realistic full brain models as begin to appear- and which boils down to Born selection on complete sets of observables.)

    With this in mind canonical quantum general relativity may be as attractive as string theory on the bayesian grounds I mention. I don’t know.

  17. Bob Levine says:

    @ Chris—

    I didn’t mean to imply (and don’t think my comment should be read as saying) that Maxwell’s extension was ‘plucked from thin air’ (at least as I think of what that expression suggests) or had no physical basis. But, at least as I’ve read the history, his main concern was to make the E and B field equations as close to mirror images as possible in the context of the evident lack of magnetic monopoles corresponding to free electrical charges. Proceeding in that fashion, without substantial experimental support, was a completely different style of physics from standard practice and, I think one could argue, anticipates a strikingly contemporary approach—much closer to Dirac’s approach to nature than Faraday’s.

  18. Noah Smith says:

    Thanks for this Peter, it seems to summarize some of the most interesting ideas in your book. Years after reading it, these ideas are what I remember – interesting no? This may be a good opportunity to try asking a question, if it’s possible to pose it properly. This is a sincere question and I’m genuinely interested in your thoughts. How about this: do you see a route by which these ideas will spread, so that they appear in undergrad physics and engineering courses? One example I think of is Lagrangian and Hamiltonian dynamics. In aerospace, it’s interesting to see where and how they appear in very practical applications. Thanks.

  19. Joel Rice says:

    From a historical perspective, there was a big bang in physics and mathematics when Hamilton discovered quaternion algebra. I read somewhere that Maxwell did his thinking in quaternions, and later translated this into matrices. And where would Einstein have gone without Maxwell ? Pascual Jordan mentioned quaternions in an early paper that led to Pauli dealing with spin, without which atoms make no sense at all. All in all, it seems that a lot rests on Hamilton’s discovery.

  20. Peter Woit says:


    I’ve listed some textbooks on the web-page for my course, and will be adding some more material there soon.

    JB and Noah,

    I’m hoping to at some point teach an undergraduate course on this material, and would write something up then. Unfortunately I don’t know of a good source now at this level (I’m thinking of “Hermann Weyl for undergrads”). There’s a long history since the early days of QM of interaction between the physics and the math of representation theory. In practice though, most physicists end up just learning some very special cases that they need to do certain calculations, and nothing about the general story. I’m not sure what will ever change this, maybe if I end up writing something like a book about this it would have a small effect.


    My argument is about representation theory, which does have to do with symmetry, not with stability. Discussing the various ways in which symmetry shows up in string theory is a huge topic. The bottom line though is that symmetry arguments get used everywhere in physics, but just because your idea uses a symmetry argument doesn’t mean that the idea does what you want it to.


    Please try and stick to some semblance of the topic. This one does lend itself to people deciding to write here about whatever they want, but, as usual, I can’t moderate a general physics discussion forum.

  21. Tim May says:

    Jumping in late, I’d say the “constancy of the speed of light” has been crucial to our understanding of how things work. General relativity and quantized gravity (and the various approaches, some wrong, some not even wrong (!), maybe one of them right?) get the attention in the comments in the Edge comments, but the combination of the constancy of c and quantum mechanics in the 1905-1930 period really had profound implications.

    Dirac’s effective predictions of anti-particles, based on light cone reasoning, must’ve been astounding to those at the time.

    And at another level, the whole view of causality and light cones that comes from the constancy of c has implications about simultaneity, the impossibility of omniscience (in our space-time), and the inability to ever, ever, communicate with distant objects in any meaningful sense.

  22. harryb says:


    First – excellent piece you started, and fascinating (192) entries I trawled – quickly – through. @Abbyyorker has a good point re Evolution, and Boltzmann’s entropy – I always think the Second Law of Thermodynamics ought to win these fests just because of its name.

    Still – the entries on elegance and symmetry not being true or useful also rang home. Jamshed Bharucha’s piece – The Beauty and Tragedy in the Mathematics of Music – had some interesting insights on the asymmetry or irrationality of the ratio of music (on strings, ho ho)

    One insight – the elegant ratios are “Close but unequal. If you tune by octaves, the fifths are out-of-tune, and vice versa.”

    So I worry about the equation : symmetry = beauty, elegance, depth.

    I see depth is the fact mathematics underpins so much of what we know.

    The deep theory of evolution, for example, in Dawkin’s entry, is offered in mathematical language:

    “The ratio of the huge amount that it explains (everything about life: its complexity, diversity and illusion of crafted design) divided by the little that it needs to postulate (non-random survival of randomly varying genes through geological time) is gigantic.”

    I guess I am noting that asymmetry may have an elegance of its own, and a reality too. To exclude asymmetry from deep truths may be to fall into the dead space we talk a lot of in these blogs re strings and so on.

