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As stated in the question, and I have looked at other questions on this topic here, I am still confused whether the equation $$ U(\Lambda,a)\hat\phi(x) U^{-1}(\Lambda,a) = R(\Lambda,a) \hat\phi(\Lambda^{-1}x + a) $$ is something that can be derived from the theory of Poincare representations, or whether it is an axiom of Quantum Field Theory?

So $\hat\phi(x)$ denotes a quantum field operator, $U(\Lambda,a)$ an infinite dimensional unitary irrep of Poincare on the Hilbert space of particle states, and $R(\Lambda,a)$ a finite-dim irrep of Poincare acting on the (components of) corresponding classical fields $\phi(x)$, eg spinors, vectors etc.

Frido
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Most text books discuss this "Wightman axiom" in the context of scalar $\hat\phi(x)$.

However, when it come to the spinor $\hat\psi(x)$ as a quantum field operator, the existence of the infinite-dimensional unitary $U(\Lambda,a)$ on the LHS of $$ U(\Lambda,a)\hat\psi(x) U^{-1}(\Lambda,a) = R(\Lambda,a) \hat\psi(\Lambda^{-1}x + a) $$ is highly questionable.

So far I don't see a single concrete example in any text book or forum which shows that the alleged infinite-dimensional unitary $U(\Lambda,a)$ actually exists for the spinor $\hat\psi(x)$.

See more details here.

MadMax
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