John Carlos Baez

Mathematicians get annoyed at how physicists take beautiful formulas and clutter them up with 'useless' constants like

𝑐 - the speed of light
ℏ - Planck's constant
π‘˜ - Boltzmann's constant
𝐺 - the gravitational constant

making it harder to see the essence of things. Mathematicians prefer units where all these constants are set equal to 1.

I used to be like that too - but right now I'm doing a project where I 𝑛𝑒𝑒𝑑 these constants to see the essence of things!

(Of course it's good to keep these constants around so you can use dimensional analysis to avoid mistakes: this is what computer scientists call a 'type discipline'. That's important, but it's NOT what I'm talking about now.)

When you're studying just one physical theory at a time, you can set dimensionful constants equal to 1 to simplify things. But often we like to study a whole π‘“π‘Žπ‘šπ‘–π‘™π‘¦ of physical theories at once - a family where those constants take different values! We can't set them to 1 if we're interested in what happens when they approach 0. For example:

As 1/𝑐 β†’ 0, special relativity reduces to Newtonian physics.
As ℏ β†’ 0, quantum mechanics reduces to classical mechanics.
As π‘˜ β†’ 0, statistical mechanics reduces to classical mechanics.
As 𝐺 β†’ 0, general relativity reduces to special relativity.

And this is just the beginning of the story: various collections of constants can approach 0 at different rates, and so on.

When we do this, we're studying what mathematicians would call a 'moduli space' of theories - or even better, a 'moduli stack'. We may want to do 'deformation theory', where we expand answers in powers of some constant. And so on.

So don't scorn those constants!

Aug 17, 2024 at 12:03pmWeb
Show threadAlexander Corby πŸ‡΅πŸ‡·Aug 17, 2024
@johncarlosbaez Hey, It's not our fault that humanity got stuck using arbitrary units of measurement that needed these insane constants to balance out the equation.
Show threadJohn Carlos BaezAug 17, 2024
@AlexCorby - so you didn't read the post, huh?
Show threadAlexander Corby πŸ‡΅πŸ‡·Aug 17, 2024
@johncarlosbaez No, I did, and I very much appreciated it. I was just remarking that I always found it funny that we invented these constants to mitigate between otherwise unrelated units and then had to tack on bizzare units for the constants themselves to make the equations work (G in terms of N*m^2/kg^2 for example). Didn't mean to come off snarky.
Show threadJohn Carlos BaezAug 17, 2024
@AlexCorby - okay, gotcha. Here's an example I love: it seems the pyramid-building Egyptians measured horizontal and vertical distances with different units! I don't know if there was ever an Egyptian Einstein who unified vertical and horizontal.
Show threadAlexander The 1stAug 17, 2024
@johncarlosbaez @AlexCorby Okay, that's impressive if they built the pyramids using vw and vh to build the pyramids using CSS.
Show threadSteve DodgeAug 17, 2024

@johncarlosbaez @AlexCorby
I present to you: the survey foot!

https://www.nist.gov/pml/us-surveyfoot

U.S. Survey Foot

NIST
Show threadPaul H 🏴󠁧󠁒󠁷󠁬󠁳󠁿πŸ‡ͺπŸ‡ΊAug 17, 2024

@johncarlosbaez

I recall as an undergraduate a particle physics lecturer who insisted on using "natural units" - c = h-bar = e = G = (any other natural constant you care to name) = 1
The course was a nightmare!

I've also encountered "Mott's Law" (attributed to Neville Mott, though I can't find a source):

"Any definite integral which you cannot evaluate for whatever reason is equal to 1"

This actually works in most cases, at least to an order of magnitude which is often sufficient...

Show threadJohn Carlos BaezAug 17, 2024

@Henrysbridge - being a mathematician I would have found it much easier to handle a physics course using natural units, and my book on gauge fields and gravity uses c = ℏ = G = e = 1. But when I recently wrote my little book on entropy I wanted to make contact with experiment, so I kept k around. And now I'm doing a project where I really need k and ℏ visible. So I think it's all a matter of what you're doing.

