Monday, May 28, 2018

What do physicists mean when they say the laws of nature are beautiful?

Simplicity in photographic art.
“Monday Blues Chat”
By Erin Photography
In my upcoming book “Lost in Math: How Beauty Leads Physics Astray,” I explain what makes a theory beautiful in the eyes of a physicist and how beauty matters for their research. For this, I interviewed about a dozen theoretical physicists (full list here) and spoke to many more. I also read every book I could find on the topic, starting with Chandrasekhar’s “Truth and Beauty” to McAllister’s “Beauty and Revolution in Science” and Orrell’s “Truth or Beauty”.

Turns out theoretical physicists largely agree on what they mean by beauty, and it has the following three aspects:


A beautiful theory is simple, and it is simple if it can be derived from few assumptions. Currently common ways to increase simplicity in the foundations of physics is unifying different concepts or adding symmetries. To make a theory simpler, you can also remove axioms; this will eventually result in one or the other version of a multiverse.

Please note that the simplicity I am referring to here is absolute simplicity and has nothing to do with Occam’s razor, which merely tells you that from two theories that achieve the same you should pick the simpler one.


A beautiful theory is also natural, meaning it does not contain numbers without units that are either much larger or much smaller than 1. In physics-speak you’d say “dimensionless parameters are of order 1.” In high energy particle physics in particular, theorists use a relaxed version of naturalness called “technical naturalness” which says that small numbers are permitted if there is an explanation for their smallness. Symmetries, for example, can serve as such an explanation.

Note that in contrast to simplicity, naturalness is an assumption about the type of assumptions, not about the number of assumptions.


Elegance is the fuzziest aspect of beauty. It is often described as an element of surprise, the “aha-effect,” or the discovery of unexpected connections. One specific aspect of elegance is a theory’s resistance to change, often referred to as “rigidity” or (misleadingly, I think) as the ability of a theory to “explain itself.”

By no way do I mean to propose this as a definition of beauty; it is merely a summary of what physicists mean when they say a theory is beautiful. General relativity, string theory, grand unification, and supersymmetry score high on all three aspects of beauty. The standard model, modified gravity, or asymptotically safe gravity, not so much.

But while physicists largely agree on what they mean by beauty, in some cases they disagree on whether a theory fulfills the requirements. This is the case most prominently for quantum mechanics and the multiverse.

For quantum mechanics, the disagreement originates in the measurement axiom. On the one hand it’s a simple axiom. On the other hand, it covers up a mess, that being the problem of defining just what a measurement and a measurement apparatus are.

For the multiverse, the disagreement is over whether throwing out an assumption counts as a simplification if you have to add it again later because otherwise you cannot describe our observations.

If you want to know more about how arguments from beauty are used and abused in the foundations of physics, my book will be published on June 12th and then it’s all yours to peruse!


  1. Thanks Dr. H. I'm looking forward to it. Of course physicists use the term in a technical way only tangentially connected to beauty as the term is commonly used and like the question "what is goodness?" or "what is truth?" Beauty in its common usage is a slippery concept. But as I've noted before, like the other two, beauty is a VALUE and its slippery nature stems from our very vague recognition of what constitutes those values

  2. The epigraph to Chandrasekhar's Mathematical Theory of Black Holes has quotes from Heisenberg and Bacon on the subject. I don't have the volume at hand, but they go something like this:

    Heisenberg: Beauty consists of the proper proportion of the parts to the whole, and to each other.

    Bacon: There is no thing of excellent beauty which hath not some strangeness in the proportion.

  3. "General relativity, string theory, grand unification, and supersymmetry score high on all three aspects of beauty. The standard model, modified gravity, or asymptotically safe gravity, not so much. "

    where does loop quantum gravity score on aspects of beauty?

  4. Every time I read something like "the laws of nature are beautiful?" someone want to sell a new book to the public :-(

  5. A beautiful equation is also one that exhibits the fewest free parameters while explaining the most physics. That's why general relativity is beautiful while the Lagrangian of the Standard Model is ugly as hell. They both work, one by itself and the other by brute force, although I would never compare one with the other.

    Looking forward to purchasing your book!

