Is representation theory worthwhile for quantum?

[Twigg]

Recently I read some comment on Sakurai's book (which I have not read) that the writer of said comment didn't understand part of the text until they understood irreducible representations. I do not know to what they were referring, but it piqued my interest in representation theory. My question is: will an understanding of representations allow me to solve problems faster/more efficiently and/or reveal experimentally relevant aspects of quantum theory that are not accessible without an understanding of representations? As far as background, I have a good grasp on generators and Lie algebra stuff as pertains to quantum, and a solid understanding of the basics like Hilbert space and linear and what'not. My background is weaker with finite groups and product states.

The cliff-notes version of irreducible representations seems clear enough, that if your matrix can be diagonalized into blocks then your system has structure you can exploit and factor into smaller chunks. It seems even a little self-evident, from a purely algebraic point of view. Does knowing about representation-y stuff let you decompose your system with less time and effort? Can someone supply an example from an applied problem? I would really appreciate it!

 

[vanhees71]

If there's a single advanced mathematical topic I'd recommend to learn then it's the theory of Lie groups and Lie algebras and their representation. For me it's the only way to understand why non-relativistic and relatistic QT looks the way it looks. Last but not least it's also great fun to learn this subject!

 

[A. Neumaier]

Representation theory is the key to understanding the finer points in much of quantum mechanics, from spin to the spectrum of hydrogen to spectroscopy in general to elementary particles. That it may dramatically simplify calculations is only one of the many benefits. My free online book discusses some of the more conceptual advantages of using Lie algebras and Lie groups in quantum mechanics.

 

[vanhees71]

I would go further and say that you cannot understand why the operators and their algebra defining a quantum system take the form they do, while it's immediately clear when the group-theoretical approach is used. E.g., the Heisenberg algebra for a one-dimensional system (usually the starting point of QM 1 lectures), $[\hat{x},\hat{p}]=\mathrm{i} \hbar$, follows from the definition of the momentum operator as the generator for spatial translations.

 

[Twigg]

Thanks for the replies!

@vanhees71 I agree with you that the study of Lie algebra adds a lot to one's understanding of quantum, though I haven't done any relativistic quantum yet (but since I Lie algebras help dissect the Lorentz group, I have some idea of what their utility is for relativistic quantum). Personally, I've used Lie theory to understand different pictures, how the time evolution (or spatial, angular, or whatever evolution operators) work and such. But I've managed to do that without knowing about representations. Do representations take it further? I already feel confident with Lie algebras and groups in and of themselves, from a couple of years of studying them for applications in symmetry methods for partial differential equations (this didn't require representation theory, hence why I'm unfamiliar).

@A. Neumaier Much appreciated! I will read the relevant parts as soon as I can.This looks like an awesome resource, overall. Again, thanks!

 

[A. Neumaier]

for applications in symmetry methods for partial differential equations (this didn't require representation theory

Unitary representation theory is only relevant in the quantum setting. The corresponding classical setting are representations in algebras of differential operators. The connection is given by geometric quantization, which roughly turns the classical representations on coadjoint orbits into unitary representations on corresponding Hilbert spaces. In many cases, the two representation theories are very close (cf. Kirillov's method of orbits).

 

[Twigg]

Unitary representation theory is only relevant in the quantum setting. The corresponding classical setting are representations in algebras of differential operators. The connection is given by geometric quantization, which roughly turns the classical representations on coadjoint orbits into unitary representations on corresponding Hilbert spaces. In many cases, the two representation theories are very close (cf. Kirillov's method of orbits).

This helps a lot. Thank you! I believe there's a lot more for me to learn than I initially thought. Your book should help a lot on that front as well. Thanks!

 

[A. Neumaier]

there's a lot more for me to learn than I initially thought.

There is a huge literature on related, physics-inspired mathematics. Since you are interested in experimental physics, one (demanding) intermediate goal for you could be to understand the theoretical basis of the fractional quantum Hall effect.