Quantum Field Theory. Note 4

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Irreducible representations of the rotation group

From the Lie algebra [I_i ,I_j]=iε_ijkI_k one can (easily) find all the irreducible representations. An irreducible representation is characterized by a number j which can have integer and half integer values. The dimension n of representation is n=j(j+1). The eigenvalues of the operator I_i are j,(j-1),(j-2),...,-j.

j=0,¹/₂,1,³/₂,...
I₃=diagonal{j,(j-1),(j-2),...,-j}

Irreducible representations of the Lorentz group

The Lie algebra of the Lorentz group is a direct product of two rotational Lie algebras. Therefore its irreducible representation is characterized by two numbers (j₁,j₂) - integer or half integer.

[A_i,A_j]=iε_ijkA_k ,

[B_i,B_j]=iε_ijkB_k ,

[A_i,B_j]=0

Direct product of two irreducible representations

The generators of the direct product are equal to the sum of the individual generators.

I_k^(1⊗2)= I_k⁽¹⁾⊗1⁽²⁾+I_k⁽²⁾⊗1⁽¹⁾

The direct product of two irreducible representations can be reduced by an orthogonal transformation into a direct sum of irreducible representations with j from |j₁-j₂| through 1 up to (j₁+j₂). The matrix of this orthogonal transformation is made of Clebsch-Gordan coefficients.

j₁⊗j2=|j₁-j₂|⊕...⊕(j₁+j₂)

Exercises

Find the matrix of finite rotation D^(½)(n,θ) for the the j=½ representation of the rotation group.
Using the additive parameter φ=½ln[(1+v)/(1-v)] of the Lorentz group find the matrix for the velocity boost along the z-axis for the (½,0) and (0,½) representations of the group.