Functional Analysis for Physics in 2024

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Discussion Overview

The discussion revolves around the relevance and necessity of functional analysis (FA) for physicists in 2024. Participants explore whether an in-depth understanding of FA is essential for current work in physics, particularly in relation to theoretical and applied aspects. The conversation touches on specific mathematical tools and concepts within FA, as well as the historical context of its application in physics.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory

Main Points Raised

  • Some participants question the necessity of a comprehensive understanding of FA for physicists today, suggesting it may be more relevant for mathematicians.
  • There is a discussion about the types of FA that might be useful, including whether a general analytical approach or a more practical one is preferable.
  • One participant notes that Wald's book on Quantum Field Theory in Curved Space Time employs more FA than typical physics texts, raising questions about the relevance of this material.
  • Another participant inquires about the specific FA concepts used in Wald's book, listing several advanced topics such as Hadamard states and Weyl algebras.
  • Some participants express curiosity about the historical context of FA's importance in physics, questioning why its relevance might differ over time.
  • There is a challenge regarding the assumptions about FA's utility, with one participant clarifying that they did not claim FA was not useful or needed.
  • Questions arise about the motivations for studying specific FA concepts, such as the Cayley transform, and whether they provide insights into physics.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the necessity of FA for physicists in 2024. Multiple competing views are presented regarding the importance and application of FA in current physics work.

Contextual Notes

Participants express uncertainty about the evolving role of FA in physics and whether historical needs for FA have changed. There are also unresolved questions about the specific mathematical tools that are currently relevant.

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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