  6. Simple, natural, elegant: The answer exits a printer after a one-hour observation.

    ... 1) Baryogenesis is post-Big Bang excess matter over antimatter violating conservation laws via selective leakage.
    ... 2) Sakharov conditions. Vacuum is neither exactly mirror-symmetric nor exactly isotropic toward quarks then hadrons.
    ... 3) Einstein-Cartan-Kibble-Sciama spacetime torsion chiral dopant.
    ... 4) Milgrom acceleration and the cosmological constant emerge. Dark matter, SUSY, and M-theory wither.
    ... 5) Extreme opposite shoes embed within a vacuum left foot with measurably different energies.
    ... 6) Measure spacetime trace chiral anisotropy ̶ one hour in a microwave spectrometer, 40,000:1 signal to noise, using molecular lollipop enantiomers.

    Enantiomeric balls ( short sticks (2-CN group for dipole moment) Divergent rotational spectra.


  7. I only recognize simplicity of the Occam's razor kind. Make only a few assumptions that can be tested empirically. Naturalness is just modern numerology of the ancient Pythagoreans. (They believed the square root of two is evil) Elegance is epistemology because no scientific theory explains itself. They explain observations. We can make theories that explain themselves or anything except the observable universe. Theoretical physicists are envious of mathematicians because their imaginations can fly without constraint from the physical world. It's flight-of-fantasy envy

  8. Folks,

    Something's wrong with the comment feature at blogger - I'm not getting comment notifications, meaning I basically don't know if anyone submitted a comment until I check the website. If anyone has an idea what's the issue, please let me know. For the rest, may I kindly ask for your patience. I'll be traveling today but will look for a fix once back home.

  9. To me, as a mathematician, beauty in physics/science is something that is almost impossible to describe and define but easy to tell when you see it. It is also very subjective and different people may disagree. for instance general relativity is beautiful, modified gravity is not. The orthodox quantum mechanics is beautiful, Bohmian mechanics is not, and so on.

  10. arxiv:1712.07969 k. How can that be empirically furiously wrong?
    XENON1T, 1300 kg active liquid xenon target of total 2000 kg. ZERO net output.
    XENONnT, 5200 kg active liquid xenon target of total 7500 kg. 2019 launch.

    Zero signal crashes physical theory. Simple, natural, elegant: The math is rigorous but empirically irrelevant. It's a curve fit. It's phlogiston.

    Newton becomes special relativity given Maxwell. Lightspeed is unchanged from all reference frames. What rubbish! No, it's true. Baryogenesis happened. Conservation laws are inexact. What rubbish! Is it true?

    "There is no reason to look. Physical theory cannot be fundamentally defective." Nothing predicts. Simple, natural, elegant: Look at the answer not at its guesses.

  11. Dr Hossenfelder:
    How do the following score in beauty,in your view?

    The C*-algebraic version of LQG (see LOST theorem).

    The spectral derivation of the standard model (see Connes's work).

    Isham and Döring's topos-theoretic formulation of the Kochen-Specker Theorem.

    Geroch's Einstein algebras

  12. Beauty is not something we can easily codify. Much of this has to do with induction nature of proposing a grand theory. There is no deductive structure to proposing some set of physical axioms or postulates as the most economical and elegant foundation to the universe. Beauty is a relative of “quality” and as Pirsig wrote in Zen in the Art of Motorcycle Maintenance is not something that can be codified.

    A good theory has a limited number of physical axioms that upon their introduction change how we think and that are then by deductive reasoning able to lead to a large set of expected results. Such theories are often considered to be beautiful, elegant and natural. Simplicity comes with the limited number of postulates, elegance is in the new structures proposed, and naturalness comes with the relative ease with which physical predictions occur.

    One thing that can happen of course is a very beautiful and elegant theory can be wrong. Supersymmetry is an intertwine between the boson and fermion structures of quantum mechanics and the structure of spacetime. It also short circuits the Coleman-Mandula “no-go” theorem on obstructions to unification of internal symmetries of gauge fields with the external symmetry of gravitation. This is a big problem with Lisi's program that includes the SL(2,C) of gravitation. Supersymmetry is though a framework more than an exact description of nature, and one must “hang” a model of gauge bosons and fermions on it. This has been the case with minimally supersymmetric standard model (MSSM). Interestingly MSSM is on the verge of being falsified, and many thousands of papers on this topic, including those by luminaries as Gordon Kane, may be completely trashed. The question is whether this is a failure of supersymmetry or a particular model.

