Is there a way to meaningfully choose a unique, finite average of a function whose graph matches the description in Wood's quote?
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Motivation:
In a magazine article on problems and progress in quantum field theory, Wood writes of Feynman path integrals, “No known mathematical procedure can meaningfully average an infinite number of objects covering an infinite expanse of space in general. The path integral is more of a physics philosophy than an exact mathematical recipe.”
This article provides a method for averaging an arbitrary collection of objects; however, the average can be any number in a proper extension of the range of these objects. (Note, an arbitrary collection of these objects is a function.)
Question: Suppose anything meaningful has an application in quantum field theory. Is there a way to meaningfully choose a unique, finite average of a function whose graph matches the description in Wood's quote?
Attempt to Define Function Whose Graph Matches Wood's Quote:
As a amateur mathematician, I know nothing about path integrals yet I know about a function with no meaningful average whose graph contains "an infinite number of objects covering an infinite expanse of space".
Suppose $f:\mathbb{R}\to\mathbb{R}$ is Borel. Let $\text{dim}_{\text{H}}(\cdot)$ be the Hausdorff dimension, where $\mathcal{H}^{\text{dim}_{\text{H}}(\cdot)}(\cdot)$ is the Hausdorff measure in its dimension on the Borel $\sigma$-algebra.
If the graph of $f$ is $G$, we define an explicit $f$, where:
- The function $f:\mathbb{R}\to\mathbb{R}$ is everywhere surjective (i.e., $f[(a,b)]=\mathbb{R}$ for all non-empty open intervals $(a,b)$)
- $\mathcal{H}^{\text{dim}_{\text{H}}(G)}(G)=0$
- We take chosen sequences of bounded functions converging to $f$ with the same satisfying and finite expected value w.r.t. a reference point, the rate of expansion of a sequence of each bounded function's graph, and a "measure" of each bounded function's graph involving covers, samples, pathways, and entropy.
Bharathk98
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it's "...a unique..." but "...an everywhere..." in the title and also in the abstract. There, you can't say "...and suppose function f: X->Y." The verb "suppose" needs something after the object, for example "is continuous" or whatever. Also an article like "a" or "the" is missing before "function". Also many many errors in the sequel, you absolutely have to correct the English if you want anyone to read the article. (Even Gmail or Google docs do that automatically, so there's no excuse!)
I typed this on latex overleaf. Overleaf offers edits but requires a premium subscription that I can’t afford. Also, I thought the English isn’t the most important part of a math article. The notations have to make sense.
I’ve written posts with worse English. I think the real reason no one is responding is because of how difficult the problem is.
Respectfully, no one responds because the problem is not well-posed. There is no clear request, and also no satisfying solution. I would just suggest you to study measure theory instead.
Alessandro Iraci, why is the problem not well-posed? If we add enough criteria, the average of an infinite expanse of space can be unique w.r.t a reference point. The magazine article should help you understand the request.
I redid the question.
Epstein and Glaser have given a mathematically rigorous theory of renormalization many years ago. Feynman way a great spirit but also limited, and a little too convinced of himself -- if he didn't know/understand, he would think it can't make sense. He rejected works on the EPR and similar paradox saying it was nonsense, but ppl got the Nobel prize for that later.
*Feynman **was** ...
@MFH What about the paper, "Integration with Filters" by Emanuele Bottazzi and Monroe Eskew? They focused especially on the quote by Wood at the beginning of the post.