/sci/ - Science & Math

>>7098232
Keywords are regularization and renormalization.

In field theories you often set up theories with parameters which yet have to be measured, like the mass or the charge of a particle.
For example, you might conjecture that there is some mechanical quantity Y associated with an energy <span class="math">E=\frac{1}{2}m_Y(x'(t))^2[/spoiler]. But that alone doesn't tell you what the value of <span class="math">m_X[/spoiler] is. There might be an experiment, say measuring the inertia of Y, which tells you it. Say the experiment says <span class="math">m_Y[/spoiler] should be about 7. If the predictions of the theories with <span class="math">m_Y=7[/spoiler] stand the test of further experiments, you might be on to something.

In the above case, we measured masses and adopted the values. A fundamental theory of elementary particles might actually predicts the masses (e.g. string theory, in principle), or at least correlates the masses/charges etc. strongly and that a priori (e.g. quantum field theories). Here the masses are determined in the way that the theory actually only works for particular values of it.

In field theories the energy terms are more complicated than just the velocity x'(t) squared, there are energy densities and whatnot.
Instead of doing actual renormalization theory here, consider the integral

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

The sign <span class="math">\int_2^\infty[/spoiler] is defined as <span class="math">lim_{a\to\infty}\int_2^a[/spoiler].
The integral over 1/x to infinity is undefined, i.e. the integral is convergent.