    Maybe supersymmetry occurs in ways completely different from what has been thought. This is my thesis. I frankly welcome the prospect that MSSM is falsified; it clears the decks for small players like me and many others. Whether following beauty has been the downfall or not is not clear to me. Obviously people tried to work light mass supersymmetric partners with the standard model. The standard model is not considered to be the most elegant theory out there, but it sure works like a top for TeV scale physics. The MSSM was built because it was what seemed most reasonable at the time, and it had some level of beauty to it. It though appears to be headed for the trash heap.

  13. You may want to check your spam folder; sometimes a setting gets broken and many things go there. I've seen it happen to others.

    Travel safely.

    sean s.

  14. Beautiful mathematics is absolutely no substitute for understanding. Dirac was the epitome of that. He had absolutely no understanding of the electron, but didn't care, and he even ignored the likes of Gustav Mie and Charles Galton Darwin. In fact, seeing as his 1962 paper an extensible model of the electron depicted the electron as a charged conducting sphere, I'd go so far as to say beauty is dangerous.

    As regards comment moderation, I notice that I can't comment using my wordpress id. It's The Universe (Google Account) or nothing. You could always try turning comment moderation off.

    John Duffield

  15. Hi all,

    Regarding the comment issue, turns out it's not a problem with my blog, but a blogger-wide issue that will supposedly be fixed next week or such. (See forum thread.) So rather than switching to a different comment widget (which would remove all existing comments), I'll wait this out. Please be warned that this means for the coming week comments will appear even slower than usual.

  16. I think the elegance aspect has to do with the sparseness of the description and its predictive power - one can get far more phenomena explained and flowing out without fudging than was put in; and preferably it happens in a way that is non-obvious and startling

  17. My professor in mathematics used to qualify beautiful equations as horny, because such equations are stimulating the faculty's survival.

  18. John Duffield,

    Dirac seems to be an example of exactly the opposite. Beauty was certainly a very important motivation for his work (one can argue it was the only one). And his contribution to physics is undeniable.

    Sabine, if you could have interviewed Dirac, would you have had a different view about beauty?


  19. I think one additional aspect (or may be you include this with the "Surprise element" under elegance) is that beautiful theories use non-intuitive concepts to explain everyday experience.
    Even Newton is rather non-intuitive (compared to Aristoteles).
    GR explaining things falling down by time running slower close to a mass gives a totally weird-seeming explanation.

    Explanatory power is also important. I remember that I definitely did not find Maxwell's equation beautiful in any way when I first saw them. I appreciated their beauty only after seeing how you can derive things like em-waves from them and how the different terms in the equation conspire to make em-waves possible. So one other aspect of beauty may be only apparent when you find that the equations are easy to operate with and reveal a rich structure of possible things to derive from them. (Complexity from simplicity.)

  20. Space Time,

    I don't know what you mean. Would I have had a different view about beauty than Dirac? Presumably. Or a different view than presently? Probably not. Or else, I don't know what you mean.

  21. @The Universe Otto Stern’s measured proton magnetic moment showed the Dirac equation is empirically wrong for composite particles. Nobel Prize.

    Stern's value was poor but sufficiently far from Dirac's calculated value. Current proton-antiproton values 1.5 ppb diverge re baryogenesis. One hour in a microwave rotational spectrometer measures overall vacuum chiral anisotropy toward hadrons, falsifying simple, natural, elegant.

    .... 2.792847350(9)μ_N proton
    ... -2.7928473441(42)μ_N antiproton
    ... DOI::10.1038/nature24048

  22. Bill,

    "A beautiful equation is also one that exhibits the fewest free parameters while explaining the most physics. That's why general relativity is beautiful while the Lagrangian of the Standard Model is ugly as hell."

    Well and compactly said.

    Space Time,

    " ... general relativity is beautiful, modified gravity is not. The orthodox quantum mechanics is beautiful, Bohmian mechanics is not, and so on."

    Huh? You're speaking as a mathematician?

