QM vs. GR

[BlackHole]

General relativity and quantum mechanics use a different language toward the universe. Relativity is a mathematical machine which uses tensors (generalization of vectors) and extensive differential geometry (including differential forms, fibre bundles and connections) to explain gravity on the large scale.

 

Quantum mechanics is also a mathematical machine but uses quite different math. QM describes the microscopic universe using linear algebra, fourier analysis, a lot of focus on abstract algebra (especially group theory and lie groups), complex analysis and functional analysis. Geometric algebra (clifford algebra with spinors) can also be used instead of differential geometry.

 

So which theory is more difficult to most physicists? Which one is the most studied?

[Klaynos]

I would say quantum is harder for most people to visulise, and therefore harder for most to understand.

[BlackHole]

I think general relativity is more difficult, especially the tensor machinery. General relativity puts more emphasis on the calculus, quantum mechanics more on algebra.

[swansont]

QM has more application to terrestrial technology, so it's studied more. Every physicist learns QM to some level. Few learn GR.

[Meir Achuz]

QM has a large number of tractable problems that have definite, testable experimental

and practical consequences. GR has very few of any of the above.

Even some seemingly simple phenomena are still intractable in GR.

The GR equations are deceptively simple.

Their solution in most cases of interest have still not been worked out.

[BlackHole]

General relativity is difficult. The theory was originally built upon ten so-called "coupled hyperbolic-elliptic nonlinear partial differential equations". Tensors and differential forms (including fiber bundles and connections) is not easy.

[J.C.MacSwell]

QM has more application to terrestrial technology, so it's studied more. Every physicist learns QM to some level. Few learn GR.

 

That's what I was thinking.

"follow the money" comes to mind.

 

They are both counterintuitive, especially when we are taught Newtonian physics first and told when young not to believe in magic.

[Tom Mattson]

David Goodstein, the Caltech professor who had the PBS series called "The Mechanical Universe", called GR "the most fiercely difficult subject in physics mathematically". I'm not so sure about it being worse than Jackson't electrodynamics, but it is sure a lot more difficult than QM.

[Severian]

I don't see how GR can be regarded as harder than QM since we really don't understand QM at all (what makes the wavefunction collapse? how do we define an operator as separate from the system?). Even if you are willing to sweep a lot under the carpet (in the name of testability) then GR is just a classical field theory while QM has a lot of new ideas like anti-commutation of operators.

 

And GR is certainly a lot less complicated than Quantum Field Theory. In fact it is turning GR into a QFT which has caused so many problems.

[J.C.MacSwell]

Hydrodynamics, pretty much Newtonian and yet it has to be pretty hard on it's "mathematical leading edge". Very difficult or simple depending on what simplifying assumptions are made. I think someone very good in mathematics (I'm pretty good in math, but I suck at "mathematics" which is a shame because it's the language of physics) would get equally overwhelmed at some point in Hydrodynamics, GR, or QM. I guess how they judged where that point came in each might determine what they thought was more difficult.

 

And then there's string "theory"...or is there?

[Tom Mattson]

When I say that GR is more difficult than QM, I mean mathematically. Take graduate level textbooks in each subject: Wald (GR) and Sakurai (QM). When I was in school and doing homework, I could immediately start writing down the solution to all of the problems in QM, and I could usually finish them in the same sitting. Not so with GR.

[BlackHole]

I think QFT is slightly more advanced. Steven Weinberg wrote 3 rigorous volumes about this. QFT uses geometric algebra instead of differential geometry. QFT is the future.

[CPL.Luke]

just because quantum field theory used a different kind of mathmatics to define space and time than GR did, you can't call it the future.

[BlackHole]

I think it's the future. Our understanding of particle physics becomes better all the time, especially after the LHC will give us new physics.