Hi, this is the author. I've been coming back to this project off and on over the past few years but I often think of these articles as mostly something I'm writing for myself, so I'm really happy to see that some other people might be getting something out of them! I'd definitely love to hear if anyone knows anything I got wrong or can think of a way any particular explanation might be made better.
I should also take this chance to mention that I work as a private tutor and I have openings for students! Much more info here: https://nicf.net/tutoring/
Nic was tutoring me in proof-based foundational maths when I worked as a research engineer and I have nothing but praise for him and his tutoring service. He created a hyper personalized syllabus and was able to focus in on the areas in which my understanding was shaky incredibly quickly. I only studied very foundational concepts with him, but you can glean from the articles on his website how far and wide his knowledge extends.
It feels very rare that someone with his level of intellectual depth is this interested in teaching others.
Nic was a colleague of mine when we were both at Google Brain and he has a rare combination of both desire and ability to really understand the fundamentals of a field. (That is especially rare in ML, haha.) He also has a knack for explaining these things clearly --- cutting away the cruft and getting to the heart of the matter. I can't recommend working with / learning from him enough!
Thanks a lot for this effort! As someone wanting to learn physics from a math background, this seems to be at the right level for discussing critical high-level ideas mathematically. The article format is great as I get distracted reading book-length treatments whether they are written by physicists or mathematicians.
If you have a Patreon or similar account, let us know!
PS: The Rieman zeta function page seems to be missing a few enclosing tags leading to non-typeset latex formulae after "This gives us a nice way to pick out terms from a Dirichlet series..." and also after the Von Mangoldt function.
What a sweet idea! I don't have a Patreon at the moment but if I get around to making one I'll definitely post it on the website.
I appreciate the heads up about the typesetting --- it looks like it's due to some macros that the web-based TeX typesetter I'm using hasn't implemented. I usually go over the PDF versions more carefully than the automatically generated web versions, and I guess this is the price I pay.
(Looks like I can no longer edit my last message, so including this as a separate reply.) You've successfully nudged me into making a PayPal donation link, which you can find right here: https://nicf.net/donate/. I do very much appreciate the kind words.
I own a copy of that book, and I also highly recommend it! (The full title is "Physics for Mathematicians: Mechanics I", but sadly we're now never going to get a "Mechanics II".) It has a different goal than my notes --- he's more interested in building up classical mechanics very, very carefully from first principles --- but it's a fun journey if you have the time to spend on it.
As a former physicist, I never understood the full math behind Schrodinger's equation. Since then I ventured into CS, so I wonder if this book will be a good refreshment.
The Schrodinger equation is just restating something you may already know if you studied physics. The Hamiltonian is the generator of time translations.
I’d really love to see a rigorous explanation of renormalization, plus why it doesn’t work for gravity, with no handwaving. As a non-physicist, that has always been the point where I hit a wall on traditional treatments of QFT.
This is actually on the agenda for this series! (At least for some value of "rigorous".) The question of why gravity isn't renormalizable was one of the questions that made me want to dig into this project in the first place.
For what it's worth, from all the various nonrigorous explanations in physics texts, the one that worked the best for me was the one in "Quantum Field Theory Lectures of Sidney Coleman".
I'm gonna give it a shot and give an analogy with a comp. science audience in mind.
All Quantum Field Theories are effective theories. Effective means that they work up to certain energy-range, they do not intend to be fundamental.
For example, the Fermi theory of beta decay is an effective theory that works only up to the energy of W and Z bosons. Quantum Electrodynamics (QED), the theory of electromagnetism and photons, is an effective theory which is valid up to the electroweak scale (~250 GeV). So on and so forth. All of them are effective, and therefore break at some point at higher energies or, equivalently, shorter distances.
Renormalizable theories are such that we can abstract away the physics beyond that breaking point where the theory doesn't make sense anymore. And capture the physics beyond that point into a redefinition of a few fundamental constants that we can take from the experiment. To rehash the basic idea, the theory doesn't work beyond certain energies. In principle physics beyond those energies impacts our predictions, because in QM you must account for all the processes. But renormalizable theories are nice enough that we can put physics beyond that energy scale behind a black box and we just have to redefine a few fundamental constants like the mass of the electron.
Comp. science audience analogy: It's as if renormalizable theories gave us a neat API that does not leak the ugly internals of what happens at super high energies (short distances). Like you don't need to know machine code or assembly to import Pytorch and build a neural network in a few lines of code.
