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They are not comparable. LSZ is actually the justification for the previous derivation, but the first is more easily introduced. You should not take the imaginary time in a physical sense. This is a computational method of regularising infinities and it is perfectly ok.


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Let's begin with $$\langle\Omega|T\{j_{\mu}(x)j_{\nu}(0)\}|\Omega\rangle=\langle\Omega|T\{[\bar{q}(x)\gamma_{\mu}q(x)][\bar{q}(0)\gamma^{\mu}q(0)]\}|\Omega\rangle=$$ $$=\langle\Omega|T\{\bar{q}(x)_a^i(\gamma^{ij})_{\mu}q(x)_a^j\bar{q}(0)_b^k(\gamma^{kl})^{\mu}q(0)_b^l\}|\Omega\rangle=\ldots$$ where I have made the $i,j,k,l$ spinor indices and the $a,b$ ...



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