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In trying to construct a special relativistic model of quantum mechanics, one has to require invariance of the probabilities under Poincare' transformations.

Then if we let the spinorial wave function to be $\Psi_{k}(x) \in L^{2}(\mathbb{R}^{4}) \otimes \mathbb{C}^{2s+1}$ it seems we should have the following equation:

\begin{equation} \Psi_{b}(\Lambda_{\nu}^{\mu} x^{\nu}) = M_{ab}(\Lambda)e^{\frac{i}{2}\hat{L}_{\mu\nu} \omega^{\mu\nu}}e^{i\hat{p}_{\mu}a^{\mu}}\Psi_{a}(x^{\mu}) \end{equation}

Where $\Lambda$ is a Poincare' transformation. Now we can see that the spinor index has to transform non-unitarily, because the $M_{ab}(\Lambda)$ are finite-dimensional representations of a compact group.

However, to require invariance of the probabilities we must have $M_{ab}(\Lambda)M^{\dagger}_{bc}(\Lambda) = \delta_{ac}$ by taking the square of both sides, but this cannot be as the representation is finite. How is this problem overcomed in QFT?

Davyz2
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1 Answers1

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The "problem" in the question is not really the main problem of relativistic quantum theory because there are other things wrong with the setup as written, starting with the idea that there is a single spin $s$ with representation space $\mathbb{C}^{2s+1}$ when in reality the finite-dimensional representations of the Lorentz group are classified by pairs of half-integers $(s_1,s_2)$ with representation spaces $\mathbb{C}^{2s_1 + 1}\otimes \mathbb{C}^{2s_2 + 1}$. Furthermore, good relativistic position operators do not exist (see, for instance, this question for a start), so a position-space wavefunction in $L^2(\mathbb{R}^4)$ is questionable, too.

The solution to the non-unitarity of the finite-dimensional representations of the Poincaré should be obvious: Stop using finite-dimensional representations! Particles in a relativistic quantum theory (and hence in QFT) must transform under unitary representations of the Poincaré group, and the list of these infinite-dimensional representations is called Wigner's classification. The most basic - the "spin-0" scalar representation - is what you get when you construct momentum-space wavefunctions on the mass shell (i.e. take the state space $L^2(\Sigma)$, where the subset $\Sigma$ of momentum space $\mathbb{R}^4$ that fulfills $p^2 = m^2$ for whatever the mass $m$ of your particle is).

ACuriousMind
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