## The Standard Model Lagrangian

UPDATE: The second installment is up:

The latest installment of “Minute Physics” (a video series which I absolutely adore) tackles a rather Herculean task: explaining the Standard Model of Physics in a minute. It is, of course, a VERY high-level overview, and alludes to future installments designed to go into more detail.

The focus of this video is the following mathematical expression:

$\mathcal{L} = \frac{-1}{4} F^2 + i\bar{\psi}\not{D}\psi + \bar{\psi}\phi\psi + h.c. + |D\phi|^2 - V(\phi)$

(Pardon the poorly aligned slash through the D.  Plain vanilla $\LaTeX$ found in WordPress doesn’t seem to handle Feynman slash notation very well.)

Unfortunately, the video doesn’t really say what this is. It is the Standard Model Lagrangian, or at least an abbreviated form of it. (More detailed forms of it can be seen here and here.)

“So, what the heck is a Lagrangian?” I hear you cry.

A Lagrangian is an expression which codifies the dynamics of a physical system, and essentially consists of the kinetic energy minus the potential energy of a system. It forms the heart of the Lagrangian formulation of classical mechanics, as well as the path-integral formulation of quantum field theory.

In essence, all of physics boils down to constructing the proper Lagrangian for a physical system. Once you have the correct Lagrangian, integrate it over time to arrive at the action. Invoking the Principle of Least Action, we know that the system will evolve in such a way as to minimize the action. That is all there is to it.

Okay, I’m obviously simplifying quite a bit here. Of course, the tricky part is constructing the correct Lagrangian.

This extremely stripped-down Lagrangian shown above includes contributions (terms with an F or a D) from gauge bosons –  photon,WZ, and gluon fields,  contributions from fermionic fields (terms with a ψ), and terms with a ϕ refer to the Higgs field.

So, how does one start with this skeleton and build a fully fleshed-out Lagrangian with which actual calculations can be performed? Well, that gets a little hairy, and the discussion will have to be spread out over time. A big hint, though: it starts with Feynman diagrams.

For excruciating detail, see the following: