# Is $U(\Lambda,a)\hat\phi(x) U^{-1}(\Lambda,a) = R(\Lambda,a) \hat\phi(\Lambda^{-1}x + a)$ an axiom or can be it derived?

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.

• If you assume your theory to be relativistic in flat space, then it must carry a representation of the Poincare group. This means that there is an action of the group on the objects of the theory, that mimics the action of the group on spacetime events. This is what the quoted equation is saying. Commented Jun 13, 2023 at 8:44
• Commented Jun 13, 2023 at 8:58
• I think it just defines the transformation property of $\phi$ under the Poincare group. A nontrivial point is that what kind of $R$s allowed depends on the underlying group. Commented Jun 13, 2023 at 8:59
• Related answers of mine: physics.stackexchange.com/a/525930/50583, physics.stackexchange.com/a/174908/50583 Commented Jun 13, 2023 at 15:53
• I thought my question was pretty straightforward: axiom or theorem (can be proved/derived)... Commented Jun 15, 2023 at 17:57

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 $$\hat\psi(x)$$.
• Since I have A Curious Mind, I would like to see a single concrete example, with the details of the unitary $U$ worked out for a given non-unitary Lorentz boost $R$ as an example. Commented Jun 15, 2023 at 18:47