By Peskin and Schroeder
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Extra info for An Introduction to Quantum Field Theory
L( 1 x): The action S , formed by integrating L over spacetime, is Lorentz invariant. 2) used for is the simplest possible transformation law for a eld. It is the only possibility for a eld that has just one component. But we know examples of multiple-component elds that transform in more complicated ways. 1. When a rotation is performed on a vector eld, it a ects the orientation of the vector as well as the location of the region containing the con guration. such as the 4-current density j (x) or the vector potential A (x).
How do we nd them? We could try to systematically nd all possible transformation laws for a eld. Then it would not be hard to write invariant Lagrangians. For simplicity, we will restrict our attention to linear transformations, so that, if a is an n component multiplet, the Lorentz transformation law is given by an n n matrix M ( ): (3:8) a (x) ! 8). 8) in the form ! M( ) : (3:9) What are the possible allowed forms for the matrices M ( )? The basic restriction on M ( ) is found by imagining two successive transformations, and 0 .
The Lagrangian formulation of eld theory makes it especially easy to discuss Lorentz invariance. An equation of motion is automatically Lorentz invariant by the above de nition if it follows from a Lagrangian that is a Lorentz scalar. This is an immediate consequence of the principle of least action: If boosts leave the Lagrangian unchanged, the boost of an extremum in the action will be another extremum. *This subject is covered, for example, in Schi (1968), Chapter 13 Baym (1969), Chapter 23 Sakurai (1967), Chapter 3.