It makes tons of sense, though, for intro physics courses to keep all these constants visible!

Show threadPaul H 🏴󠁧󠁒󠁷󠁬󠁳󠁿πŸ‡ͺπŸ‡ΊAug 17, 2024

@johncarlosbaez

The big problem this lecturer had was that not only were all constants = 1, but they were consequently ignored in equations and calculations - as you note, it made it impossible to keep track of what was physically going on in the formal mathematics!

Show threadJohn Carlsen πŸ‡ΊπŸ‡ΈπŸ‡³πŸ‡±πŸ‡ͺπŸ‡ΊAug 17, 2024

@Henrysbridge @johncarlosbaez

I recall some politicians wanted to define the value of pi to be exactly 3.

Personally, I like that we calculate natural phenomena using symbols, though I now hesitate to call them "constants" because the values are often refined as our understanding of them improves.

https://en.wikipedia.org/wiki/Indiana_pi_bill

Indiana pi bill - Wikipedia

Show threadMichael VilainAug 17, 2024
@Henrysbridge @johncarlosbaez My professor for Quantum Mechanics was a student of Dirac. He didn't lecture from the book and used Dirac's notation for everything. I didn't understand much of what he taught and ended up taking the course again from another professor who I'd had before for Thermo. Great class and I got a B the 2nd time.
Show threadPaul H 🏴󠁧󠁒󠁷󠁬󠁳󠁿πŸ‡ͺπŸ‡ΊAug 18, 2024

@mvilain @johncarlosbaez

Sometimes it's a good idea to be at a distance from genius to learn stuff. A notable exception - Richard Feynman.

Show threadΠ‘Π°ΠΊΡƒΡ€Π°Aug 18, 2024
@Henrysbridge @johncarlosbaez
My physics lectures at the Uni were both the most fascinating and frustrating things at the same time. Remain to be somewhere in top 10 in my life.
Fascinating because you can calculate *everything* with a few (seemingly) simple techniques. Frustrating because I had no idea how the lector was doing it. The stuff written on the blackboard had absolutely no correlation with anything I could find in the textbooks. Dropped constants, different units, "let's say it's 0"
Show threadPaul H 🏴󠁧󠁒󠁷󠁬󠁳󠁿πŸ‡ͺπŸ‡ΊAug 18, 2024

@pradd @johncarlosbaez

I know what you mean. As I was more of an experimental mindset, I got on with the "we can forget about those terms" idea pretty easily - first order approximations all the way!

Show threadJohn Carlos BaezAug 18, 2024

@pradd @Henrysbridge -

"Fascinating because you can calculate *everything* with a few (seemingly) simple techniques. Frustrating because I had no idea how the lector was doing it."

It's sort of like learning to do gymnastics by watching videos of

Show threadyour auntifa liza πŸ‡΅πŸ‡· πŸ¦› 🦦Aug 17, 2024
@johncarlosbaez JFC you made me look and then read and then learn something about maths and physics. well played sir, well play. you are the real deal as a gentleman and a scholar πŸ€“ πŸ‘πŸ½
Show threadSally StrangeAug 17, 2024
@blogdiva @johncarlosbaez Same! A fairly large shift in my understanding of the relationships between math and physics, as well as several different layers of existence, just took place. And I haven't even had any coffee yet!
Show threadJohn Carlos BaezAug 17, 2024
@SallyStrange - that's great! It's so fun when reality changes and suddenly makes *more* sense, not less. But for me that usually requires coffee.
Show threadJohn Carlos BaezAug 17, 2024
@blogdiva - thanks, that means a lot to me!
Show threadRoberto RossiAug 17, 2024
@johncarlosbaez I think you still miss the most wierd constant of all (that’s my personal opinion) \alpha = 1/137. It keeps appearing in many calculations related to electroweek and strong force interactions. As it was defined by Sommerfeld in your book it would be equal to 1…
Show threadJohn Carlos BaezAug 17, 2024

@roberossi - the fine structure constant is a dimensionless constant, so it can't be set to 1 without affecting the physical predictions. My post was about dimensionful constants, which can be set to 1 by choosing appropriate units, without changing any physical predictions.

