Find Algebraic Physics Courses Starting with Classical Mechanics - Dan

In summary, this online quantum mechanics course by Frederic Schuller uses an algebraic approach in an advanced algebraic manner. There are textbooks (mentioned in the original post) that use this approach, and they might be more suitable for what you are looking for.
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topsquark
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I just finished an online YouTube Quantum Mechanics course by Frederic Schuller. The course was in two parts, the one Schuller was doing was theory; it was an approach I had been trying to develop myself and had never found anywhere. I don't know what exactly to call the course label: it was QM Mathematics approached in an (Advanced) Algebraic manner. Here's a link to give you an idea, if you need it.

For lack of a better term I'm referring to it as "Algebraic Physics." I am looking for more courses using Algebra, say, starting with Classical Mechanics. Does anyone know of any textbooks using this approach?

Thanks!

-Dan
 
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  • #3
It looks standard! And it is not algebra but analysis.
 
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  • #4
malawi_glenn said:
https://www.amazon.com/dp/0750306068/?tag=pfamazon01-20
There is supposedly a new 3rd edition comming soon (hopefully)

https://www.amazon.com/dp/9401781982/?tag=pfamazon01-20
There is also a part II
Actually, it does look like a good text. But what I am after is somewhat different. For example, the QM treatment by Schuller starts off with functions acting on Banach spaces and proceeds from there to Borel measures, Lebesgue integrals, and so on and so on. His lecture on the Algebra of whether we can actually define a self-adjoint momentum operator was fascinating. I am reasonably sure this was a graduate Mathematics course, but I'm not certain. There was a practical part of the class that wasn't in the videos that seemed like it was a more standard exposition.

-Dan
 
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  • #7
Koopman-von Neumann mechanics is operator based.
https://en.wikipedia.org/wiki/Koopman–von_Neumann_classical_mechanics

There are differential geometry based approaches to machanics if that is what you are looking for.
 
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  • #8
Have you looked at Analysis and Functional analysis books?
 
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  • #9
martinbn said:
Have you looked at Analysis and Functional analysis books?
Yes. They give the background, it's the application to Physics that seems to be what's confusing me. (What that probably means is that I do indeed need to spend more time with the actual Mathematics.) But there are Physical conditions that enter when applying it and those are what is whetting my interest the at the moment. I can reasonably handle the Intro level stuff (ie. the Intro level texts on Algebra, Topology, and Differential Geometry.) For the more advanced material, I'm seeing what I need to learn to apply then I'll try to take it from there.

(For example, I already know that I need to spend more time with homotopy groups in QFT, but the main part of the topic is further than I've been able to work with so far.)

-Dan
 
  • #10
topsquark said:
For lack of a better term I'm referring to it as "Algebraic Physics." I am looking for more courses using Algebra, say, starting with Classical Mechanics. Does anyone know of any textbooks using this approach?
Do you really mean algebraic? Or do you mean some combination of (abstract) algebra, differential geometry, and (functional) analysis?

I am going to interpret your request very loosely.

Mathematical books on classical mechanics include:
"Mathematical Methods of Classical Mechanics" by Arnold;
"Classical Mathematical Physics" by Thirring;
"Mechanics" by Scheck.

For non-relativistic quantum mechanics, there is the outstanding "Quantum Field Theory for Mathematicians" by Hall, which assumes that the reader has undergrad background in measure theory and Hilbert spaces. It teaches grad-level functional analysis alongside QM. Did I mention that this book is outstanding?

Mathematical books on quantum field theory:
"What Is a Quantum Field Theory? A First Introduction for Mathematicians" by Talagrand;
"Quantum Field Theory: A Tourist Guide for Mathematicians" by Folland.

The books by Talagrand and Folland use mathematical rigour where possible, and where physicists' quantum field theory calculations have yet to be made mathematically rigourous (by anyone), they state the mathematical difficulties, and then formally push through the physicists' calculations.

Folland is more condensed than Talgrand, and covers more advanced physics. Folland treats QED, while Talagrand treats models that are not complicated by spin and gauge invariance. One of Talagrand's goals, inspired by Hall's QM book, was to give a treatment of QFT that is more introductory, readable, and detailed than Folland.

A book that applies applies algebraic structures and differential geometry (fibre bundles) to the standard model and non-supersymmetric grand unified theories (GUTs) of particle physics is "Mathematical Gauge Theory With Applications to the Standard Model of Theoretical Physics" by Mark Hamilton.
 
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  • #11
George Jones said:
Do you really mean algebraic? Or do you mean some combination of (abstract) algebra, differential geometry, and (functional) analysis?

I am going to interpret your request very loosely.

Mathematical books on classical mechanics include:
"Mathematical Methods of Classical Mechanics" by Arnold;
"Classical Mathematical Physics" by Thirring;
"Mechanics" by Scheck.

For non-relativistic quantum mechanics, there is the outstanding "Quantum Field Theory for Mathematicians" by Hall, which assumes that the reader has undergrad background in measure theory and Hilbert spaces. It teaches grad-level functional analysis alongside QM. Did I mention that this book is outstanding?

Mathematical books on quantum field theory:
"What Is a Quantum Field Theory? A First Introduction for Mathematicians" by Talagrand;
"Quantum Field Theory: A Tourist Guide for Mathematicians" by Folland.

