By S S Schweber
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Extra info for An introduction to relativistic quantum field theory
15) kl This has a simple interpretation: The one-particle operator destroys a particle in state l and creates a particle in state k. The “weight” with which this happens is given by the one-particle matrix element okl . We give now a very simple example for this formalism. As the one-particle basis we chose the generalised position-spin basis |xσ . As a typical one-particle operator we take the momentum of the particle. 16) which is already known from elementary quantum mechanics. For a reason we will give in the next section below the creation and annihilation operators for this basis will be denoted by ψ(x, σ).
So for the matrix dˆ we have dˆt dˆ = dˆdˆt = 1. Such scalar product invariant: (dx)( matrices are called orthogonal matrices. These matrices describe rotations or the composition of a rotation and a reflection. e. the group of orthogonal transformations of the Euclidean space Ê3 . The pure rotations are given by orthogonal matrices with detdˆ = +1 and form a subgroup of O(3), namely the so called special orthogonal group SO(3). From this reasoning it follows that the whole Lorentz group has O(3) and SO(3) as subgroups.
3 Noether’s Theorem (Classical Part) As shown above the classical field theory is defined by an action functional S[φ]. 27) which is the equation of motion for the fields. The action functional is given as the fourdimensional integral of the Lagrange density, which is a function of the fields φ and their gradients ∂µ φ: S[φ] = d4 xL (φ, ∂µ φ). 28) The only constraint on the field is that it must vanish in the infinity of four-dimensional space such that S exists. 27) gives the Euler-Lagrange equations for the fields: ∂L ∂L δS[φ] = − ∂µ = 0.
An introduction to relativistic quantum field theory by S S Schweber