  23. Mathematics is required , but the way it is done to do physics for funds and just survival is wrong. One can go to the screen in " The Man who knew infinity " where Professor Hardy tells S Ramanujan probably in the hospital " I want rigor Ramanujan" when Ramanujan writes the problem and just the right solution without the steps. Ramanujan responds well , he gives rigor ti his solutions with Prof hardy's suggestion. Mathematics can be used purposefully in physics if there is rigor.

    Even in engineering and experimental work many do not report error bars, even in high ranking journals which can be done only when they do the experiments atleast thrice. The rush to publish is the primary cause for this.

  24. "For the multiverse, the disagreement is over whether throwing out an assumption counts as a simplification if you have to add it again later because otherwise you cannot describe our observations."

    I think you mean that the collapse postulate is removed, and then splitting is added later. Splitting is metaphorical. Splitting never happens. You don't need to know when or where a universe splits. The Schrödinger equation describes everything.

  25. Surely the idea of beauty, as applied to a physical theory, only makes sense if it is assumed that you are looking at the fundamental theory. Unless that is the case, ugly equations are the norm.

    For example, I was stunned as a teenager by the gas law equation PV=nRT - then I learned it is only an approximation, and more accurate, but vastly uglier equations do better!

    Is physics at the depth to find a fundamental theory - who knows, but I'll bet every generation thinks it is!

  26. "Huh? You're speaking as a mathematician?"

    t h ray, are you surprised that the examples I gave were from physics rather than mathematics? Well, it is hard (in my opinion impossible) to find an example of ugly in mathematics.

  27. I think an important aspect of beauty that is actually a valid guideline for physics, is that a physical description should be as free as possible from arbitrary choices that are made by us humans to provide for a description. Usually this means being closer to the "Copernican principle" that the world is less centred around you and that if you make a choice at a point in space time (e.g. a frame of reference), there is no trivial way to communicate that choice to the rest of the world. It also usually means symmetric, with the symmetry being the group acting on the possible choices that provide descriptions. This often makes it more technical to write things down (although this is mostly a matter of what you are used to), but it also hides lots of distracting information, that you could use to make write down variants that, however, can't be physically relevant, because they depend on an arbitrary choice you made. It is not unlike abstractions in a programming language.

    General relativity is more beautiful than field theory + gravitons, because general relativity does not assume a background metric. Of course in every point of space time, you can _choose_ a frame such that $g_{\mu\nu} = \eta_{\mu\nu}$ but now the description involves a choice. Realising that there is a choice to be made, makes life technically harder but pre relativity you made the tacit assumption that your "inertial frame of reference" was trivially agreed upon in the rest of the universe. Not making that assumption is i.m.o. the heart of general relativity.

    Sections of U(1) line bundle are better than wave functions, because only phase differences and absolute values at a point make sense. Once you think about it in this way the operator $\partial_\mu = \frac{\partial}{\partial x^\mu}$ makes no sense anymore: you need a U(1) connection $\nabla$ which in a local trivialisation looks like $\nabla_\mu = \partial_\mu + A_\mu$ with (i times) the vector potential $A$. The curvature is (i times) the Faraday tensor, because EM works in exactly this way on the wave function. Even the Bohm Aharanov effect now falls out naturally as the holonomy of a flat connection. The gauge group $\psi \to e^{i\phi} \psi$ can be seen actively as a symmetry, but more naturally passively as the result of different choices of trivialisation of the line bundle.

    The Maxwell equations in Heaviside as written today are more beautiful than the equations written in x,y,z that he wrote down himself because the former are manifestly independent of the choice of a frame of reference. LIkewise, the index notation used by physicists is computationally useful, but it is also horrible in that it makes it impossible to even say what you mean without making a choice of reference frame, not to mention actively encouraging not to say what kind of object you are dealing with "because it is "clear" from the indices, and making people think in terms of components and operations on indices. Weinbergs book on quantum field theory starts by writing down the gamma matrices he uses. It always leaves a lingering feeling of unease about what depends on conventions and what not. Mathematicians (or at least geometers) take pride in writing down coordinate independent intrinsic entities whenever possible, and it greatly helps in only writing down expressions that make sense independent of choices, of which there tend to be very few, so it is a great guiding principle.


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