The books by Talagrand and Folland use mathematical rigour where possible, and where physicists' quantum field theory calculations have yet to be made mathematically rigourous (by anyone), they state the mathematical difficulties, and then formally push through the physicists' calculations.

Folland is more condensed than Talgrand, and covers more advanced physics. Folland treats QED, while Talagrand treats models that are not complicated by spin and gauge invariance. One of Talagrand's goals, inspired by Hall's QM book, was to give a treatment of QFT that is more introductory, readable, and detailed than Folland.

A book that applies applies algebraic structures and differential geometry (fibre bundles) to the standard model and non-supersymmetric grand unified theories (GUTs) of particle physics is "Mathematical Gauge Theory With Applications to the Standard Model of Theoretical Physics" by Mark Hamilton.
Halls sounds pretty much like what I'm looking for. Thanks!

-Dan
 
  • #12
Frabjous said:
Have you considered emailing the prof?
https://people.utwente.nl/f.p.schuller?tab=contact

He has also self-published some lecture notes.
https://www.amazon.com/Books-Frederic-Schuller/s?rh=n:283155,p_27:Frederic+Schuller

I don't think the Schuller in the second link is the Schuller of lectures. The self-published guy seems French, while Schuller is German.

@topsquark

Schuller has a course on classical mechanics, but it is in German. I think one who does not understand German can make do with YouTube's automatically translated subtitles. I have seen one lecture in the German series, and I was able to understand it using subtitles. Link:

As others have commented, the quantum mechanics course was mainly analytical, not algebraic. If you want an algebra heavy treatment of classical mechanics, you can take a look at the book _Classical Dynamics: A Modern Perspective_ by Sudarshan and Mukunda. I haven't read it. The book takes a group theoretic approach. Link: https://www.amazon.in/dp/9380250770/

Also, have you mailed Prof. Schuller? Did he reply?
 
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Apoorv Potnis said:
I don't think the Schuller in the second link is the Schuller of lectures. The self-published guy seems French, while Schuller is German.

@topsquark

Schuller has a course on classical mechanics, but it is in German. I think one who does not understand German can make do with YouTube's automatically translated subtitles. I have seen one lecture in the German series, and I was able to understand it using subtitles. Link:

As others have commented, the quantum mechanics course was mainly analytical, not algebraic. If you want an algebra heavy treatment of classical mechanics, you can take a look at the book _Classical Dynamics: A Modern Perspective_ by Sudarshan and Mukunda. I haven't read it. The book takes a group theoretic approach. Link: https://www.amazon.in/dp/9380250770/

Also, have you mailed Prof. Schuller? Did he reply?

Thanks for the information.

No, I have not received a reply. However, the videos are some 9 years old. For all I know, he's working somewhere else, or even retired, and isn't using that e-mail any more.

About a month ago I discovered "Differential Geometry and Mathematical Physics" by Rudoph and Scmidt, parts I and II. They are essentially what I am looking for. After Dr. Schuller's lectures I am fit to study them. And are they ever expensive! Yeesh! But they are exactly what I'm looking for.

I'm finding that my apparent "block" about Abstract Algebra goes away somewhat upon application to Physics, no matter how abstract, so I have more hopes about self-study in this area now.

Thanks again!

-Dan
 
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  • #14
topsquark said:
Thanks for the information.

No, I have not received a reply. However, the videos are some 9 years old. For all I know, he's working somewhere else, or even retired, and isn't using that e-mail any more.

About a month ago I discovered "Differential Geometry and Mathematical Physics" by Rudoph and Scmidt, parts I and II. They are essentially what I am looking for. After Dr. Schuller's lectures I am fit to study them. And are they ever expensive! Yeesh! But they are exactly what I'm looking for.

I'm finding that my apparent "block" about Abstract Algebra goes away somewhat upon application to Physics, no matter how abstract, so I have more hopes about self-study in this area now.

Thanks again!

-Dan
Just in the very unlikely case that you don't know, Prof. Schuller has a lecture course on the geometrical anatomy of theoretical physics: . It covers some material that the Rudolph--Schmidt volumes cover. Prof. Schuller also has a general relativity course, with some overlap with the geometrical anatomy course. As with the quantum mechanics course, lecture notes for all the courses are available as well.

Also, just for info, he is a professor at the University of Twente.
 
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  • #15
Apoorv Potnis said:
Just in the very unlikely case that you don't know, Prof. Schuller has a lecture course on the geometrical anatomy of theoretical physics: . It covers some material that the Rudolph--Schmidt volumes cover. Prof. Schuller also has a general relativity course, with some overlap with the geometrical anatomy course. As with the quantum mechanics course, lecture notes for all the courses are available as well.

Also, just for info, he is a professor at the University of Twente.

Yes, I've seen "anatomy" series, as well as his Quantum Mechanics lectures. I haven't yet looked into the GR course, nor his Topology videos. (I'm not a Relativist, though I'm curious, so I'll get to that eventually. As for Topology, that's another field I can't seem to get too far into on my own, so I'll definitely be watching that one to see if he can work the same "magic" with me that he has with Differential Geometry. I like his style and wish that I could've attended one of his courses.)

Apparently, then, he's still at Twente. In that case, he's decided not to respond to my e-mail. 😢 Ah, well. I'm neither a colleague, nor a student, so I get it.

Thanks again for the info!

-Dan
 

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