For gravity this doesn't work. Gravity is mediated by a massless spin-2 field (Einstein's theory). If you try to quantize this theory you will find that it explodes at second order. We can calculate the first quantum corrections to Einstein's gravity but then it explodes. And when we attempt our tricks to hide all the short distance/high energy stuff inside a black box (renormalization) it just doesn't work. It's as if the API was leaky. It leaks the internals to the high level.
To give you a couple of examples of this leakiness. Dark energy, the energy of the vacuum, causes an expansion to our universe at the largest scales possible. So it's a phenomena of super loooong distances and yet, it's dominated by the super small distance (high energy) interactions that contribute to this energy of the vacuum. Another example, black holes are typically hyper massive, huge beasts and yet... they are inherently quantum gravitational objects for which we need the full theory of quantum gravity to understand them.
This is a blessing and a curse. The curse is that it makes our jobs of getting a theory of quantum gravity so much harder. The blessing is that it gives us a chance of peeking at a more fundamental theory of physics. If we could have worked out everything with effective theories we may never know what's beyond those black boxes that hide the internals. Gravity gives us the chance to peek through and understand something deeper.
Skimmed some of the articles, particularly those nearer to my field. Seems like a generally good set of informal notes.
Random comments:
>when the states evolve in time and the observables don’t we are using Liouville’s picture; when the observables evolve in time and the states don’t we are using Hamilton’s picture.
I have never heard this terminology, I have only heard Schrodinger's picture vs. Heisenberg's picture.
>This means that, very unlike on a Riemannian manifold, a symplectic manifold has no local geometry, so there’s no symplectic analogue of anything like curvature.
Perhaps the only enlightening comment I have ever heard about the tautological 1-form/symplectic approach to Hamiltonian mechanics.
> I have never heard this terminology, I have only heard Schrodinger's picture vs. Heisenberg's picture.
I wrote the QM article a very long time ago at this point, and I actually can't reconstruct at the moment why I used those two names! I've also heard Schrodinger and Heisenberg much more frequently. Might be worth an edit.
I've never gotten a satisfactory explanation of what sort of mathematical object a physical unit (meter, kilo, second etc) is. There are plenty of bones of contention between maths and physics, but this one bothers me the most.
Anyone interested in coming at physics from a mathematics perspective should read Arnold's mechanics book.
One concern is that measures, out of the box, have issues in 3+ dimensions. Concretely due to paradoxes such as Banach-Tarski, that arise from the Zermelo Fraenkel (ZF) + Axiom of Choice (AC) = ZFC axiomatic formulation for set theory.
Since things need to conserve in pyhsics, one has to account for this issue and doing so is harder than it may seem as AC is part of the "fabric" of most mathematics which, at large, chooses to ignore the problem.
This is very very strongly not a concern for physics. Banach-Tarski and similar "paradoxes" require dividing things up into unmeasurable sets. These sets are pretty wild and aren't ever going to turn up when you're doing physics.
Speaking as a physicist, we don't care at all about stuff like Banach-Tarski, and there is essentially zeo expectation that stuff like ZF vs ZFC will have any impact on physics.
Could you explain what you think Gibbs style vectors have to do with Banach-Tarski or the axiom of choice?
As an aside, I'd like to emphasise that geometric algebra gives exactly the same physics outcomes as doing the maths with vectors or tensors or whatever else you like. The difference is essentially just notation. Some things look prettier.
Specific to building _intuitions_ for why the Banach-Tarski arises in ZF+AC.
GA gets rid of the external conventions for coordinate and chirality and also uses SU(2) which is simply connected vs SO(3) which is not. Rotors in GA can be used as elements of the algebra like any number avoiding the complexity of Euler angles, gimbal lock, etc....
GA's rotors are geometrically intuitive and can do rotations around an arbitrary axis, where quaternions are limited to an axis through the origin.
As Banach-Tarski is not physically realizable and because physics uses the computable reals, rationals and other aleph naught numbers it doesn't cause a problem there.
Lots of important work resulted _from_ the Banach-Tarski paradox but really it is just a cautionary tail about ZF+AC and on-measurable sets as far as physics goes.
What I was talking about is tools about building intuitions on why it arises.
Note that the maths aren't exactly the same, as an example Maxwells equations require four separate formula to express in Vector Calculus vs just one in GA. I don't think I fully comprehended the connection before learning GA.
