On p. 32 of Quantum field theory in a nutshell, Zee tries to derive the propagator for a spin 1 field:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]D_{\nu\lambda} = \frac{-g_{\nu\lambda} + k_\nu k_\lambda /m^2}{k^2 - m^2}[/itex]

using the Lorentz invariance of the equation [itex]k^\mu \varepsilon_\mu^{(a)}=0[/itex] where [itex]\varepsilon_\mu^{(a)}[/itex] denotes the three polarization vectors [itex]a = x,y,z[/itex].

I understand that if we have two electromagnetic 4-currents [itex]J_\lambda , J_\nu[/itex], then the amplitude for a particle with momentum [itex]k[/itex] to propagate from one to the other is going to be proportional to

[itex]\sum_{a} \varepsilon_\mu^{(a)}\varepsilon_\lambda^{(a)}[/itex]

which is in turn proportional to the propagator [itex]D_{\nu\lambda}[/itex].

According to Zee, Lorentz invariance as well as [itex]k^\mu \varepsilon_\mu^{(a)}=0[/itex] enfornces the condition that

[itex]\sum_{a} {\varepsilon_\nu}^{(a)}{\varepsilon_\lambda}^{(a)}[/itex]

is proportional to [itex]g_{\nu\lambda} - k_\nu k_\lambda /m^2[/itex] and that evaluating for k at rest with [itex]\nu = \lambda = 1[/itex] changes the sign.

I'm having trouble understanding each step. I apologise for the lengthiness of this question.

1. Why should [itex]\sum_{a} {\varepsilon_\nu}^{(a)}{\varepsilon_\lambda}^{(a)}[/itex] be Lorentz invariant just because [itex]k^\mu \varepsilon_\mu^{(a)}[/itex] is?

2. Why exactly does Lorentz invariance dictate that it must be a linear combination involving only the metric and [itex]k_\nu k_\lambda[/itex]?

3. Why does

[itex]k^\nu{\varepsilon_\lambda}^{(a)} = 0 \implies \sum_{a} \varepsilon_\nu^{(a)}{\varepsilon_\lambda}^{(a)} \propto g_{\nu\lambda} - k_\nu k_\lambda /m^2[/itex]

4. How does ``evaluation of k at rest with [itex]\nu = \lambda = 1[/itex]'' change the sign?

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# Homework Help: Understanding Lorentz invariance

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