✎ Dimensional Analysis [The Physics Travel Guide]

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====== Dimensional Analysis ======

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
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<tabbox Student>

<blockquote>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".

<cite>[[http://www.physics.rutgers.edu/~zrwan/thesis/physics/Towards-Final-Laws.txt|Towards the final laws of physics]] by Steven Weinberg</cite></blockquote>

**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_theory:quantum_field_theory|quantum field theory]]. 
 
<tabbox Researcher>

**Recommended Resources:**

* [[http://www.sciencedirect.com/science/article/pii/0003491681900725|Dimensional Analysis in field theory]] by P.M Stevenson

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