G Parity Operator Calculation for Neutral and Charged Pion States

[kakarukeys]

why G parity = $$(-1)^I C$$?
C is the Charge conjugation number of the neutral member.

G parity of $$\pi^0$$ is very obvious. Given $$e^{i\pi I_2} |I\ 0\rangle = (-1)^I |I\ 0\rangle$$

How do you compute the G parity of $$\pi^+$$?

G parity operator
$$G = Ce^{i\pi I_2}$$

 

[marlon]

Look at http://ej.iop.org/links/q19/gde,d+XMMgTmN65bCrJ,UA/ejv11i2p99.pdf
or
http://www.phys.uAlberta.ca/~gingrich/phys512/latex2html/node64.html

Charge conjugation is determined by how physical entities (like the E field) change if you replace a charge by its opposite(this is what the charge conjugation operator does)...For example E will be come -E if you replace q by -q

marlon

to see how it is done : www.physics.ohio-state.edu/~kass/P780_L6_sp03.ppt

 

[sir-pinski]

K

The G parity operator is used to describe the behavior of particles under the combined operation of charge conjugation (C), parity (P), and time reversal (T). In the case of neutral and charged pion states, we are interested in the G parity operator because it allows us to determine the overall parity of these states.

The calculation for G parity of a neutral pion state is straightforward, as shown in the content. However, for a charged pion state, we need to take into account the charge conjugation number (C) of the neutral member, as mentioned in the content.

The charge conjugation number of a particle is a quantum number that describes how it transforms under charge conjugation. In the case of a neutral pion, its charge conjugation number is 1, meaning that it remains unchanged under charge conjugation. On the other hand, for a charged pion, its charge conjugation number is -1, meaning that it changes sign under charge conjugation.

Taking this into account, the G parity operator for a charged pion state becomes G = (-1)^I C e^{i\pi I_2} K, where I is the isospin quantum number and K is the operator for time reversal. This is because the charge conjugation operator, when acting on a charged pion state, will change its sign due to the charge conjugation number of -1, while the time reversal operator K will not change its sign.

Therefore, the G parity of a charged pion state can be calculated as G |I\ 1\rangle = (-1)^I C e^{i\pi I_2} K |I\ 1\rangle = (-1)^I C (-1)^I |I\ 1\rangle = (-1)^{2I} |I\ 1\rangle = 1, which is consistent with the fact that charged pions have positive G parity.

In summary, the G parity operator calculation for neutral and charged pion states takes into account the charge conjugation number of the neutral member in order to properly determine the overall parity of these states. This is why G parity = (-1)^I C, where C is the charge conjugation number of the neutral member.