A. Zee, Quantum Field Theory in a Nutshell · Problem I.5.1

Quantizing Canonically

Physics — QFT

Problem

Write down the most general form for

\sum_a \varepsilon^{(a)}_{\mu\nu}(k)\varepsilon^{(a)}_{\lambda\sigma}(k)

using symmetry repeatedly. For example, it must be invariant under the exchange

\{\mu\nu \leftrightarrow \lambda\sigma\}

. You might end up with something like

AG_{\mu\nu}G_{\lambda\sigma} + B(G_{\mu\lambda}G_{\nu\sigma} + G_{\mu\sigma}G_{\nu\lambda}) + C(G_{\mu\nu}k_{\lambda}k_{\sigma} + k_{\mu}k_{\nu}G_{\lambda\sigma})

+ D(k_{\mu}k_{\lambda}G_{\nu\sigma} + k_{\mu}k_{\sigma}G_{\nu\lambda} + k_{\nu}k_{\sigma}G_{\mu\lambda} + k_{\nu}k_{\lambda}G_{\mu\sigma}) + Ek_{\mu}k_{\nu}k_{\lambda}k_{\sigma}

with various unknown

A, \ldots, E

. Apply

k^{\mu} \sum_a \varepsilon^{(a)}_{\mu\nu}(k)\varepsilon^{(a)}_{\lambda\sigma}(k) = 0

and find out what that implies for the constants. Proceeding in this way, derive (13).

Intuitive explanation

The sum over polarization tensors must respect Lorentz covariance and the transversality constraint $k^\mu \varepsilon^{(a)}_{\mu\nu}(k) = 0$ that comes from gauge invariance. Build the most general rank-four tensor from the available building blocks ( $G_{\mu\nu}$ and $k_\mu$ ), then use transversality—by contracting with $k^\mu$ and $k^\lambda$ —to eliminate overcounting and fix the coefficients. This systematic approach turns a seemingly arbitrary sum into a unique, constrained form.

Intuitive explanation

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