Exercise 3: Representation of Poincar´ e Group on functions

Quantum Field Theory

Set 6

Exercise 1: Tensor product of SU(2) representations

The statement that two particles have spin 1/2 reflects the fact that each particle can exist in two different states corresponding to the values of thez-component of the angular momentum. These two states define a vector space on which the group SU(2) is represented. Consider the total spin of the bound state of the two particles: this corresponds to taking the tensor product of twoj= 1/2 representations. Show that the resulting vector space,

V =

|s1i ⊗ |s2i, si=±1 2

,

can be decomposed in a direct sum of two spaces on which act irreducible representations ofSU(2). Find these representations.

Exercise 2: Lorentz Group and Lorentz Algebra

Consider the set of the Lorentz transformations in space-time. Show that they correspond to the group SO(1,3) =

Λ∈GL(4,R)|Λ^TηΛ =η ,

whereη is the metric with Minkowski signatureη= diag(1,−1,−1,−1). Starting from the above definition

• identify the Lie algebra;

• compute the dimension of the Lie algebra;

• show that the following expression represents a basis of the algebra (J^µν)^ρ_σ =i(η^µρδ^ν_σ−η^νρδ_σ^µ) ;

• show that the Lie algebra is

J^µν,J^αβ

=i(η^ναJ^µβ−η^µαJ^νβ+η^µβJ^να−η^νβJ^µα);

• define the quantities

Jⁱ=1

2^ijkJ^jk , Kⁱ=Jⁱ⁰, and compute the commutation relations between them:

Jⁱ, J^j

=?,

Jⁱ, K^j

=?,

Kⁱ, K^j

=?.

Exercise 3: Representation of Poincar´ e Group on functions

CallH the set of functions of a four-vectorx^µ and denote φ(x) a generic element of it. Take an element of the Poincar´e group acting onx^µ as:

g≡(Λ, a) : x^µ−→Λ^µ_νx^ν+a^µ ≡P_(Λ,a)(x^µ).

Define the action ofg onH as follows

D_(Λ,0)[φ](x) =φ(Λ⁻¹x), D_(0,a)[φ](x) =φ(x−a), D(Λ,a)[φ](x) =D(0,a)

D(Λ,0)[φ]

(x) =φ(Λ⁻¹(x−a)).

Show that this definition respects the group composition law and defines therefore a representation.