Quantum Field Theory I

Chapter 2
Free Scalar Quantum Field
In this chapter the simplest QFT, namely the theory of the free scalar field, is
developed along the lines described at the end of the previous chapter. The keyword
is the canonical quantization (of the corresponding classical field theory).
2.1 Elements of Classical Field Theory
2.1.1 Lagrangian Field Theory
mass points nonrelativistic fields relativistic fields
q a (t) a = 1,2,... ϕ(⃗x,t) ⃗x ∈ R 3 ϕ(x) x ∈ R(3,1)
S = ∫ dt L(q, ˙q) S = ∫ d 3 x dt L(ϕ,∇ϕ, ˙ϕ) S = ∫ d 4 x L(ϕ,∂ µ ϕ)
d ∂L
dt ∂ ˙q a
− ∂L
∂q a
= 0 −→ ∂ µ
∂L
∂(∂ µϕ) − ∂L
∂ϕ = 0
Thethirdcolumnisjustastraightforwardgeneralizationofthefirstone,with
the time variablereplacedby the correspondingspace-timeanalog. The purpose
of the second column is to make such a formal generalization easier to digest 1 .
For more than one field ϕ a (x) a = 1,...,n one hasL(ϕ 1 ,∂ µ ϕ 1 ,...,ϕ n ,∂ µ ϕ n )
and there are n Lagrange-Euler equations
∂ µ
∂L
∂(∂ µ ϕ a ) − ∂L = 0
∂ϕ a
1 The dynamical variable is renamed (q → ϕ) and the discrete index is replaced by the
continuous one (q a → ϕ x = ϕ(x)). The kinetic energy T, in the Lagrangian L = T − U,
is written as the integral ( ∑ a T (˙qa) → ∫ d 3 x T ( ˙ϕ(x))). The potential energy is in general
the double integral ( ∑ a,b U (qa,q b) → ∫ d 3 x d 3 y U (ϕ(x),ϕ(y))), but for the continuous
limit of the nearest neighbor interactions (e.g. for an elastic continuum) the potential energy
is the function of ϕ(x) and its gradient, with the double integral reduced to the single one
( ∫ d 3 x d 3 y U (ϕ(x),ϕ(y)) → ∫ d 3 x u(ϕ(x),∇ϕ(x)))
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