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Quantum Field Theory
... no spooky action at a distance (Einstein)
Early Results
Relativistic Quantum Mechanics Klein-Gordon Equation
Dirac Equation
The Dawn of QFT Spinors
Spin
Feynman Slash Notation
Antimatter
Klein-Gordon Field
Dirac Field
Renormalisation
Grassman Variable
Conformal Field Theory
Countdown to the Standard Model
From a framework to a model Yang-Mills Theory
Quantum Electrodynamics
Quantum Chromodynamics
Electroweak Theory
Higgs Mechanism
Standard Model
Semi-Classical Gravity and the Dark Age Hawking Radiation
Chandrashekhar Limit
Inflation
Problems with the Standard Model
Outlook
Beyond the Standard Model Beyond the Standard Model
Quantum Gravity
Theory of Everything
Related
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:

$\sum_{k=1}^\infty k$

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,

$\zeta(s)= \sum_{k=1}^\infty \frac{1}{k^{-s}}$

We see that this is just $\zeta (-1)$. 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 $\zeta(-1)$, then we see that the result is $\frac1{12}$. Therefore, we renormalise this sum to $\frac1{12}$.

Of course, it is obvious that this is not rigorous, however, it is very useful from a Physics point of view.

Ramanujam summation

The Ramanujam Summation is essentially a property of the partial sums, rather than a property of the entire sum, as that doesn't exist. If we take the Euler–Maclaurin summation formula together with the correction rule using Bernoulli numbers, we see that:

\begin{align} {} &\frac{1}{2}f\left( 0\right) + f\left( 1\right) + \cdots + f\left( n - 1\right) + \frac{1}{2}f\left( n\right) \\ = &\frac{1}{2}\left[f\left( 0\right) + f\left( n\right)\right] + \sum_{k=1}^{n-1}f\left(k\right)\\ = &\int_0^n f(x)\,dx + \sum_{k=1}^p\frac{B_{k + 1}}{(k + 1)!}\left[f^{(k)}(n) - f^{(k)}(0)\right] + R_p \end{align}

Ramanujan[1] wrote it for the case $p$going to infinity:

$\sum_{k=1}^{x}f(k) = C + \int_0^x f(t)\,dt + \frac{1}{2}f(x) + \sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k - 1)}(x)$

where $C$ 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 $R$ tends to 0 as $x$ tends to infinity, we see that, in a general case, for functions $f (x)$ with no divergence at $x$ = 0:

$C(a)=\int_0^a f(t)\,dt-\frac{1}{2}f(0)-\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(0)$

where Ramanujan assumed $\scriptstyle a \,=\, 0$. By taking $\scriptstyle a \,=\, \infty$ we normally recover the usual summation for convergent series. For functions $f ( x )$ with no divergence at $x =1$ , we obtain:

$C(a) = \int_1^a f(t)\,dt+ \frac{1}{2}f(1) - \sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}f^{(2k-1)}(1)$

$C(0)$ was then proposed to use as the sum of the divergent sequence.

In particular, the sum of all positive integers is

$1+2+3+\cdots = -\frac{1}{12}\ (\Re)$

where the notation $\scriptstyle (\Re)$ indicates Ramanujan Summation.