Index Concordia

Quantum field theory – Week 02

Posted in Quantum Field Theory by Index Guy on September 11, 2007

Since the state schools here in New York observe religious holidays, this week is a short one because of Rosh Hashanah celebrated from Wednesday night through Friday.

Today we continue with the brief exposition of classical field theory. We mentioned the fact that one can have transformations on the spacetime coordinates (for example elements of the Poincaré group) and transformations among different fields (a phase transformation being an example of these). First we consider for now the transformations between two different frames of coordinates x,y. If we define

\bar{\mathcal{L}} = \mathcal{L}\displaystyle\frac{d^4 x}{d^4 y},

then the action is guaranteed to remain invariant under a coordinate transformation.

We have Form Invariance when the Lagrange density in one frame has the same functional dependence on the fields and their spacetime derivative (defined in that same frame) as the Lagrange density’s functional dependence on the fields and their spacetime derivatives defined in the other frame. In other words

\mathcal{L} = \bar{\mathcal{L}}.

Next we considered infinitesimal transformations. We can specified a general transformation by N parameters \beta_{a}. We can treat them (very informally) as infinitesimal numbers \delta \beta_{a}. Then a coordinate transformation will look like

\bar{x}^{\mu} = x^{\mu} + \delta x^{\mu}\delta\beta_{a} = x^{\mu} + \displaystyle\sum^{N}_{a = 1} \frac{\partial x^{\mu}}{\partial \beta_{a}} \delta \beta_{a}.

Similarly, the field will transform like

\bar{\phi}_{i} = \phi_{i} + \delta \phi_{i}\delta\beta_{a} = \phi_{i} + \displaystyle\sum^{N}_{a = 1} \frac{\partial \phi_{i}}{\partial \beta_{a}} \delta \beta_{a}.

In the end we mentioned Noether’s theorem: if a Lagrange density \mathcal{L} is invariant under the transformations mentioned above, then there exist N conserved currents J^{\mu}_{a},

\partial_{\mu}  J^{\mu}_{a} = 0.

If we use the divergence theorem, then the volume integral of the time component of the current gives a conserved quantity.

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