# Index Concordia

## Quantum field theory – Week 05

Posted in Quantum Field Theory by Index Guy on October 5, 2007

During this week we studied the Poincaré group and we started with the quantization of scalar fields.

The Poincaré group consists of translations in the four spacetime dimensions and Lorentz transformations. Since for the Lorentz group we had 6 independent parameters (3 boosts and 3 rotations), adding in the four translations gives 10 parameters for Poincaré. We will argue that translations involve changes in coordinates and so the generators should be differential operators. The defining representation will be obtain by acting on coordinates.

Denote an arbitrary Poincaré transformation by $D(\Lambda, a)$ with $\Lambda$ the Lorentz transform and $a$ the constant vector whose magnitude gives the amount that is translated. How do scalar fields transform under Poincaré? Well, by definition

$\varphi (x) = \bar{\varphi}(\bar{x}) \Rightarrow \bar{\varphi}(x) = \varphi (\Lambda^{-1}(x - a)).$

Scalar fields represent particles with zero spin. To generalize this for fields of arbitrary spin we use $\phi_{a}$ with $a$ a discrete index over spacetime components. We argue that a Poincaré transformation will mix the components in a linear way:

$\bar{\phi}_{a}(x) = S_{ab}(\Lambda) \phi_{b}(\Lambda (x - a)).$