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Re: Philosophy of Mathematics

December 19th, 2019, 2:18 pm

Screen Shot 2019-12-19 at 3.16.40 PM.png
"not just a half-dozen experts, but graduate students -" and useful for the non grad elderly and young too.
 
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katastrofa
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Re: Philosophy of Mathematics

December 19th, 2019, 6:14 pm

Some would argue that diagramming mathematical reasoning is a crime. One of the reasons is that, being an artificial representation of the problem, diagrams are an additional source of errors in calculations and can give false predictions, e.g. I once knew (I actually pretended that I didn't) a scientist, who wasn't too sharp, but produced a series of mathematical papers about a miraculous effect in semiconductors - using Feynman diagrams. They formed a research group under the scientist's lead and gave him lucrative funding. After a few years of celebrations and wasted experimental endeavour, a renown expert in the field (old guy, imagine Balrog) looked into the articles sceptically and spotted that the calculus omitted a vertex correction - it cancelled out the effect.
From my experience, Feynman diagrams in solid state physics obscure rather than clarify it.
 
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Re: Philosophy of Mathematics

December 19th, 2019, 6:47 pm

what are those diagrams trying to illustrate? particle dance steps (Feynman's got idea from his visits to the clubs) ?? 
 
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Alan
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Re: Philosophy of Mathematics

December 19th, 2019, 7:06 pm

To calculate the theoretical outcome of experiments in QED, you use perturbation theory: a power series in the dimensionless fine structure constant [$]\alpha[$][$]\approx \frac{1}{137}[$]. The coefficient of each power of [$]\alpha[$] is associated with various diagrams, and, given a diagram, there is a cookbook recipe for writing the corresponding mathematical expression. Roughly, each term is a (multiple) integral times a combinatoric factor, and each "loop" generates a (4d)-integral. So the complexity of any term is more or less associated with number of loops that you see in the diagram. 

For example, an electron has what's called an  anomalous magnetic dipole moment and the leading term in the expansion to calculate it has the diagram shown at the link.

Just ran across this, looks like a nice account: 
Physics and Feynman's diagrams
Last edited by Alan on December 19th, 2019, 7:40 pm, edited 1 time in total.
 
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Cuchulainn
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Re: Philosophy of Mathematics

December 19th, 2019, 7:39 pm

Some would argue that diagramming mathematical reasoning is a crime.
It's too imprecise. 
The only thing you can do is look at them.

Roger Penrose's book .. Roads to Reality is full of diagrams. 
 
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katastrofa
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Re: Philosophy of Mathematics

December 19th, 2019, 8:30 pm

To calculate the theoretical outcome of experiments in QED, you use perturbation theory: a power series in the dimensionless fine structure constant [$]\alpha[$][$]\approx \frac{1}{137}[$]. The coefficient of each power of [$]\alpha[$] is associated with various diagrams, and, given a diagram, there is a cookbook recipe for writing the corresponding mathematical expression. Roughly, each term is a (multiple) integral times a combinatoric factor, and each "loop" generates a (4d)-integral. So the complexity of any term is more or less associated with number of loops that you see in the diagram. 

For example, an electron has what's called an  anomalous magnetic dipole moment and the leading term in the expansion to calculate it has the diagram shown at the link.

Just ran across this, looks like a nice account: 
Physics and Feynman's diagrams
Another example of what I was talking about: "Robert Karplus and Norman Kroll first attempted this type of calculation using Feynman’s diagrams in 1949; eight years later several other physicists found a series of algebraic errors in the calculation, whose correction only affected the fifth decimal place of their original answer. Since the 1980s, Tom Kinoshita (at Cornell) has gone all the way to diagrams containing eight vertices—a calculation involv-ing 891 distinct Feynman diagrams, accurate to thirteendecimal places!"

The only usefulness I can see in Fd is as a source of nerdy jokes to pick up ladies in clubs (I wouldn't be surprised if that was the real Feynman's motivation): Penguin diagram
Image
Last edited by katastrofa on December 19th, 2019, 9:28 pm, edited 1 time in total.
 
