Quantum Field Theory I

2.1. ELEMENTS OF CLASSICAL FIELD THEORY 53
Example: Lorentz transformations — field transformations ϕ(x) → ϕ(Λx).
Infinitesimal transformations ϕ(x) → ϕ(x) − i 2 ωρσ (M ρσ ) µ ν xν ∂ µ ϕ(x) (sorry) 4 ,
i.e. δ ρσ ϕ = −i(M ρσ ) µ ν xν ∂ µ ϕ = ( δ µ ρη σν −δ µ ση ρν
)
x ν ∂ µ ϕ = (x ρ ∂ σ −x σ ∂ ρ )ϕ.
The Lagrangian density L[ϕ] = 1 2 ∂ µϕ∂ µ ϕ− 1 2 m2 ϕ 2 − 1 4! gϕ4 again only as a specific
example, important is that everything holds for any scalar Lagrangian density.
Symmetry δL = − i 2 ωρσ (M ρσ ) λ ν xν ∂ λ L = ω λν x ν ∂ λ L, is processed further to
δL = ω λν ∂ λ (x ν L) − ω λν η λν L = ω λν ∂ λ (x ν L) = ω λν ∂ µ (g λµ x ν L). So one can
write δL = ω λν ∂ µ J µλν where 5 J µλν = g λµ x ν L
j µλν = ∂L (
x ν ∂ λ −x λ ∂ ν) ϕ−η λµ x ν L
∂(∂ µ ϕ)
rotations
boosts
j µij = ∂L (
x j ∂ i −x i ∂ j) ϕ−η iµ x j L
∂(∂ µ ϕ)
∫
Q ij = − d 3 x ∂L (
x i ∂ j −x j ∂ i) ϕ
∂ ˙ϕ
j µ0i = ∂L (
x i ∂ 0 −x 0 ∂ i) ϕ−η 0µ x i L
∂(∂ µ ϕ)
∫
Q 0i = − d 3 x ∂L (
x 0 ∂ i −x i ∂ 0) ϕ+x i L
∂ ˙ϕ
In a slightly different notation
∫
rotations QR ⃗ =
∫
boosts QB ⃗ = t
d 3 x ∂L
∂ ˙ϕ ⃗x×∇ϕ
d 3 x ∂L
∂ ˙ϕ ∇ϕ− ∫
( ) ∂L
d 3 x ⃗x
∂ ˙ϕ ˙ϕ−L
and finally in our specific example
∫
rotations QR ⃗ = − d 3 x ˙ϕ ⃗x×∇ϕ
∫ ∫
boosts QB ⃗ = −t d 3 x ˙ϕ ∇ϕ+
d 3 x ⃗x ( ˙ϕ 2 −L )
These bunches of letters are not very exciting. The only purpose of showing them
is to demonstrate how one can obtain conserved charges for all 10 generators of
the Poincarè group. After quantization, these charges will play the role of the
generators of the group representation in the Fock space.
4 For an explanation of this spooky expression see 1.3.1. Six independent parameters ω ρσ
correspond to 3 rotations (ω ij ) and 3 boosts (ω 0i ). The changes in ϕ due to these six transformations
are denoted as δ ρσϕ with ρ < σ.
5 The index µ is the standard Lorentz index from the continuity equation, the pair λν
specifies the transformation, and by coincidence in this case it has the form of Lorentz indices.