But I agree that Ξ± is weird! We would all love to know why it has the value it does, but almost everyone has given up trying to answer that.

Show threadWeekend EditorAug 17, 2024

@johncarlosbaez

Occasionally, I get confused in a calculation and, looking at an expression, ask: "What *is* this thing, anyway?"

Being able to put the units back temporarily to discover the type of object I'm looking at is useful. (Very much like types in computer languages, and for that matter, in some varieties of logic.)

Sure, taking the unit scales out makes stuff look more elegant, but being able to put them back in occasionally helps check that one hasn't made a mistake.

Show threadGrant_HAug 17, 2024

@johncarlosbaez my question is what sort of scale do the graph axes have to get G = 1/c = hbar? 😜

Thanks for the alt text reference to Bronshtein - they are all fascinating.

Show threadWillem JanssenAug 17, 2024
@johncarlosbaez That’s a beautiful graph/cube I hadn’t seen before, thanks.
There’s plenty physicists who take these constants as 1 too. And then there are cgs astrophysicists 🀷.
Show threadJohn Carlos BaezAug 17, 2024

@Wlm - This cube is called Bronstein's cube. I'm glad you liked it! Here are some sources for it:

https://hsm.stackexchange.com/questions/14181/source-documents-for-bronsteins-cube-of-physics

Source documents for Bronstein's Cube of Physics

The "cube of physics" is a quite useful summary of physics, for historical$^1$ and teaching$^2$ purposes, that is best explained (as far as I know) in "Physics On A Cube" by Jer...

History of Science and Mathematics Stack Exchange
Show threadCenobyte Aug 18, 2024
@johncarlosbaez I’ve got to keep this in mind. I usually play in Natural units and set it all to 1, but sometimes I regret that. I can totally see wanting to connect it to more easily measurable things like spacetime curvature (1/Length^2)
Show threadThis account has movedAug 18, 2024
@johncarlosbaez Don't forget to consider dimensionless constants such as fine-structure constants, and see where those fit into the picture!
Show threadStijn De BaerdemackerAug 18, 2024
@johncarlosbaez one is only one if the unit for one is one
Show threadJohn BarrettAug 20, 2024
@johncarlosbaez I find it useful to think of physical quantities like length, times, masses etc. as elements of a line (a one-dimensional vector space). Then a unit is a basis vector for the line. Equations with units become vector equations. So, for example, 1kg=1000g is a relation between the two vectors kg and g. Lines can be multiplied and divided, so c is a canonical element of L/T. You can put the units into equations with confidence and cancel them, e.g. km/m=1000.
Show threadJohn Carlos BaezAug 20, 2024

@JohnBarrett - I agree! Lines are important!

Jim Dolan promulgated the idea of 'dimensional categories', which are symmetric monoidal k-linear categories where every object is a 'line object': an object with a tensor inverse, whose self-braiding is the identity. In applications to dimensional analysis, each object X is a 'dimension' (like lengthΒ³ Γ— time), and morphisms

f: I β†’ X

are 'quantities' of dimension X.

A really great example is the category of complex line bundles over some variety. Here 'quantities' are global sections.

Todd Trimble explains the math better here:

https://ncatlab.org/toddtrimble/published/Notes+on+conversations+with+James+Dolan#case_study_the_doctrine_of_dimensional_theories

All this is supposed to link up to what I mentioned about deformation theory. For example, special relativistic theories are isomorphic for different values of the speed of light 𝑐, except for 𝑐=∞ (Newtonian mechanics) and 𝑐=0 (which people now call Carrollian relativity). The quantity 𝑐 should be a partial section of a line bundle on β„‚PΒΉ or at least ℝPΒΉ. It becomes ill-defined when 𝑐=∞, but its inverse is defined there.

Anyway, sorry for talking your ear off, but what I say in my articles on Mathstodon is the intersection of what I want to say and what I think readers will follow, and you just massively expanded that intersection.