Role of real & functional analysis in physics?

[Fizicks1]

I know complex analysis is of immense help in physics at it aids us in calculating certain integrals much more easily.
But what about real analysis and functional analysis? Are these branches of mathematical analysis of much use in physics? If so, in what branches of physics and how?

 

[micromass]

Functional analysis provides the fundamentals of QM. That doesn't mean that it's necessary to study functional analysis in order to understand QM. But if you want a deep understanding and if you want to find out why the mathematics checks out, then functional analysis is very necessary.

 

[dextercioby]

Real analysis..., well, let's say calculus is everything for theoretical physics. All sorts of calculus.

 

[Fizicks1]

Functional analysis provides the fundamentals of QM. That doesn't mean that it's necessary to study functional analysis in order to understand QM. But if you want a deep understanding and if you want to find out why the mathematics checks out, then functional analysis is very necessary.

Thanks. What about real analysis?

Real analysis..., well, let's say calculus is everything for theoretical physics. All sorts of calculus.

Thanks for the reply, but that was...kinda vague, to say the least. Yes, calculus plays a vital part in physics, but if I can carry out all the computation, why do I need to prove all the theorems? Some more detail would be much appreciated!

 

[Fredrik]

Thanks for the reply, but that was...kinda vague, to say the least. Yes, calculus plays a vital part in physics, but if I can carry out all the computation, why do I need to prove all the theorems? Some more detail would be much appreciated!

If all you're interested in is to calculate stuff, then you don't need to prove many theorems. However, if you adopt the strategy to ignore proofs too early, you will fail to understand basic concepts, and this will make it harder to calculate stuff as well.

Science has taught us that the only way to understand how the world works is to study the theories that make accurate predictions about results of experiments. Since understanding the theory is the only thing that can be considered "understanding reality", a person with a deeper understanding of the theory by definition has a deeper understanding of reality.

 

[Fizicks1]

If all you're interested in is to calculate stuff, then you don't need to prove many theorems. However, if you adopt the strategy to ignore proofs too early, you will fail to understand basic concepts, and this will make it harder to calculate stuff as well.

Science has taught us that the only way to understand how the world works is to study the theories that make accurate predictions about results of experiments. Since understanding the theory is the only thing that can be considered "understanding reality", a person with a deeper understanding of the theory by definition has a deeper understanding of reality.

You are right, going through proofs and learning how to prove is very important as a physicist. In freshman year, all the proofs of the theorems I learned in multivariable calculus and linear algebra I've gone through (unless the proof was beyond the scope of the course). I also took a course on abstract algebra and am now taking a course on computational physics, which is quite proof heavy too.

But all these proofs I talked about above are very different from those in real analysis. How does, say, doing epsilon-delta proofs help a physicist? So far I'm still not getting a solid answer on what (if any) role real analysis specifically plays in physics. "Calculus is important" and "proofs help our understanding" just doesn't cut it. Micromass described the usefulness of functional analysis- it provides us a deeper understanding of QM on a mathematical level. Now what about real analysis?

 

[WannabeNewton]

How the heck do you expect to know functional analysis without knowing real analysis in the first place? It's utility should be obvious just from that.

 

[Fredrik]

To understand the mathematics of QM, you have to be very good at functional analysis. To understand functional analysis, you have to be very good at topology. Most people first encounter topology in a course on real analysis, and then they take a course on topology. Then they study integration theory, and finally functional analysis. Real analysis is simply the natural first step on the path towards topology and functional analysis.

Topology is also useful when you learn differential geometry, which is needed to understand GR.

 

[micromass]

To understand the mathematics of QM, you have to be very good at functional analysis. To understand functional analysis, you have to be very good at topology. Most people first encounter topology in a course on real analysis, and then they take a course on topology. Then they study integration theory, and finally functional analysis. Real analysis is simply the natural first step on the path towards topology and functional analysis.

Topology is also useful when you learn differential geometry, which is needed to understand GR.

There do exist books on functional analysis which don't do topology. Kreyzig is an example of such a book. Of course, if you go to more advanced books, then topology is very necessary. And of course topology is needed for differential geometry too, so there is no reason not to study it (it is insanely beautiful too!)

In my opinion, real analysis (such as found in Rudin) is completely useless. But you can't study functional analysis (or even differential geometry) without knowing it.

 

[Fizicks1]

How the heck do you expect to know functional analysis without knowing real analysis in the first place? It's utility should be obvious just from that.

Well, I wasn't aware that real analysis was a prerequisite of functional analysis. Thanks for pointing that out. Your condescending tone wasn't appreciated though.

To understand the mathematics of QM, you have to be very good at functional analysis. To understand functional analysis, you have to be very good at topology. Most people first encounter topology in a course on real analysis, and then they take a course on topology. Then they study integration theory, and finally functional analysis. Real analysis is simply the natural first step on the path towards topology and functional analysis.

Topology is also useful when you learn differential geometry, which is needed to understand GR.

There do exist books on functional analysis which don't do topology. Kreyzig is an example of such a book. Of course, if you go to more advanced books, then topology is very necessary. And of course topology is needed for differential geometry too, so there is no reason not to study it (it is insanely beautiful too!)

In my opinion, real analysis (such as found in Rudin) is completely useless. But you can't study functional analysis (or even differential geometry) without knowing it.

Thanks for the detailed responses, Fredrik and Micromass! It was very informative, as well as helpful to my future choices in math courses.

 

[Minizhu]

Ok, I have to say real analysis is the base of the functional analysis. Since you know that functional analysis is important to QM, of course real analysis is important. Without the knowledge in real analysis, you really can't understand functional analysis fully. Actually, real analysis is the foundation of all advanced course in mathematics and physics.

 

[Bachelier]

Functional analysis provides the fundamentals of QM. That doesn't mean that it's necessary to study functional analysis in order to understand QM. But if you want a deep understanding and if you want to find out why the mathematics checks out, then functional analysis is very necessary.

Would studying QM make one a better Mathematician? I'm assuming it would be a great application of one's math knowledge.

 

[micromass]

Would studying QM make one a better Mathematician? I'm assuming it would be a great application of one's math knowledge.

It's a real awesome application. But I don't know if it would make you a better mathematician. If you want to advance yourself mathematically, then you should study mathematics texts. If you happen to be interested in physics, then you can do QM. But don't do QM if your only reason is to be a better mathematician.

I'm quite interested in QM because I want to see how mathematics is applied in physics. I don't think it really increased my math knowledge so far, but it did give me an entire new perspective.

 

[Bachelier]

It's a real awesome application. But I don't know if it would make you a better mathematician. If you want to advance yourself mathematically, then you should study mathematics texts. If you happen to be interested in physics, then you can do QM. But don't do QM if your only reason is to be a better mathematician.

I'm quite interested in QM because I want to see how mathematics is applied in physics. I don't think it really increased my math knowledge so far, but it did give me an entire new perspective.

Great answer. Thank you Sir.