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Alan
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Re: Philosophy of Mathematics

December 19th, 2019, 9:19 pm

The diagrams themselves are not the source of the errors. Each diagram represents an integral. It's the evaluation of the integrals that's the hard part and so the leading source of errors. The diagrammatic expansion is exact -- in the sense that it's an exact 1-1 representation of an (asymptotic) power series. Of course, you could miss a diagram or get a combinatoric factor wrong. But, it's like a cookbook -- you can't blame Feynman if somebody didn't follow the recipe correctly or missed an ingredient. For the important calculations, eventually any errors get found and corrected.
 
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katastrofa
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Re: Philosophy of Mathematics

December 20th, 2019, 3:33 am

Someone must have thought similar when giving Americans firearms ;-)

I just think the calculus of predicates is safer. Diagrams apply to specific problems to which they were designed - they refer to objects which are meaningless to mathematical logic. As Cuchulainn would say, it's imprecise.

As the famous tobacco advertisement reads:
Image
 
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Re: Philosophy of Mathematics

December 21st, 2019, 4:04 pm

I would prefer to look at Feynman's work rather than Feynman the showman.

Let's take his original paper on negative probability that caused all the confusion/harm.

https://cds.cern.ch/record/154856/files/pre-27827.pdf

I just wonder how many researchers actually critiqued this off-the-cuff article?

I'll try to distill my understanding soon..
 
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Re: Philosophy of Mathematics

December 22nd, 2019, 1:44 am

Dirac introduced negative probabilities long before Feynman

Already in 1960 Vigier and Terletski seems to be critical to the interpretation of negative probability. The cause is the  silly negative energy as they and other have pointed out "Negative 'probability' densities appeared to be inevitable in relativistic quantum mechanics which includes states with negative energies." 

Dec 2019: Nature Scientific Reports:  "The essence of his idea is that a negative probability results in much less mathematical complications in intermediate steps for the analysis of a given physical event"

"However, these negative probabilities are distinct from the definition of probability with respect to the relative frequency of events, in which case, an operational interpretation is not necessarily straightforward"


who is going to clean up this Feynman mess  

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Cuchulainn
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Re: Philosophy of Mathematics

December 22nd, 2019, 12:48 pm

Dirac introduced negative probabilities long before Feynman

Of course, It doesn't resolve the issue. Should we blame Dirac? :-)

Dirac also had the brain wave for delta function which evolved into the theory of distributions by Laurent Schwartz and S.L. Sobolev. Great strides in PDE were made.
 
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Re: Philosophy of Mathematics

December 22nd, 2019, 1:10 pm

Blame the sea?  (the Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy). 

"Forced to make a recommendation about who should receive the prize first, Einstein put Schrödinger slightly ahead of Heisenberg. Once again, Dirac was not mentioned and, in fact, was never nominated by Einstein."

" Schrödinger received 41 nominations; Heisenberg, 29; and Dirac, just 3. "

Paul Dirac and the Nobel Prize in Physics

.
 
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Alan
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Re: Philosophy of Mathematics

December 22nd, 2019, 5:32 pm

who is going to clean up this Feynman mess

First of all, I will admit to either having never read, or forgot ever reading, Feynman's paper on Negative Probability

That said, having just briefly gone through it, I don't see what the big deal is.
As Feynman makes clear, the issue is mainly one of mathematical convenience in intermediate calculations.

The "final" probabilities are never negative.
Suspect any mess is a post-Feynman Game of Telephone effect.
 
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Re: Philosophy of Mathematics

December 22nd, 2019, 6:36 pm



Suspect any mess is a post-Feynman Game of Telephone effect.
"Just the place for a Snark!" the Bellman cried,
   As he landed his crew with care;
Supporting each man on the top of the tide
   By a finger entwined in his hair.

"Just the place for a Snark! I have said it twice:
   That alone should encourage the crew.
Just the place for a Snark! I have said it thrice:
   What I tell you three times is true."
 
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Alan
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Re: Philosophy of Mathematics

December 22nd, 2019, 7:38 pm

So, negative probabilities as per Feynman are clearly kosher. Any others are a terrible heresy and the speaker is likely a witch (gender neutral). I suggest the ordeal by water test to find out when in doubt. 
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