Index Concordia

Quantum field theory – Week 01

Posted in Quantum Field Theory by Index Guy on September 6, 2007

During this week professor Sterman kicked off the course by mentioning a brief historical background on relativistic quantum field theory. Then we review some notions from quantum mechanics and relativity and finally came to the Klein-Gordon equation. The last part of the class was dedicated to discussing some notational convention and units.

During the second meeting of the class we discussed local field theories, by this meaning theories where the Lagrangian function Lis given by the volume integral over a Lagrange density \mathcal{L}. Then we mentioned the action principle and found the Lagrange equations

\displaystyle\frac{\partial \mathcal{L}}{\partial \phi_{i}} - \partial_{\mu}\left(\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi_{i})}\right) = 0,

by requiring a variation of the action to vanish under variations of the fields \phi_{i}.

Once we know the equations that the Lagrange density satisfies (and where the equation of motion for the fields come from) we discussed the case of the real scalar field. Since we already saw that a real scalar particle obeyed the Klein-Gordon equation, we can try to construct the Lagrange density by using the Lagrange equations. These will involve real scalar fields obeying the Klein-Gordon equation. Now if we consider two real scalar fields \phi_{1}, \phi_{2}, it turns out that the Lagrange density can be written as one for a single complex scalar field \Phi. Written this way, we have \Phi and \Phi^{\dagger} both satisfying the Klein-Gordon equation.

Since the Klein-Gordon equation is linear in the fields, we can use the superposition principle. We also have no scattering between two different solutions. In other words, no interactions. This is an example of a free field. In order to introduce interactions we add (or subtract) a potential V(\phi) to the Lagrange density. Typically this potential is some sort of polynomial.

The last topic we discussed was a brief mentioned of transformations of fields and the Lagrange density. Our aim is to obtain Noether’s theorem and discuss symmetries and conserved quantities.

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