Why is it interesting?

Layman

Student

Looking at the Lagrangian density in (1), we can easily work out what the units of the constant e, $\mu$, G, etc., are. All terms in the Lagrangian density must have units [mass]$^4$, because length and time have units of inverse mass and the Lagrangian density integrated over spacetime must have no units. From the $m\bar{\Psi}\Psi$ term, we see that the electron field must have units [mass]$^{3/2}$, because $\frac{3}{2}+\frac{3}{2}+1 = 4$. The derivative operator (the rate of change operator) has units of [mass]$^1$, and so the photon field also has units [mass]$^1$. Now we can work out what the units of the coupling constants are. As I said before, the electric charge turns out to be a pure number, to have no units. But then as you add more and more powers of fields, more and more derivatives, you are adding more and more quantities that have units of positive powers of mass, and since the Lagrangian density has to have fixed units of [mass]$^4$, therefore the mass dimensions of the associated coupling constants must get lower and lower, until eventually you come to constants like $\mu$ and G which have negative units of mass. (Specifically, $\mu$ has the units of [mass]$^{-1}$, while G has the units [mass]$^{-2}$.) Such terms in (1) would completely spoil the agreement between theory and experiment for the magnetic moment of the electron, so experimentally we can say that they are not there to a fantastic order of precision, and for many years it seems that this could be explained by saying that such terms must be excluded because they would give infinite results, as in (4).

Of course, that is exactly what we are looking for: a theoretical framework based on quantum mechanics, and a few symmetry principles, in which the specific dynamical principle, the Lagrangian, is only mathematically consistent if it takes one particular form. At the end of the day, we want to have the feeling that "it could not have been any other way".

Towards the final laws of physics by Steven Weinberg

Recommended Resources:

• https://particlephd.wordpress.com/2008/12/08/dimensional-analysis-for-animals/
• see http://math.ucr.edu/home/baez/renormalizability.html for how dimensional analysis is used in quantum field theory.

Researcher

Recommended Resources:

• Dimensional Analysis in field theory by P.M Stevenson

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