|Quantum Field Theory|
|... no spooky action at a distance (Einstein)|
|Relativistic Quantum Mechanics|| Klein-Gordon Equation|
|The Dawn of QFT|| Spinors|
Feynman Slash Notation
Conformal Field Theory
Countdown to the Standard Model
|From a framework to a model|| Yang-Mills Theory|
|Semi-Classical Gravity and the Dark Age|| Hawking Radiation|
Problems with the Standard Model
|Beyond the Standard Model|| Beyond the Standard Model|
Theory of Everything
|Related||De Donder-Weyl Theory|
Renormalisation refers to replacing infinite results of a an infinite summation with a finite number through logically sort-of sensible means. A classical, intuitive example of Renormalisation is a Ramanujam Sum.
ExamplesEdit this section
Ramanujam SummmationEdit this section
Take, for example, the classical example, of the sum of all positive integers:
Clearly, thus sum diverges. However, notice that if one would use the (Wrongly, actually, to speak rigorously) definition Riemann Zeta Function that only applies to positive integers, that is,
We see that this is just . In reality, however, the zeta function is not defined this way for negative integers. If we use the actual definition of the Riemann Zeta Function to calculate this , then we see that the result is . Therefore, we renormalise this sum to .
Of course, it is obvious that this is not rigorous, however, it is very useful from a Physics point of view.
Ramanujan wrote it for the case going to infinity:
where is a constant specific to the series and its analytic continuation and the limits on the integral were not specified by Ramanujan, but presumably they were as given above. Comparing both formulae and assuming that tends to 0 as tends to infinity, we see that, in a general case, for functions with no divergence at = 0:
where Ramanujan assumed . By taking we normally recover the usual summation for convergent series. For functions with no divergence at , we obtain:
was then proposed to use as the sum of the divergent sequence.
In particular, the sum of all positive integers is
where the notation indicates Ramanujan Summation.
External LinksEdit this section
- Neumaimer, Arnold. Renormalization without inﬁnities – an elementary tutorial. http://www.mat.univie.ac.at/~neum/ms/ren.pdf.
- Motl, Lubos (2014). "Sum of integers and oversold common sense". The Reference Frame. http://motls.blogspot.in/2014/01/sum-of-integers-and-oversold-common.html?m=1.