Functional Analysis for Physics in 2024

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SUMMARY

Functional analysis (FA) remains a relevant topic for physicists in 2024, despite debates about its necessity. Key concepts such as Cayley transforms, extensions of operators, and the use of Lebesgue integrals are discussed, with Wald's "Quantum Field Theory in Curved Space Time" cited as a resource that employs advanced FA techniques. The conversation highlights the distinction between theoretical and applied physics, questioning the depth of FA knowledge required for contemporary research. Ultimately, the discussion emphasizes the ongoing importance of FA in understanding modern physics.

PREREQUISITES
  • Understanding of functional analysis concepts, including Cayley transforms and extensions of operators.
  • Familiarity with Lebesgue integrals and their applications in analysis.
  • Knowledge of quantum mechanics and its mathematical foundations.
  • Basic comprehension of algebraic structures such as Banach algebras and Weyl algebras.
NEXT STEPS
  • Study "Quantum Field Theory in Curved Space Time" by Wald to explore the application of functional analysis in physics.
  • Learn about the Cayley transform and its significance in functional analysis.
  • Research the Stone von Neumann Theorem and its implications in quantum mechanics.
  • Investigate the role of Banach algebras in modern theoretical physics.
USEFUL FOR

This discussion is beneficial for physicists, mathematicians, and students interested in the intersection of functional analysis and theoretical physics, particularly those exploring advanced topics in quantum mechanics and field theory.

psterphysics
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TL;DR
How much functional analysis is needed in 2024?
Physicists provided the motivation for studying functional analysis (FA) 100 years ago. But is an in depth understanding of FA necessary in 2024? A slightly different way of putting it would be: is there any important work being done by physicists that requires working knowledge of all the machinery, e.g., Caley transforms, extensions of operators, etc.? Or is it more something that is better left to the mathematicians who want to dot all the i's and cross all the t's?

Wald's Quantum Field Theory in Curved Space Time makes use of considerably more FA machinery than most physics books, but even there a lot of it seems fifty years old. And what kind of FA does one want to spend time learning?

Does want to spend ones time learning about FA in general analysis terms (Lebesgue integrals)? Or is a breezier approach (without Lebesgue) like Kreyszig sufficient?

For the record, I took a course with baby Rudin, but I've never seen a physicist use a Lebesgue integral. Even a lot of mathematicians think that FA is best thought of as algebraic rather than analytical.

What kind of FA are people using beyond the old von Neumann basics that everybody knows?
 
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psterphysics said:
TL;DR Summary: How much functional analysis is needed in 2024?

Physicists provided the motivation for studying functional analysis (FA) 100 years ago. But is an in depth understanding of FA necessary in 2024? A slightly different way of putting it would be: is there any important work being done by physicists that requires working knowledge of all the machinery, e.g., Caley transforms, extensions of operators, etc.?
By "important", do you mean applied versus theoretical? A lot of theoretical physics eventually becomes applied in significant ways.
 
I explain what I meant by "important" in the next question. Of significant interest to working physicists rather than mathematicians.
 
Why would 2024 be different than 2023 or 1996 or any other time?
 
Of interest to theoretical mathematicians regarding "dot all the i's and cross all the t's" would be the proofs of mathematical theorems. Physicists might also be interested in the proofs if they lend some insight into the subject that translates to a better understanding of the physics. I am not a physicist. Does Wald's book concentrate on mathematical proofs? If so, do the proofs increase your insight onto the physics.
Of course, you might not be interested in the subject of Wald's book, but that is another issue.
 
Just out of curiosity, what FA is used in Wald's book?
 
martinbn said:
Just out of curiosity, what FA is used in Wald's book?
Hadamard states, Weyl algebras, Banach algebras, Stone Theorem, Stone von Neumann Theorem
 
martinbn said:
Why would 2024 be different than 2023 or 1996 or any other time?
100 years ago there was a need to put the math of QM on a firmer foundation. Some might assume that that has largely been accomplished. Physics advances. Not sure I understand what you are getting at.
 
psterphysics said:
100 years ago there was a need to put the math of QM on a firmer foundation. Some might assume that that has largely been accomplished. Physics advances. Not sure I understand what you are getting at.
100 years ago to study general relativity you needed to learn some differential geometry. Now, 100 years later, physics has advanced and to study general relativity you still need to learn some differential geometry.

My questions was: why do you think that FA was useful and needed then but not now in 2024?
 
  • #10
Where did I say FA was not "useful" or "needed"? I only asked a question. I've elaborated with a bunch of buzzwords. Do you know what a Caley transform is? Have you ever asked yourself whether it was worth your time to find out? (Not rhetorical questions. Please answer yes/no.)
 
  • #11
psterphysics said:
Where did I say FA was not "useful" or "needed"? I only asked a question. I've elaborated with a bunch of buzzwords. Do you know what a Caley transform is? Have you ever asked yourself whether it was worth your time to find out? (Not rhetorical questions. Please answer yes/no.)
Yes. (Cayley transform)
 
  • #12
Why did you decide it was worth your time to study it? What book did you use?
 

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