This is also digging deep into the implications of your chosen groups and resulting algebra but as an example:
A Tensor can't represent a spinor but an even multivector can.
A pure grade multivector can only completely represent antisymmetric tensors.
You can look into Dirac's belt trick as a physical example showing that SO(3) isn't simply connected but it arises in E(3) in that particular case too.
I wish this site had latex support, so I apologize for the above which is probably of little value in reality.
I don't want this to come across as insulting, but this message sounds more like someone trying to sell me something than an objective scientist. You can just say no, there is no link (intuitive or not) between geometric algebra and Banach-Tarski.
To answer some of the other points
1. No physicist is particuarly confused by Euler angles, and gimbal lock is not a problem in physics.
2. I'm not sure I agree that physics even uses computable reals or rationals. I would say in reality we use fuzzy confidence intervals mostly and not exact numbers.
3. Maxwell's equations look different when written in geometric algebra style, you get one neat looking equation rather than the traditional 4, but its just a difference of notation, the same stuff is happening just written in a slightly different way.
4. A tensor can represent a spinor if you let it transform under the correct transformation rule. A basic spinor just looks like (https://en.wikipedia.org/wiki/Dirac_spinor) a complex vector which you let the Clifford algebra act on.
More generally everything I've seen from Geometric Algebra enthusiasts is just a weird way of doing fairly standard stuff in special cases of Clifford algebras in slightly weird old-fashioned notation. Pretty much everyone I've seen doing real work just does stuff in the Clifford algebra.
Unfortunately, even though it is said to be a "generalization" of these things, mathematical measure theory has nothing to do with physical units of measure or dimensional analysis.
Because its focus is mostly on measures defined on "non-physical" sets, such as various functional spaces (with applications to integrating differential equations, for instance).
A meter is a displacement vector with a basis vectorthe length of the path travelled by light in a vacuum during a time interval of 1/299,792,458 of a second.
In physics the length of the basis vector is set to 1 if possible which is called 'natural units'
But the SI system is the domain of Metrology, not physics.
The physical quantities are of 2 kinds, as already classified by Aristotle, discrete quantities and continuous quantities.
Examples of discrete quantities are the amount of substance and the electric charge.
The discrete quantities are just counted, so their values are integer numbers. They have a natural unit. Nevertheless, for those that are expressed in very large numbers it may be convenient to choose a conventional unit that is a big multiple of the natural unit, for instance the mole and the coulomb in the SI system of units.
All the continuous physical quantities are derived in some way from the measures of space and time, which is the reason for their continuity. For instance the electric charge is discrete, but the electric current is continuous, because it is the ratio between charge and time and time is continuous.
In order to measure a continuous physical quantity, a unit must be chosen. The unit may be chosen arbitrarily or it may be chosen in such a way as to eliminate universal constants from the formulae that express the relationships between physical quantities.
In either case, the value of a measurement is the result of a division operation between the measured value and the chosen unit, which is a real number, though it is normally approximated by a rational number.
In order to be able to define a division operation on the set of values of a physical quantity that has as a result a scalar, the minimum algebraic structure of that set of values is an Archimedean group.
That means that it must be possible to add and subtract and compare the values of the physical quantity and given two values it is always possible to add one of them with itself multiple times and eventually there will be a multiple greater than the second value (which will determine that the second value lies between two consecutive multiples of the first).
Based on the axioms of Archimedean groups it is possible to devise an algorithm that can multiply a value by a rational number and which can determine that a second value lies between two rational multiples that are as close as desired, producing by passing to the limit a real scalar. Thus any value can be divided by another value chosen as unit.
In practice, all the continuous physical quantities have richer algebraic structures, they are vector spaces over the real numbers, so the division of two collinear vectors is the scalar that multiplies one to give the other.
Nevertheless, the fact that the continuous physical quantities form vector spaces over the real numbers can be demonstrated based only on the supposition that they are Archimedean groups.
So the units of continuous physical quantities are just arbitrarily chosen values of those physical quantities, which are normally vector spaces with one dimension or with more dimensions, while the measured values are just rational approximations of the scalars obtained by division.
