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In summary, the gamma (or beta) function(s) have various physical applications, particularly in string theory and quantum physics. While these applications primarily involve real numbers, there are some cases where complex variables are used, such as in dimensional regularization in QFT. However, even in these cases, the focus is still on real arguments of the gamma function.
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nomadreid
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- TL;DR Summary
- All the applications I have found for physical applications of the gamma seem to be, to my untrained eye, only for real z (and mostly positive), with the Riemann zeta function being the only obvious exception. What am I missing?
An example of physical applications for the gamma (or beta) function(s) is
http://sces.phys.utk.edu/~moreo/mm08/Riddi.pdf
(I refer to the beta function related to the gamma function, not the other functions with this name)
The applications in Wikipedia
https://en.wikipedia.org/wiki/Gamma_function#Applications
https://en.wikipedia.org/wiki/Beta_function#Applications
seem also to be for real numbers (without non-positive integers), and mostly positive, except of course for the Riemann zeta function.
This is likely due to a lack of depth on my part, so could anyone indicate at least briefly where non-real complex values might be in the domain of a gamma function used in a physical application? Associated links with fuller explanations, while not absolutely necessary, are always an appreciated extra bonus. Thanks.
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Demystifier
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The Veneziano amplitude in physics is a Beta function of real variables (momenta of particles). However, in string theory this amplitude is derived from certain integrals over the 2-dimensional string world-sheet naturally described in terms of complex analysis.
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nomadreid
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Thanks, Demystifier.
If I understand correctly, the string theory description of the world-sheet includes the Beta function over complex variables?
[That is, it is not the case that the connection is only indirect:
classical description: Beta (but no complex)
string theory description: complex (but no Beta).]
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Demystifier
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nomadreid said:
Thanks, Demystifier.
If I understand correctly, the string theory description of the world-sheet includes the Beta function over complex variables?
[That is, it is not the case that the connection is only indirect:
classical description: Beta (but no complex)
string theory description: complex (but no Beta).]
No, I would say the connection is only indirect. It's probably not what you are interested in. But I'm not sure, because I don't know why are you interested in complex variables in the first place. Physically measurable quantities are always real, so it's hard to expect a "direct connection" (whatever that means) of complex variables to physics.
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nomadreid
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Thanks, Demystifier. But as far as I can see, variables over ℂn seem to be all over the place in quantum physics. That is, obviously the observables are real, but the connection doesn't have to be that direct. In other words, are there equations in which Gamma must range over ℂ (or Beta over ℂ2)?
For some examples as to how indirect I would be satisfied with:
https://physics.stackexchange.com/q...ons-of-a-complex-variable-used-for-in-physics
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Demystifier
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nomadreid said:
Thanks, Demystifier. But as far as I can see, variables over ℂn seem to be all over the place in quantum physics. That is, obviously the observables are real, but the connection doesn't have to be that direct. In other words, are there equations in which Gamma must range over ℂ (or Beta over ℂ2)?
For some examples as to how indirect I would be satisfied with:
https://physics.stackexchange.com/q...ons-of-a-complex-variable-used-for-in-physics
In the link someone mentioned dimensional regularization in QFT, which involves analytic continuation in complex plane. These dimensional regulated quantities are often expressed in terms of Gamma functions of complex variables. But in the end, it seems that one again is only interested in real arguments of the gamma function.
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nomadreid
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Thanks again, Demystifier. I suspect that if you do not know of any, then they probably don't exist. (Mark of high esteem.)
FAQ: Applications of complex gamma (or beta) functions in physics?
1. What are complex gamma and beta functions?
Complex gamma and beta functions are mathematical functions that extend the concept of factorial to complex numbers. They are denoted as Γ(z) and B(z,w) respectively, where z and w are complex numbers.
2. How are complex gamma and beta functions used in physics?
Complex gamma and beta functions are used in physics to solve various problems related to quantum mechanics, statistical mechanics, and fluid dynamics. They are particularly useful in calculating probabilities, wave functions, and energy levels in quantum systems.
3. Can you provide an example of an application of complex gamma and beta functions in physics?
One example is the calculation of the partition function in statistical mechanics. The partition function is given by the integral of the Boltzmann factor, which involves the complex gamma function. This integral can be solved using the residue theorem, which simplifies the calculation and allows for faster and more accurate results.
4. Are there any limitations to the use of complex gamma and beta functions in physics?
Yes, there are limitations to the use of complex gamma and beta functions in physics. They are not defined for all complex numbers, and their values can become infinite or undefined for certain values of z and w. Additionally, they may not always provide physically meaningful solutions, so they should be used with caution and in conjunction with other methods.
5. Are there any other fields of science where complex gamma and beta functions are used?
Yes, complex gamma and beta functions are also used in other fields of science such as engineering, economics, and computer science. In engineering, they are used to solve problems related to signal processing and control systems. In economics, they are used in the study of financial models and pricing options. In computer science, they are used in the development of algorithms and data analysis techniques.
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