5
$\begingroup$

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)?

Improve this question
$\endgroup$
4
  • $\begingroup$ 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. $\endgroup$
    – David Z
    May 5, 2013 at 19:00
  • $\begingroup$ Just that a time-reversed electron is a positron. Sorry, I wasn't clear about that. $\endgroup$
    – FatCat0
    May 5, 2013 at 19:10
  • $\begingroup$ 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. $\endgroup$
    – David Z
    May 5, 2013 at 19:18
  • $\begingroup$ Related: physics.stackexchange.com/q/391/2451 and physics.stackexchange.com/q/55440/2451 $\endgroup$
    – Qmechanic ♦
    May 5, 2013 at 19:58

1 Answer 1

Reset to default
5
$\begingroup$

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.

Improve this answer
$\endgroup$
4
  • $\begingroup$ CPT is an exact symmetry but not the identity I suppose? So $1 \neq PCT$ when $1$ means the identity. E.g. an electron moving linearly to the right without angular momentum. C changes it into a positron. P and T change the motion of direction, so they cancel each other in this case. Together CPT gives a positron moving to the right? $\endgroup$
    – Gerard
    Nov 15, 2021 at 21:12
  • $\begingroup$ @Gerard I used a common abuse of notation here, where I don't specify that $PCT \vert \phi \rangle = 1 \vert \phi \rangle$ for all quantum field states $\vert \phi \rangle$. In addition, I believe your notion of time reversal is a little naive here, time reversal does not only change direction of motion, but also "flips" an electron to a positron. $\endgroup$
    – Neuneck
    Nov 17, 2021 at 11:03
  • $\begingroup$ Now I am lost. Not saying that you are wrong! How about the active and passive point of view? E.g. after applying P: did only the coordinate system flip while the particles remain at their original physical position or did the particles flip and the coordinate system not? Time reversal: does the electron become a positron moving forward in time or is the coordinate system of time reversed and the electron still an electron? $\endgroup$
    – Gerard
    Nov 19, 2021 at 15:30
  • $\begingroup$ @Gerard Things start to get subtle here and you need to carefully construct the action of the symmetry operators on your quantum fields. Parity does not just flip the coordinate system, it also flips spins and transforms left-handed weyl spinors into right-handed weyl spinors (which are coupled, iff they form a massive dirac spinor). Also, time reversal is unusual in that it is an anti-unitary operator. I've been out of physics for a few years now, so at this level, I'll have to refer you to text books on the topic. $\endgroup$
    – Neuneck
    Mar 22, 2022 at 10:17

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.