# What is the math showing that the time reversed version of an electron is a positron? (+general time reversal question)

As in Wheeler's One Electron Universe idea, how do you show that electrons and positrons are time-reversed versions of each other? Do you just apply time reversal to an electron and out pops a positron? Maybe a more accurate question would be "How do you describe a particle moving backwards in time?", out of which the transformation from electron -> positron should become apparent.

Edited to add: Also, what is the difference between time reversal and moving backwards in time? Is time reversal the observer moving backwards in time (in which case we would see an electron as an electron) vs. the electron moving backwards in time (when we would see a positron)?

• To clarify: do you just want to see the math showing that the time-reversed version of an electron is a positron? Or did you want to see math supporting the idea that there is only one electron/positron in the universe? Because the latter is just an inspiration for the former and doesn't actually have any math behind it, as far as I know. – David Z May 5 '13 at 19:00
• Just that a time-reversed electron is a positron. Sorry, I wasn't clear about that. – FatCat0 May 5 '13 at 19:10
• OK, I made an edit to clarify that - I figure it's probably better not to feature the one electron universe in the title, so people know that's not quite what you're asking about. But please feel free to edit again if you would like to change the wording to better reflect what you want to ask. It would also help if you include (in the question, by editing) some mention of what research you have already done on the topic, so people know where to start in answering. – David Z May 5 '13 at 19:18
• – Qmechanic May 5 '13 at 19:58

The the easiest way to see that time reversal transforms electrons into positrons relies on the fact that PCT (parity, charge conjugation and time reversal) combined are a symmetry of every Lorentz-invant QFT. Using $P^{-1} = P$, $C^{-1} = C$, $T^{-1} = T$, i.e. a parity transformation is undone by a second parity transformation etc. you can see that $$1 = PCT = (PC)^{-1}T \Rightarrow T = PC$$ so time reversal has the same effect as a parity transformation (under which electrons stay electrons) followed by charge conjugation (which takes electrons to positrons). Therefore, time reversal turns electrons into positrons.