# Does the Dirac Spinor live in the complexification of the Lorentz group?

In this question I learned that when working with quaternionic representations of (the double cover of) our relevant orthogonal group, we cannot avoid working in complex vector spaces. However it is well known that the $$(\frac{1}{2},0)\oplus(0,\frac{1}{2})$$ representation, is a real representation of the Lorentz group. This leads me to believe, just as we do not use complex vectors in the vector reps $$(\frac{1}{2},\frac{1}{2})$$ and $$(1,0)\oplus(0,1)$$, there is no need for complex spinors in this representation. On the flip side, I understand that the Standard Model requires complex representations of the Lorentz group, and this is of course accomplished via the fact that individually, the $$(\frac{1}{2},0)$$ and its conjugate $$(0,\frac{1}{2})$$ are complex representations, and the weak force couples to a field which transforms via the $$(\frac{1}{2},0)$$ rep.

All this is to say, given that the entirely analogous adjoint representation $$(1,0)\oplus(0,1)$$ is irreducible and real, I do not see what justifies constantly declaring a Dirac spinor having $$8$$ real degrees of freedom, instead of $$4$$, as it lives in a real representation of the Lorentz group (and if we are doing physics we care about objects which transform under the Lorentz group, and not its complexification).

To say "The Dirac spinor splits into $$2$$ Majorana spinors: $$\psi_D = \psi_{M_1}+i\psi_{M_2}$$" appears to concede that the Dirac spinors must live in a complexified vector space. To argue from the other point of view that since they come from the direct sum of complex representations they retain all those degrees of freedom of the Weyl spinors, would imply Field Strength Tensors $$\in (1,0)\oplus(0,1)$$ are naturally complex just as Dirac spinors, which it seems nobody believes.

What justifies the seemingly inconsistent treatment of field strength tensors vs Dirac spinors when it comes to their real degrees of freedom? Is it merely convenient to work in the complexificiation, since spinor fields on their own are not directly observable?

• As you said yourself, we could replace every Dirac spinor with two Majorana spinors. For instance, QED could be formulated with two Majorana fields. It’s perfectly equivalent, just less convenient. Mar 24, 2022 at 19:06
• You could just as well ask, why would anybody ever use complex scalar fields when they’re equivalent to a pair of real scalar fields? The answer is again convenience depending on the physical situation. Mar 24, 2022 at 19:07
• @Knzhou Well that seems to imply Dirac spinors are not elements of the vector space which Lorentz transforms act on, but its complexificiation. Yes? This might all be mathematical pedantry, I'm not sure, but it seems important to know where objects of the theory live. Mar 24, 2022 at 19:09
• Correct. Dirac spinors are not elements of the vector space which Lorentz transforms act on. They are vectors of the complex vector space which Dirac matrices act on. To me it seems a coincidence that spacetime has four components, and so do the Dirac spinors. Mar 24, 2022 at 19:20
• @Kurt G. Sorry, maybe I was loose with my Language in the previous comment. I know Spinors, whether Majorana or Dirac, are not elements of the vector space which we call Minkowski Space. By my comment I meant each representation acts on a vector space, and that vector space in the case of $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ is where Majorana spinors live (and hence live where elements of the double cover of the Lorentz group act "Lorentz transforms on spinors"), but this vector space must be complexified to fit Dirac Spinors. Mar 24, 2022 at 19:29

For future people who stumble upon this question, the confusion lies in the distinction between "representations of the Lorentz group over a complex vector space" and "representations of the complexification of the Lorentz group".

We are used to representing the Lorentz group over real vector spaces, and the representation theory of Lie groups over real vs. complex vector spaces in general can be quite different.

The Lorentz Lie algebra and hence Lie group are real, even if the generators of rotations and boosts are not composed of purely real matrices, because we only allow real linear combinations of the generators in the algebra. When we choose to allow generally complex linear combinations of generators, this action is known as complexifying, and we are now working in the complexified Lie algebra/group.

The incredibly useful fact which confused this, at least for me, is that representations of the complexified Lie group, are in $$1-1$$ correspondence with representations of the real Lie group over complex vector spaces.

Notice this says nothing about representations over real vector spaces, which is good because the adjoint representation had better be real and irreducible, which as a representation over a real vector space, it is.

Though we may find it natural to work only with real Majorana spinors, when QED demands our spinors be complex, we are now free to perform a change of basis in which the complex spinor is decomposable into left and right handed parts. Though rigorously speaking we are still not allowed to add complex combinations of generators, certain unitary changes of basis will essentially do that for us.

Thus Dirac spinors are not elements which transform under the action of the complexified Lorentz group, but merely transform under the Lorentz group as usual, and being complex is what 'breaks' the analogy between them and the adjoint rep.