# Observables of Dirac equation

So I learned about the Dirac equation which describes a relativistic free particle with spin $$\frac{1}{2}$$. I get the mathematics but what i can't find nowhere:

What are the observables of this theory?

Surely since the aim is to describe a relativistic particle there has to be some form of obtaining the probability of finding the particle in some region. Similarily, how can one obtain the momentum (or rather its probability distribution) given that the particle is in a fixed state described by some spinor satisfying the Dirac equation?

• Without the right context, the Dirac equation doesn't make sense as a physical theory. The right context is quantum field theory (QFT). Without QFT, you probably won't find a satisfying answer to your question. Many texts try to introduce the Dirac equation as a physical theory without using QFT, introducing it instead as a relativistic analog of the non-relativistic Schrödinger equation, but that doesn't work, and that's part of what led to the development of QFT. – Chiral Anomaly May 17 at 19:37
• @ChiralAnomaly Ah that is very interesting. So could one say that the notion of a relativistic free particle doesn't make much sense? Are there any sources which talk about this? I have only seen as you described some derivations of the Dirac equation which is presented as a physical theory, but only the mathematics is presented. – Jannik Pitt May 17 at 19:46
• See, Chapter $1$, Lectures on Quantum Field Theory by Sidney Coleman. He beautifully proves (in a physicist's sense of the word) why you cannot construct a relativistic quantum theory of a single particle. In particular, you're asking the right question, what's the observable? The most basic observable one has to endow a theory of a single particle with is the position operator. And, it turns out, that there is no way to construct one in a manner that is consistent with both quantum mechanics and relativity. – Dvij D.C. May 17 at 23:39
• There is a notion of "relativistic free particle" that does make sense, but like @DvijD.C. said, it can't involve a (strict) position observable. The idea that particles can be counted can still make sense in the relativistic context, but even that idea becomes fuzzy when interactions are involved, when non-inertial observers are considered, or when spacetime is curved. In hindsight, we should think of particles as phenomena -- things we might get out of the theory under the right conditions, not things we write into the theory's postulates. – Chiral Anomaly May 18 at 2:10