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I'm following Wong's Introductory Nuclear Physics, 2e. In section 2-4 there's a discussion of charge conjugation I don't fully follow. Something simple is escaping me. The excerpt is produced below, but here's a quick run down:

  • The proton state is given as a second-quantization real-particle creation operator acting on the vacuum: $|p\rangle = a^\dagger_{1/2,+1/2} |0\rangle$, where the subscripts refer to the isospin $t=1/2$ of a nucleon and the eigenvalue $t_0=+1/2$ of the third-component of isospin (to distinguish the proton from the neutron $a^\dagger_{1/2,-1/2}|0\rangle$).
  • Antiprotons are given as an anti-particle creation operator $|\bar p\rangle=b^\dagger_{1/2,-1/2}$
  • It is stated without proof (which is okay) that the two creation operators must be related by $b^\dagger_{t, t_0}=(-1)^{t-t_0}a_{t,-t_0}$
  • Then the part I don't understand is how charge conjugation (particle-antiparticle) is given for the proton as $$|p\rangle\xrightarrow[\ C\ ]{}(-1)^{1/2+1/2}|\bar p\rangle$$

It's unclear to me how the creation operators and the operator relations are being used together to produce that charge conjugation mapping. If this other Exchange question addresses it, I don't see how or it is beyond me.

What exactly is the mapping $C$ they are using and how is it related to what was previously given? More specifically, what are the detailed steps between 2-23 through 2-25 and 2-26?

I tried starting with the vacuum, then applying the particle-creation, particle-destruction, and anti-particle creation operators but couldn't get their relation; I wasn't sure of my particle destruction operator. Perhaps they are relying on other physics not given in the discussion.

Wong

BMS
  • 11,493
  • Maybe reading this will help on the definition of operators http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qmoper.html . http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qmoper.html "Associated with each measurable parameter in a physical system is a quantum mechanical operator." charge and charge conjugation are measurable parameters. also this history of the operation https://www.britannica.com/science/charge-conjugation – anna v Nov 25 '23 at 05:55
  • Hi @annav. I'm familiar with the ideas of observables and operators. But it's unclear to me how Eqn 2-26 follows from 2-23 through 2-25. – BMS Nov 25 '23 at 06:36
  • You are asking about C in the text? I think it represents the act of the writer/reader in changing the charges of the particles. The same for parity. When one does a parity operation it is a change in the coordinates, a spatial inversion. IMHO – anna v Nov 25 '23 at 10:22
  • in addition to the above of course, the P operator es dependent on the coordinates used to model the interactions among the charges. – anna v Nov 25 '23 at 14:27
  • Closely related: https://physics.stackexchange.com/q/110719/44126 – rob Nov 29 '23 at 01:29
  • I think I have answered this question long ago here https://physics.stackexchange.com/questions/110719/phase-factors-under-rotations-of-strong-and-weak-isospin You just need to replace everywhere u--> p and d-->n and you will be done – TwoBs Dec 04 '23 at 16:42
  • @TwoBs So is the idea that Wong could have said "it can be shown that..."? Meaning that it's not a simple step or two to show, and something along the lines of your previous answer is required? I wrote this question assuming I was missing something at the surface level. – BMS Dec 05 '23 at 07:31
  • The paragraph in my old answer from long ago that starts with ‘However’ it’s enough to get the gist and it isn’t that complicated after all. – TwoBs Dec 05 '23 at 21:33
  • I don't see Eqn. 2-25 in that answer. How would Wong explain the steps using what was developed? – BMS Dec 06 '23 at 18:34

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