This division process is very obvious in the structure of the analog-digital converters used to measure voltages. These ADCs have two inputs, the voltage to be measured and the reference voltage, which is the arbitrarily chosen unit. The ADCs produce a rational number that is the approximate result of the division of the measured voltage by the reference voltage. If the reference voltage is not equal to the conventional unit, i.e. 1 V, the measurement result will be converted by multiplying with an appropriate conversion factor. The division operation can be done in the ADC for example by successive approximation, i.e. by binary search of the two multiples of a fraction of the reference value between which the measured value lies. The fraction of the reference voltage may be generated by a resistive or capacitive divider, while its multiples can be generated by a multiplying DAC.
> The presence of the negative signs in (1) may seem surprising at first, but this is due to the fact that (1) is describing the effect of a passive change of units rather than an active change of the object {x}.
This is where the limits of my brain were reached. Is there a translation of this into category theory terms? Is this where category theory could help formalize units in physics?
However, his paragraph after that is pretty interesting, which I read as sort of treating units as variables since you couldn't combine them, and he only has length, mass, and time for these examples. But then there's an exponent piece? Okay now I'm lost again.
Maybe this is obvious and not what you are asking, but he's just saying that if you increase your unit of measurement by a factor of X, then the number of such units that comprise your object decreases by the same factor.
(Also, this quote is from the Terry Tao blog post that dang links below, not the OP, right?)
I didn't get that impression and it's not really a natural idea. A book about German for English speakers would be at best marginally useful for German speakers trying to learn English. Sachs and Wu's book about GR for mathematicians (which assumes a background in differential geometry) also cautions against expecting the book to work backwards like that.
Okay...I think this might be interesting. I've seen and read a lot of "math for dumb physicists" works, which as a physicist...yeah, I see their point. This could help me understand the math wizards a little better.
I was more math oriented during my studies, and I hated physics (couldn't openly admit that). I still don't get a lot of the physics I was taught, but I did juggle my way out of it using math, learning some formulas and getting a passing grade. Deep inside I admire physicist more, because for them the things that never clicked for me are natural.
imho, high-level physics is harder than pure math. With math you can specialize and focus on some formulas or areas of interest, but this is not really possible with physics. With physics you have to know all the areas of math very well--group theory, differential equations, differential geometry, etc. You have to have know all the math well and all the physics from Maxwell and beyond. It's just much more material involved. To be on the frontier of physics is essentially pure math, plus hundreds of years of physics.
Computational Physics Ph.D. here...I don't know about that. I have written lots of code (not just using off-the-shelf packages) to solve Hamiltonian mechanics and Quantum reactive scattering. OMG, I spent about 30 minutes going through the Hamiltonian mechanics chapter from the point of view of a mathematician and I got lost about half way through. I feel like in my fairly long career I learned just enough of the math to make it work, but don't really understand the math at a fundamental level like I do the physics.
This introduction must assume that the reader already understands physics deeply, right? For instance, the page on Hamiltonian mechanics stated that force is the derivative of momentum with respect to time. I can't imagine how one will understand the intuition behind the definition without having already learned at least college-level physics.
This is derivation, which is different from using momentum to define force. That definition actually has many benefits in Hamiltonian mechanics. The key challenge is to intuitively understand why such definition makes intuitive sense.
Florian Scheck's textbooks in the Springer Graduate Texts in Physics series could also serve as a bridge. Though very challenging for the non-mathematician I've grown quite fond of "Mechanics: From Newton's laws to Deterministic Chaos" and plan to read the other Scheck books in the series as well.
I know mathematics and I hate his series of books.
The thing that irks me most is using higher-level concepts, like the existence of a atom, to illustrate lower level concepts that led to the discovery of the atom in the first place.
A bit on a tangent, I have been always curious as to how much of the modern theoretical physics is just math. That is, how little print space the standard treatment of physical theory could be compressed into if all the purely mathematical stuff is assumed known (or delegated to a separate text).
euler is the last titan of pure raw 'classic' mathematics because gauss was a pretty strong 'theoretical' physicist.
how have the mathematical contributions of quantum physics affected mathematics? have they??
maybe the field that's really lagging in recognizing the implications of "recent" scientific revolution (QM) is philosophy?
finally, I wonder how will the schizm in mathematics that is the IUT (mochizuki's theory) will finally pan out. apparently euler also left stuff behind that took over 70 years to be understood so I ain't holding my breath.
I should also take this chance to mention that I work as a private tutor and I have openings for students! Much more info here: https://nicf.net/tutoring/