In his book "QFT" (vol. 1) Weinberg writes the expression for an arbitrary spin massive field: $$ \hat {\Psi}_{a}(x) = \sum_{\sigma = -s}^{s} \int \frac{d^{3}\mathbf p}{\sqrt{(2 \pi)^{3}2 \epsilon_{\mathbf p}}}\left( k_{1}F_{a}(\mathbf p) \hat {a}^{\sigma}(\mathbf p )e^{-ipx} + k_{2}G_{a}(\mathbf p) {\hat {b}^{\sigma}}^{\dagger}(\mathbf p )e^{ipx} \right). $$ By assuming very long way of derivations (which consists of lorentz transformation law for fields) before that he concludes that $$ F_{A}^{\sigma}(\mathbf p ) = (-1)^{s + \sigma}G_{A}^{-\sigma}(\mathbf p). $$ How to prove this faster than Weinberg do it? Let's assume that equations of motion and degrees of freedom decreasing relations for corresponding field are known.


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