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rick1138
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I am sure that everyone here is familiar with the story of Gabriel Veneziano's initial push into string theory by noticing Euler's beta function in a math book. Does anyone happen to know what the book was?
What Book Inspired Gabriel Veneziano's Discovery of String Theory?
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In summary, Gabriel Veneziano noticed Euler's beta function in a math book and realized that it was a form of the gamma function. He then used the gamma function to help him understand renormalization calculations in the theory of the SM.
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selfAdjoint
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rick1138 said:
Gee, almost any advanced calculus book or table of integrals would have it. The beta function is an important one, just like the gamma function from which it derives.
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I'm sorry,SA,but it's the other way around,at least the names would indicate that:
[tex] B(p,q)=:\int_{0}^{1} x^{p-1}(1-x)^{q-1} \ dx \ ,\mbox{Re(p)}>0,\mbox{Re(q)}>0 [/tex]
is called Eulerian Integral of the First Kind ("Beta Euler").
[tex] \Gamma (z)=:\int_{0}^{\infty} e^{-t}t^{z-1} \ dt \ , \mbox{Re(z)}>0 [/tex]
is called Eulerian Integral of the Second Kind ("Gamma Euler").
[tex] \Gamma (z)=\lim_{n\rightarrow +\infty} n^{z}B(z,n+1) \ , \ n\in\mathbb{N} [/tex]
[tex] B(p,q)=\frac{\Gamma (p)\Gamma (q)}{\Gamma (p+q)} \ , \ \mbox{Re(p)}>0,\mbox{Re(q)}>0 [/tex]
To the OP:there's only one way to find out:a(n) (auto)biography of Mr.Veneziano.
Daniel.
[tex] B(p,q)=:\int_{0}^{1} x^{p-1}(1-x)^{q-1} \ dx \ ,\mbox{Re(p)}>0,\mbox{Re(q)}>0 [/tex]
is called Eulerian Integral of the First Kind ("Beta Euler").
[tex] \Gamma (z)=:\int_{0}^{\infty} e^{-t}t^{z-1} \ dt \ , \mbox{Re(z)}>0 [/tex]
is called Eulerian Integral of the Second Kind ("Gamma Euler").
[tex] \Gamma (z)=\lim_{n\rightarrow +\infty} n^{z}B(z,n+1) \ , \ n\in\mathbb{N} [/tex]
[tex] B(p,q)=\frac{\Gamma (p)\Gamma (q)}{\Gamma (p+q)} \ , \ \mbox{Re(p)}>0,\mbox{Re(q)}>0 [/tex]
To the OP:there's only one way to find out:a(n) (auto)biography of Mr.Veneziano.
Daniel.
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selfAdjoint
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Your last relation was the one I was thinking of. Gamma is actually better known than Beta, and that formula for B(p,q) is the one I have usually seen in nonspecialist contexts. And I believe it was this form that Veneziano got his idea from too.
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I guess i'll have to look into Whittaker & Watson.Maybe they give a short history,too.The formulas i posted were from some notes taken a long time ago.
Daniel.
P.S.I knew you meant that formula
Daniel.
P.S.I knew you meant that formula
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I strongly reccomend Whittaker E.T. and Wattson G.N. "A course of Modern Analysis",CUP (i have the 4-the edition,1927),for the best account on Euler functions I've ever seen*.
These functions were probably looked upon in a math book (maybe even W & W) by Veneziano,when he tried to clear up for himself all those renormalization calculations in the SM...
Daniel.
P.S.(as an edit) *On page 259 (of the IV-th ed.),as the first reference,the book by N.Nielsen "Handbuch der Theorie der Gamma-Funktion" (Leipzig,1906) is said to give a complete bibliography.
These functions were probably looked upon in a math book (maybe even W & W) by Veneziano,when he tried to clear up for himself all those renormalization calculations in the SM...
Daniel.
P.S.(as an edit) *On page 259 (of the IV-th ed.),as the first reference,the book by N.Nielsen "Handbuch der Theorie der Gamma-Funktion" (Leipzig,1906) is said to give a complete bibliography.
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rick1138
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This was a trivia question, not a question of the type, "What are the beta and gamma functions?".
1. What is String Theory?
String Theory is a theoretical framework in physics that attempts to reconcile the principles of quantum mechanics and general relativity. It proposes that the fundamental building blocks of the universe are tiny, vibrating strings rather than point-like particles.
2. How did String Theory originate?
String Theory originated in the late 1960s and early 1970s from the study of dual resonance models of particle interactions. It was further developed in the 1980s by physicists such as John Schwarz, Michael Green, and Edward Witten.
3. What are the main principles of String Theory?
The main principles of String Theory include the concept of extra dimensions (beyond the three dimensions of space and one of time), the idea of strings as the fundamental building blocks of the universe, and the notion of supersymmetry (that each particle has a "superpartner").
4. How is String Theory different from other theories in physics?
String Theory differs from other theories in physics in that it attempts to provide a unified framework for understanding the fundamental forces of nature (gravity, electromagnetism, strong and weak nuclear forces) rather than treating them as separate entities. It also incorporates concepts from both quantum mechanics and general relativity.
5. What are some current challenges and criticisms of String Theory?
Some current challenges and criticisms of String Theory include the lack of experimental evidence to support its predictions, the complexity and difficulty of its calculations, and the fact that it has not yet produced any unique testable predictions that can be confirmed or disproven by experiments. Additionally, some argue that it may be impossible to ever prove or disprove String Theory due to the extremely small scales at which it operates.
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