# Time evolution of a massive fermion produced in a state of definite chirality

But for massive particles like an electron, the chirality is not conserved in time i.e. if an electron is produced in the state $e_L$ at time t=0, at a later time it becomes a mixture of left-handed and right-handed components.

How can the evolution of a purely left-chiral or right-chiral state of a massive fermion (like the electron) be quantified? In other words, given that the electron was produced in a state of definite chirality at $t=0$, what will it become at a later time $t>0$?

I previously asked a question here but I did not get an answer that I was looking for. Let me explain the answer I'm looking for. We know that a neutrino created in a define flavour state becomes a mixture of flavours at a later instant of time. In the same manner, I want to understand how a state of definite chirality become a state which is a mixture of both chiralities.

• The left- and right-handed parts are simply coupled through the Dirac equation, see also this answer of mine. Can you be more precise as to what you want to know about this? – ACuriousMind May 24 '18 at 19:10
• @ACuriousMind I want to show that an electron which is produced as a chirality eigenstate does not remain a chirality eigenstate under time evolution, and I want to find exactly how it evolves with time. – SRS May 29 '18 at 12:27
• I think I've commented this before, but: under standard definitions, there is no such thing as the chirality of a particle, as I write in detail here and here. @ACuriousMind also links to a nice answer to a question I asked years ago where I had the same misconception; that's where I got the issue cleared up. – knzhou Jun 15 '18 at 16:42
• If you wish to talk about the chirality of the particle, you'll have to tell us what definition you have in mind. I imagine you're thinking about the "first-quantized" picture where the solution the Dirac equation is the wavefunction of a particle. There you can indeed define chirality, but first quantization has so many conceptual pitfalls I wouldn't recommend using this picture at all! In particular, nobody uses first quantization in the phenomenology literature; it is only used by accident, or out of confusion. That's presumably why you haven't seen credible sources addressing this! – knzhou Jun 15 '18 at 16:43
• @knzhou No. I have QFT in mind. In QFT, we have chiral fields such as left-handed and right-handed electron fields i.e., $\hat{e}_L$ or $\hat{e}_R$. They are chirality projection operators $(1\pm\gamma^5)$ acting on the electron field $\hat{e}$. Agree? Why can't we have chirality eigenstates in the following way? Presumably, when chiral fields act on the vacuum $|0\rangle$ state will produce a one-particle state with definite chirality i.e., an eigenstate of either $(1+\gamma^5)$ or $(1-\gamma^5)$. – SRS Jun 16 '18 at 17:45