/sci/ - Science & Math

(cont.)

Say you have a physical situation for which no established theory works.
To explain what's going on, you might conjecture that maybe there is a yet undiscovered quantity Z, interacting with the system (historically: neutrinos, anti-particles) and you come up with a physical theory where the situation actually looks like

<span class="math">E=\int_2^\infty \left(\frac{2}{x+2}+\frac{1}{(1+x)^2}-\frac{1}{x}m_Z\right)\,dx[/spoiler]

And now the integral has the chance to be convergent, but only if <span class="math">m_Z=2[/spoiler], in this case. Not that it matters here, but in that case the integral has the value 1/4-log(4).
You’re in the situation now that you can say:
>Guys, guys, I predict further experiments will hint at E being -log(4)+1/4=-1.136…!
That value is kinda random, so if this happens they will be even convinced that your Z-quantity theory with mass equal to 2 is correct.
The idea that there is a higgs is from the 70’s, i.e. 40 year before it was possible to make that particle scattering measurement. It was just conjectured to be there because it was the simplest solution to complete the energy expression of the standard model.

http://en.wikipedia.org/wiki/Renormalization#Renormalized_and_bare_quantities

Assining -1.136… to an integral over a positive integrant might seem completely arbitrary without any context, of course.
But as math isn’t arbitrary, there aren’t so many theories to modify self-consistent theories.
At least if you want to keep things simple. Ramanujan came up with a theory assigning -1/12 to 1+2+3+… somewhere in India after having read age old books.
Here is a computation in quantum electrodynamics using that result

http://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension