Show that (in the transverse gauge) the completeness relation is ∑λ=R, L(ελ)i^*(ελ)j=δi j-q̂i q̂

step 1 For this problem on the topic of de Brogley's postulate, you want to show that in order to be ab

Show that (in the transverse gauge) the completeness...

Show that (in the transverse gauge) the completeness relation is $$ \sum_{\lambda=R, L}\left(\varepsilon_\lambda\right)_i^*\left(\varepsilon_\lambda\right)_j=\delta_{i j}-\hat{q}_i \hat{q}_j $$ If $\varepsilon$ were along $q$, it would be associated with a helicity-zero photon. This state is missing because of the transversality condition, $\mathbf{q} \cdot \boldsymbol{\varepsilon}=0$. It can only be absent because the photon is massless. We return to a further discussion of photon polarization vectors in Section 6.13.

Show that (in the transverse gauge) the completeness relation is
$$
\sum_{\lambda=R, L}\left(\varepsilon_\lambda\right)_i^*\left(\varepsilon_\lambda\right)_j=\delta_{i j}-\hat{q}_i \hat{q}_j
$$

If $\varepsilon$ were along $q$, it would be associated with a helicity-zero photon. This state is missing because of the transversality condition, $\mathbf{q} \cdot \boldsymbol{\varepsilon}=0$. It can only be absent because the photon is massless. We return to a further discussion of photon polarization vectors in Section 6.13.

Key Concepts

Polarization Vectors

Polarization vectors describe the orientations of the oscillating electric (and magnetic) fields in electromagnetic waves. For massless photons, only polarization vectors transverse to the direction of propagation are physical, corresponding to the observable degrees of freedom of the photon.

Transversality Condition

The transversality condition is the requirement that the polarization vector be orthogonal to the momentum vector (q·? = 0). This condition is essential in removing non-physical degrees of freedom and ensuring that only the two transverse polarization states contribute, which is a direct consequence of the massless nature of the photon.

Helicity

Helicity refers to the projection of a particle's spin onto the direction of its momentum. For massless particles like the photon, only two helicity states (typically labeled + and - or right and left) are physically present, with the helicity-zero (longitudinal) state being absent as a consequence of gauge invariance and the transverse gauge constraint.

Transverse Gauge

The transverse gauge is a choice of gauge in electrodynamics where the divergence of the vector potential is set to zero. This choice simplifies analyses by restricting allowed photon polarizations to those perpendicular to the momentum vector, thereby eliminating unphysical gauge degrees of freedom.

Completeness Relation

The completeness relation is a mathematical statement that the sum over an orthonormal set of polarization vectors spans the subspace of physical states. In the context of transversal photons, it implies that the sum of projected outer products of the two allowed polarization vectors reconstructs the identity in the transverse subspace, effectively projecting out the longitudinal component.

Transcript

Please give Ace some feedback