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 $\mathbf{q}$, 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 $\mathbf{q}$, 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.

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Key Concepts

Transverse Gauge

The transverse gauge is a specific gauge choice in electrodynamics where the divergence of the vector potential is set to zero (?·A = 0). This condition ensures that the photon polarization vectors are purely transverse to the momentum vector, thereby eliminating any unphysical longitudinal components.

Completeness Relation for Polarization Vectors

The completeness relation for polarization vectors shows that the sum over the physical polarization states forms a projection operator on the space perpendicular to the direction of propagation. It guarantees that any vector in the transverse plane can be reconstructed by its projections onto the physical polarization states, thereby ensuring consistency in calculations involving photon states.

Photon Helicity

Photon helicity is defined as the projection of the photon’s spin along its direction of motion. For a massless photon, only the two helicity states (right-handed and left-handed) are physically allowed. The absence of a helicity zero state is a direct consequence of the transversality (gauge) condition and reflects the inherent property of massless particles having fewer degrees of freedom.

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