Quantum Field Theory. Note 3

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Lie algebra of the rotation group

For the parametrization {n,θ}, where n is the rotation axis and θ is the rotation angle, the matrix R of the small rotation dθ is

R(n,dω) = 1 + i Indθ .

The corresponding Lie algebra for the three generators I is the same as for the quantum mechanical angular momentum operators

[I_i, I_j] = i ε_ijk I_k.

Lie algebra of the Lorentz group

The small transformation with rotation dθ≡ndθ and velocity boost dv is represented via generators J and K

Λ(dθ,dv) = 1 + iJdθ + iKdv

with the Lie algebra

[J_i, J_j] = i ε_ijk J_k , [J_i, K_j] = i ε_ijk K_k , [K_i, K_j] = -i ε_ijk K_k.

For the parametrization dη = dθ + idv the small transformation is

Λ(dη) = 1 + iAdη + iBdη^*

where

A = ¹/₂ (J - iK) , B = ¹/₂ (J + iK)

with the Lie algebra

[A_i, A_j] = i ε_ijk A_k , [B_i, B_j] = i ε_ijk B_k , [A_i, B_j] = 0 .

Reconstruction of the group element R(n,θ) in terms of generators I

Since θ is an additive parameter we can write

R(n,θ+dθ) = R(n,dθ)R(n,+dθ) = R(n,dθ)(1 + iIndθ) .

That is the matrix R satisfies the differential equation

^dR/_dθ = iR In

with the boundary condition R(n,θ=0)=1. Apparently the solution is the Taylor series

R(n,dθ) = exp(iInθ) ≡ 1 + iInθ + ¹/_2! (iInθ)² + ...

Irreducible representations of the rotation group

Schur's lemma. Kazimir operator of the rotation group.

Exercises

Suppose that for a certain parametrezation α_i the generators of a certain group G are I_i , wich form a Lie algebra with structure constants Cⁱ_jk . We introduce now a new set of parameters β_i which are nonlinear functions of the old parameters β=β(α). Find the new generators and new structure constants.
From the commutation relations [I_i,I_j]=iε_ijkI_k find the 2x2 representation of the rotation generators I. Assume that I₃ is diagonal.
Find the rotation matrix R(n,θ) of the 2x2 representation of the rotation group.