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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.

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    $\begingroup$ 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. $\endgroup$
    – QuantumFieldMedalist
    Commented Jun 13, 2023 at 8:44
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    $\begingroup$ Wikipedia. $\endgroup$
    – Tobias Fünke
    Commented Jun 13, 2023 at 8:58
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    $\begingroup$ 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. $\endgroup$
    – Keyflux
    Commented Jun 13, 2023 at 8:59
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    $\begingroup$ Related answers of mine: physics.stackexchange.com/a/525930/50583, physics.stackexchange.com/a/174908/50583 $\endgroup$
    – ACuriousMind ♦
    Commented Jun 13, 2023 at 15:53
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    $\begingroup$ I thought my question was pretty straightforward: axiom or theorem (can be proved/derived)... $\endgroup$
    – Frido
    Commented Jun 15, 2023 at 17:57

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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.

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  • $\begingroup$ I have to admit I'm back to being confused. So you are saying that for scalar field there is a constructive proof (in which case I think the term axiom should be replaced by theorem) whilst for spinor fields the axiom really is still an axiom as there is no proof yet of a realization of a unitary infinite dim irrep U? $\endgroup$
    – Frido
    Commented Jun 15, 2023 at 17:15
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    $\begingroup$ The (Wigner) classification of irreducible unitary representations of the Poincaré group is well-known, see e.g. Weinberg's QFT book for constructions via the method of little groups ("theory of induced representations" for mathematicians). This answer is simply wrong. There are Wightman axioms that are difficult to show rigorously, this is not one of them. $\endgroup$
    – ACuriousMind ♦
    Commented Jun 15, 2023 at 17:19
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    $\begingroup$ 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. $\endgroup$
    – MadMax
    Commented Jun 15, 2023 at 18:47

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