    Daniel Kahneman (nobel laureate in Economics) discusses “theory-induced blindness” and over-adherence to “inside views” as ways in which groups stunt their progress. He also famously crafted an elegant theory on “loss-aversion” in economics which showed as humans we have an asymmetric view between losses and gains (reacting to losses typically twice as much as equivalent gains).

    Its elegant, and asymmetric.

    Close but unequal.

    So, excellent blog forcing me to look at alternative deep thoughts out there – truly insightful.

  23. Chris W. says:

    1. The symmetry of time translation gives energy*
    2. The symmetries of spatial translation gives momentum*
    3. Rotational symmetry gives angular momentum*
    4. Phase transformation symmetry gives charge*
    [* … conservation]

    Whereas #1 – #3 can be approached in a strictly classical way, #4 has always struck me as the most interesting and mysterious. Why should quantum mechanical phase, a notion which is not bound prima facie to any assumptions about the forces affecting the evolution (propagation) of particle probability amplitudes, have anything to do with electric charge or charge conservation? The way the connection is made—through the global gauge invariance of the electromagnetic field—is clear enough, but the outcome is still strange; from early on the electromagnetic field seemed to have a particularly intimate connection to the structure of quantum mechanics. One feels there must be more to this story….

  24. Mike J. says:

    I am a non-physic under-graduate. Can you introduce a book/paper/article etc. that describes these symmetry in a more understandable way. Thanks.

  25. Peter Woit says:

    Mike J.,

    My book “Not Even Wrong” has a chapter on this. But, honestly, if you haven’t studied physics a bit seriously, these ideas aren’t going to make much sense.

  26. Thomas Larsson says:

    Conformal field theory – using the representation theory of the Virasoro algebra to explain the spectrum of critical exponents in 2D. QED is often hailed as the most successful theory, because it predicts a handful of quantities to ten decimal places or so. CFT correctly predicts infinitely many quantities to infinite precision. 2D phase transitions is admittedly not a an extremely important field, but within its limited domain of validity, CFT is the ultimate theory.

  27. Geoff says:

    I’m profoundly biased in this regard as my first and foremost love has been and will always be general relativity. It is a never ending source of fascination that when translated literally the field equations simply say that physics is geometry and geometry is physics. I find it incredible that a mathematical discipline originally tasked with drawing maps of curved surfaces on flat pieces of paper has been so successful in describing the universe at large. Of all the theorems that Gauss proved, the one that he named Theorema Egregium turns out to be critical to the whole theory. (A beautiful article by Kuchar explaining some of this can be found here – )


    In a completely different field I always enjoyed listening to the recordings of Sidney Coleman where he would repeatedly and passionately say that if the PCT Theorem was wrong he’d have to go back to day one and start over. I must confess I haven’t appreciated that fact in any meaningful way just yet.

    Peter, as a suggestion perhaps you could give us some of the history behind the development of Noether’s Theorem. It’s my limited understanding that it came up in the context of General Relativity, specifically trying to understand gravitational radiation.

  28. Per says:

    Thomas Larsson:

    Wouldn’t you say that’s a consequence of the symmetry argument? The reason CFT is so successful is of course because of its symmetries (which survives quantization).

  29. I quickly browsed the list of answers, and not surprisingly, the atomic theory (in one form or another), natural selection, Maxwell’s theory of em field, and GR come several times. It’s an opportunity to remember how much we owe to nineteenth century science regarding the explanations of natural phenomena.

    I’m a bit surprised though that Kaluza-Klein theory does not appear on the list. To me that’s still the best idea one ever had about unification. It is also at the root of the set of ideas that eventually led to gauge theories.

  30. Chris W. says:

    Following up on Fabien’s comment, see arXiv:gr-qc/0012054 (“On Pauli’s invention of non-abelian Kaluza-Klein Theory in 1953”), which seems to be widely cited.

    I seem to recall reading somewhere that Pauli offered some rather strong criticisms of the Kaluza-Klein approach over a decade before the work described in this paper. I didn’t find exactly what I had read previously, but I did find this paper [PDF, at cdsweb.cern.ch], containing excerpts from a correspondence between Einstein and Pauli in the late 1930s. Pauli and Einstein came to agree on the futility of the Kaluza-Klein approach in finding a classical basis for quantum discreteness. Pauli’s 1953 work, and most subsequent work inspired by the Kaluza-Klein approach, had a rather different motivation.

  31. D R Lunsford says:

    Chris W,

    Pauli began his career in the early 20s working with Veblen – or at least contemporaneously with him – on something that was known as “projective relativity”, based on the “projective geometry of paths”, where the geodesics in a Riemann space are given a sort of projective structure, and one works in a sort of system of homogeneous coordinates. He later showed that Kaluza’s ansatz was just PR in a particular coordinate system. Since he had abandoned PR some time earlier, this reduced the Kaluza ansatz to the realm of the non-dynamical, and really, invalidated it as a road to unification. He also pointed out, in addition to this weighty argument, that the particular form of the action in Kaluza’s work was arbitrary. These arguments, as far as I can tell, have never been superseded by more subtle ones.


