I hope to replicate my previous thread in QFT which was started three years ago from reading Srednicki's textbook and solution manual and also the problem book that I read (by some serbian fellow).(adsbygoogle = window.adsbygoogle || []).push({});

This time I am planning to read several books, so the the title of this thread is general.

Anyway, I'll start with my first question from Bjorken's and Drell's classic fields book.

My question refers to pages 153-155:

I'll quote the passage:

My two questions: As before, we introduce the spectral amplitude by grouping together in the sum over ##n## all states of given four momentum ##q##:

[tex] (16.97)\rho_{\alpha \beta}(q) = (2\pi)^3\sum_n \delta^4(p_n-q)\langle 0 | \psi_\alpha (0) |n \rangle \langle n|\bar{\psi}_\beta(0)| 0\rangle [/tex]

and set out to construct its general form from invriance arguments.

##\rho(q)## is a ##4\times 4## matrix and may be expanded in terms of the 16 linearly independent products of the ##\gamma## matrices:

[tex] (16.98)\rho_{\alpha \beta}(q) =\rho(q)\delta_{\alpha \beta}+\rho_\mu(q)\gamma_{\alpha\beta}^\mu + \rho_{\mu\nu}(q)\sigma_{\alpha\beta}^{\mu\nu}+\tilde{\rho}(q)\gamma_{\alpha\beta}^5+\tilde{\rho}_\mu(\gamma^\mu\gamma^5)_{\alpha\beta}[/tex]

...

[tex](16.102)\rho(q)=S^{-1}(a)\rho(qa^{-1})S(a)[/tex]

where the matrix ##S## is defined by:

[tex] (16.100)S^{-1}\gamma^{\mu}S= a^\mu_\nu \gamma^\nu[/tex]

Equation (16.102), together with the general expansion of ##\rho(q)## given in (16.98), determines the structure of the coefficients ##\rho, \rho_\mu##, etc. For instance, if (16.98) is inserted into (16.102), it follows that

[tex]\rho(q)=\rho(qa^{-1})[/tex]

that is, ##\rho## transforms as a Lorentz scalar. Similarly, ##\rho_\mu(q)=a_\mu^\nu \rho_\nu(qa^{-1})## transforms as a Lorentz 4-vector, and so on.

...

In this way the form (16.98) is limited to

[tex](16.103)\rho_{\alpha\beta}(q)=\rho_1(q^2)\not{q}_{\alpha\beta}+\rho_2(q^2)\delta_{\alpha\beta}+\tilde{\rho}_1(q^2)(\not{q}\gamma^5)_{\alpha\beta}+\tilde{\rho}_2(q^2)\gamma_{\alpha\beta}^5[/tex]

To reduce the form of ##\rho_{\alpha\beta}(q)## further, we must require invariance of the theory under the parity transformation ##\mathfrak{P}##, which has the property:

[tex](16.104)\mathfrak{P}\psi_\alpha(0)\mathfrak{P}^{-1} = \gamma_{\alpha \lambda}^0 \pi_\lambda(0)[/tex]

Inserting (16.104) into (16.97) and carrying out the steps analogous to those used for proper Lorentz transformations, we arrive at the analogue of (16.102):

[tex](16.105)\rho_{\alpha \beta}(q,q_0) = \gamma^0_{\alpha \beta} \rho_{\lambda \delta}(-q,q_0)\gamma_{\delta \beta}^0[/tex]

Inserting (16.103) into (16.105) one finds

[tex](16.106) \tilde{\rho}_1 = \tilde{\rho}_2=0[/tex]

1. How to rigorously derive the two equations for Lorentz scalar and 4 vector in the quote above ?

2. How to rigorously derive equations (16.105) and (16.106)?

Thanks in advance, I plan on reading also Boyarkin's book and Hatchinson's and other books in QFT and QCD.

Cheers!

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# A Questions from textbooks in QFT

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