Question

Consider a generic Spin-1 vector field (massless)Aμ, we know that the solution for its equations of motion can be built as Aμ(x)=∫d3kNk(ϵμ(→k,λ)ˆak,λe−ik⋅x+ϵ∗μ(→k,λ)ˆa†k,λeik⋅x)whereϵμ(→k,λ) are the polarizations ofAμ. By construction they satisfy the following completeness relation 3∑λ=0gλλϵμ(→k,λ)ϵν(→k,λ)=gμν.

Is it true also for the combination∑3λ=0gλλϵμ(→k,λ)ϵ∗ν(→k,λ)?

Answer 1

Yes, the completeness relation also holds for the combination ∑3λ=0gλλϵμ(→k,λ)ϵ∗ν(→k,λ).

Step-by-step explanation:

Yes, the completeness relation also holds for the combination ∑3λ=0gλλϵμ(→k,λ)ϵ∗ν(→k,λ). The completeness relation is a property of the polarization vectors ϵμ(→k,λ), which satisfy the equation 3∑λ=0gλλϵμ(→k,λ)ϵν(→k,λ)=gμν. This equation guarantees that the polarization vectors span the entire vector space of the polarization states. Since the combination ∑3λ=0gλλϵμ(→k,λ)ϵ∗ν(→k,λ) is still a linear combination of the polarization vectors, it will also satisfy the completeness relation.