There is always the classical lecture notes by Dolgachev on this. https://dept.math.lsa.umich.edu/~idolga/physicsbook.pdf
I didn't enjoy them, even though having a math background
Physics for Mathematicians – Introduction
51–60 of 63 posts
Re: Physics for Mathematicians – Introduction
#52Earlier quoted context omitted.
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.
Why?
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).
Re: Physics for Mathematicians – Introduction
#53Hi, 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 tak…
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.
Re: Physics for Mathematicians – Introduction
#54A 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).
Re: Physics for Mathematicians – Introduction
#55Hi, 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 tak…
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…
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.
EDIT: Should be fixed now!
Re: Physics for Mathematicians – Introduction
#56Skimmed 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…
> 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.
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Re: Physics for Mathematicians – Introduction
#57Earlier quoted context omitted.
IMHO that is the result of Gibbs style vectors and the cross product only being validated in R^3 Lie groups and geometric algebra remove a lot of problems. It also applies to differential calculus and ML methods like back propagation and gradient decent. Gibbs style vectors and the cross are convenient as they tend to match our visual intuitions. But lots of the 'physics isn't real math' claims just don't understand…
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.
Re: Physics for Mathematicians – Introduction
#58This 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.
If you know F=ma, p=mv, and by calculus that a=v', all of which is high school level, then the time derivative of momentum is clearly force.
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.
Re: Physics for Mathematicians – Introduction
#59Hi, 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 tak…
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…
(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.
Re: Physics for Mathematicians – Introduction
#60Hi, 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 tak…
Man oh man ... this is the good stuff. Thank you.