I'm still stuck on this. I tried to go for the former of the two approaches mentioned above. I let \psi\rightarrow\psi_L and \overline{\psi}\rightarrow\overline{\psi}_L, then:
L=c\overline{\psi}_L\left(\mathrm{i}\hbar\gamma^\mu\partial_\mu-mc\right)\psi_L
Splitting this into two terms and...
Homework Statement
From Mandl and Shaw (exercise 4.5):
Deduce the equations of motion for the fields:
\psi_L(x)\equiv{1 \over 2} (1-\gamma_5)\psi(x)
\psi_R(x)\equiv{1 \over 2} (1+\gamma_5)\psi(x)
for non-vanishing mass, and show that they decouple in the limit m=0. Hence show that the...
Homework Statement
Given that, for operators A and B:
\mathrm{e}^{\mathrm{i} \alpha A} B \mathrm{e}^{-\mathrm{i} \alpha A} = \sum_{n=0}^{\infinity} {(\mathrm{i}\alpha)^n \over n!} B_n
where B_0 = B and B_n = [A, B_{n-1}] for n=1,2,...
show that:
P_1...
Actually no worries I've sorted it: applying charge conjugation twice takes one back to the original state. It's so simple I don't know why I didn't think of it!
I have another question. It can be shown that C a(\mathbf{k})C^{-1}=\eta_c b(\mathbf{k}) and C b(\mathbf{k})C^{-1}=\eta_c^* a(\mathbf{k}).
I can show the first one fairly easily by just substituting the full expressions of the fields into the transformation condition above. I'm a little...
Homework Statement
Show that the complex Klein-Gordon Lagrangian density:
L=N\left(\partial_\alpha\phi^{\dagger}(x)\partial^\alpha\phi(x)-\mu^2\phi^{\dagger}(x)\phi(x)\right)
is invariant under charge conjugation:
\phi(x)\rightarrow C\phi(x)C^{-1}=\eta_c \phi^\dagger (x)
Where C...
Okay, I've pretty much arrived at the answer, except I still need to remove a factor of V somehow. After I integrate I get this:
\omega_{\mathbf{k'}} \sum_{\mathbf{k}} \sqrt{\frac{2 \hbar c^2 }{V \omega_{\mathbf{k}}}} \left( a(\mathbf{k}) \delta_{kk'} + a^\dagger (\mathbf{k})...
Awesome. That's really helpful! I feel I have enough information to progress now. Thanks a lot!
Actually one last thing. How would you represent the second of the Dirac deltas (with the plus in) in Kronecker delta format? Or would it be eliminated in the integral by default?
EDIT: actually how...
Right, I've had another go at this. Here's my working so far.
First I'll just outline the Fourier transform I'll be using (in one dimension).
\int^\infty_{-\infty} \mathrm{e}^{\mathrm{i}a x} \mathrm{e}^{-\mathrm{i}w x}\mathrm{d}x=2\pi \delta(w-a)
And the inverse:
\int^\infty_{-\infty}...
Except mine apparently. I did Fourier transforms but the course skipped over delta functions. Thanks for the help. I'll read around a bit more, have another stab at it and let you know if I run into any more problems.
I'm self-teaching from Mandl and Shaw, and haven't seen anything like that yet :-S. I'm basically attempting exercise 3.1, from chapter 3 on the Klein Gordon field.