Compton Scattering Sum Rules for Massive Vector Bosons

Chapter 1
Physical method
1.1 Introduction to Field Theory
The mathematical framework of particle physics is quantum field theory (QFT). One
can transit from the formalism of classical mechanics to field theory by replacing the
n trajectories with—here, quantized—fields:
q i (t) → φ(x, t). (1.1)
While in classical mechanics the n generalized coordinates of a system represent
n degrees of freedom (d.o.f.s), the transformation i → x induces infinite d.o.f.s, with
φ(x, t) being one d.o.f. at a given point x. In (quantum) field theory, the generalized
variables are the fields (operators) and derivatives thereof, so that the Lagrange density
can be written as
L = L (φ(x), ∂ µ φ(x)) . (1.2)
The four-dimensional spatial integral over L is the action:
∫
S = d 4 x L, (1.3)
assuming that the fields vanish at infinity. Since the action is a Lorentz invariant
function and it is stationary under arbitrary variations of the field δφ i (x) (action
principle),
∫
δS =
{ ∂L
d 4 x
∂φ i (x) − ∂ µ
∂L
∂(∂ µ φ i (x))
}
δφ i (x) ! = 0, (1.4)
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