Question 3.10: Antiquark Solutions in a Bag Show that a quark wave function......
Antiquark Solutions in a Bag
Show that a quark wave function obeying the equation
\mathrm{i}n\cdot\gamma(x){\bf q}(x)=-{\bf q}(x){\bigg|}_{R=R(\vartheta,\phi)} (1)
corresponds to antiquark solutions that fulfill
in\cdot\gamma(x)\tilde{\mathrm{q}}(x)=\tilde{\mathrm{q}}(x)~~. (2)
We have to recall that antiparticle wave functions are {\hat{C}}{\hat{P}}{\hat{T}} transforms of the corresponding particle solutions:
\tilde{\mathrm{q}}(x)=\hat{C}\hat{P}\hat{T}_{\mathrm{q}}(x)\ . (3)
Indeed {\hat{C}}{\hat{P}}{\hat{T}} transforms a spinor Ψ(x) into a spinor Ψ(−x) up to a phase factor, i.e. a particle moving forward in space and time is transformed into one moving backward in space and time. This corresponds to the Feynman–Stueckelberg interpretation of antiparticles.^{11}
In the standard representation
\hat{C}=\mathrm{i}\gamma_{2}\hat{K},\qquad\hat{P}=\gamma_{0},\qquad\hat{T}=\mathrm{i}\gamma_{1}\gamma_{3}\hat{K},\hat{K} being complex comjugation, and therefore
\hat{C}\hat{P}\hat{T}\cdot\cdot\cdot=\gamma_{0}\mathrm{i}\gamma_{2}\Big(\mathrm{i}\gamma_{1}\gamma_{3}(\cdot\cdot\cdot)^{\ast}\Big)^{\ast}\ . (4)
Applying (4) to (1) we obtain
\tilde{\mathfrak{q}}(x)=\hat{C}\hat{P}\hat{T}\Big[-\mathrm{i}n\cdot\gamma\mathrm{q}(x)\Big]\\ =\gamma_{0}\mathrm{i}\gamma_{2}\biggl[\mathrm{i}\gamma_{1}\gamma_{3}(\mathrm{i}n\cdot\gamma^{*})\bigl(\mathrm{q}(x)\bigr)^{*}\biggr]^{*}\\ =\gamma_{0}\mathrm{i}\gamma_{2}\biggl[(-\mathrm{i}n\cdot\gamma)\mathrm{i}\gamma_{1}\gamma_{3}\bigl(\mathrm{q}(x)\bigr)^{*}\biggr]^{*}\\=\gamma_{0}\mathrm{i}\gamma_{2}(\mathrm{i}n\cdot\gamma^{\star})\bigg[\mathrm{i}\gamma_{1}\gamma_{3}\big(\mathrm{q}(x)\big)^{\star}\bigg]^{\star}\\=\mathrm{i}n\cdot\gamma\gamma_{0}\mathrm{i}\gamma_{2}\biggl[\mathrm{i}\gamma_{1}\gamma_{3}\bigl(\mathrm{q}(x)\bigr)^{\ast}\biggr]^{\ast}\\
= in· \gamma \tilde{q}(x). (5)
Here we have repeatedly made use of the fact that \gamma_{1} and \gamma_{3} are real and that \gamma_{2} is purely imaginary.
The antiparticle solutions therefore fulfill the modified boundary condition (2), i.e. we have only to determine the solutions of (1) in order to obtain the quark spectrum in the MIT bag.
^{11}See also W. Greiner and J. Reinhardt: Field Quantization (Springer, Berlin, Heidelberg 1996) and W. Greiner: Relativistic Quantum Mechanics – Wave Equations, 3rd ed. (Springer, Berlin, Heidelberg 2000).