# Lorentz transformations for spinors

This is a common misconception.

Lorentz group $SO(3,1)$ (or its double-cover $SL(2,\mathbb{C})$ if you want to follow Wigner's analysis of symmetries in QM) is non-compact. This means that it has no finite-dimensional unitary representations (the inability to choose hermitian Clifford basis elements is just a consequence of this).

When you're constructing classical Dirac spinors, you don't need unitarity. Indeed, there's no need for $S(\Lambda)$ to be a unitary representation. We are dealing with a classical field here, and unitarity is not required in classical physics.

In QFT we are dealing with a quantum Dirac field. The state space (fermionic Fock space) of the free QFT is infinite-dimensional and unitary. This doesn't contradict the original claim precisely because of infinite dimensionality.

Classification of the infinite-dimensional unitary irreps of the Poincare group (inhomogenous Lorentz group if you wish) was done by Wigner. It uses finite-dimensional nonunitary representation theory of $SL(2,\mathbb{C})$ (which is equivalent to the finite-dimensional representation theory of the complexified Lie algebra $\mathfrak{so}(4)\sim\mathfrak{D}_2$) heavily.

Summarizing: the difference is that Poincare generators of the QFT act unitary on the infinite-dimensional Fock space. The classical transformation of the spinor field is respected by this action, but does not have to be and is not unitary.

The most famous theorem by Wigner states that, in a complex Hilbert space $H$, every bijective map sending rays into rays (a ray is a unit vector up to a phase) and preserving the transition probabilities is represented (up to a phase) by a unitary or antiunitary (depending on the initial map if $\dim H>1$) map in $H$.

Dealing with spinors $\Psi \in \mathbb C^4$, $H= \mathbb C^4$ and there is no Hilbert space product (positive sesquilinear form) such that the transition probabilities are preserved under the action of $S(\Lambda)$, so Wigner theorem does not enter the game.

Furthermore $S$ deals with a finite dimensional Hilbert space $\mathbb C^4$ and it is possible to prove that in finite-dimensional Hilbert spaces no non-trivial unitary representation exists for a non-compact connected semisimple Lie group that does not include proper non-trivial closed normal subgroups. The orthochronous proper Lorentz group has this property. An easy argument extends the negative result to its universal covering $SL(2, \mathbb C)$.

Non-trivial unitary representations of $SL(2,\mathbb C)$ are necessarily infinite dimensional. One of the most elementary case is described by the Hilbert space $L^2(\mathbb R^3, dk)\otimes \mathbb C^4$ where the infinite-dimensional factor $L^2(\mathbb R^3, dk)$ shows up.

This representation is the building block for constructing other representations and in particular the Fock space of Dirac quantum field.