  32. D R Lunsford says:

    Fabien B,

    Kaluza theory and gauge theory could not possibly have had any more different origin. Kaluza was non-dynamical and was based on Riemannian geometry with a restricted (cylinder) geometry. Gauge theory emerged from Weyl’s gauge invariant extension of Riemannian geometry, in which calibration changes enter alongside coordinate changes as logically independent operations. These two things have really nothing in common. In particular, the Weyl ansatz is completely local and needs no global assumptions, such as the cylinder ansatz.


  33. Thomas Larsson says:

    Per: Of course CFT is all about symmetry, but I would not say that the symmetries survive quantization, for two reasons.

    1) The quantum theory (Hilbert space, correlation functions) is constructed directly, without classical crutches. So there is no quantization.

    2) If there were a classical theory, it would presumably have zero central charge. So the classical form of the symmetry has not survived.

    I like both of these aspects.

  34. Chris W. says:

    Much thanks for that. On reflection, I think I was recalling a comment by you posted a long time ago on another N.E.W. post. (I tried locating it just now, without success.)

  35. jorge says:

    “…string theory is the first science in hundreds of years to be pursued in pre-Baconian fashion…”

    Philip W. Anderson

  36. Jess Riedel says:

    One of the most mysterious basic aspects of quantum mechanics is that observable quantities correspond precisely to such self-adjoint operators, so these infinitesimal generators are observables.

    One of the reasons I decided to come work for Wojciech Zurek is because he more-or-less solved this mystery. More precisely, he showed [PRA, ArXiv] that what is often taken to be a very mysterious postulate

    Observables are represented by Hermitian operators

    can be derived from a much, much more natural postulate

    Outcomes are identified with vector components of the global wavefunction

    along with the natural definition that an outcome must be amplified to count as a measurement. By the linearity of quantum mechanics, only orthogonal vectors can be amplified. If we label each vector with a real number, we are left with a Hermitian operator being the natural mathematical object to identify with an observation, i.e. the catalog of all possible measurement outcomes.

    (Actually, from this we can see that it is better to identify observables with the normal operators, which include Hermitian operators as a subset. Indeed, the sensible operator when we measure the amplitude and phase of an electromagnetic wave is normal but not Hermitian.)

    Incidentally, this argument would undoubtably be my choice for most elegant explanation in physics.

  37. D R L, of course you are right to stress the importance of Weyl’s idea. Nonetheless dimensional reduction did play a role in this story, in particular in the works of Klein himself and also Pauli. You can consult O’Raifeartaigh’s book “the dawning of gauge theory” about this, notably chapters 3, 6 and 7.

  38. harryb says:

    A good try, but no cigar?


    Picked up from Lawrence Krauss’s latest work – A Universe from Nothing – interested in what you think of the book.

  39. Alberto says:

    As a first year student in physics in the seventies, I discovered the relation of symmetry and conservation laws in the first pages of the first volume of Laudau’s physics course. An instructor told me to look for it (in those times, in Europe, the books of the famous Soviet physicists and mathematicians circulated widely among penniless students, in the translations into French printed by the state publisher MIR). I thought this was the closest one could get to seeing the hand of god in action.

  40. D R Lunsford says:

    There are two amazing things that come from the same person, Felix Klein, and I can’t imagine their strange mystery or depth ever being exceeded.

    One was the association of Platonic solids with the solution of equations of the fifth degree.

    The other was the mathematical theory of the top, in which the best treatment necessitated the introduction of a 4d space of indefinite metric. Klein however pointed out that the required non-Euclidean geometry should not be taken seriously (!)


  41. Peter Woit says:


    That drawing pretty much does sum it up, and it has become very famous.

    I confess that the cosmological “why did something come from nothing” question never interested me at all (for one thing, the big bang singularity doesn’t seem like “nothing” to me), and the arguments about religion really don’t interest me either. So, I think I’m going to pass on reading that book carefully or writing about it.

  42. harryb says:


    Understood – but just to be clear Krauss’s works including this latest one are very much negative on any religious, quasi-religious or even anthropic arguments (with a – supportive – Afterword by Richard Dawkins you can see where he is coming from.)

    He is negative on string theory too as you can imagine from the cartoon for its lack of scientific rigour or evidence – a theory of anything, which is not science.

    His recent biography on Feynman – Quantum Man – may be a better option.

  43. jorge says:

    Why do we need to postulate an infinite number of universes with all sorts of different properties just to explain our one?

    Paul Steinhardt
    Edge 2005: What do you belive is true even though you cannot prove it?

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  45. chiz says:

    Geoff – Noether’s original paper is here and it includes some discussion of the non-conservation of energy in GR.

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