Problems QFT 2

1. Show that the invariance of scalar field theory w.r.t. rotations leads to angular momentum conservation. Give the expression of the angular momentum of a scalar field.
2. In the case of two scalar hermitian fields [math]\displaystyle{ \Phi_a(x),\ a=1,2 }[/math] with the same mass, show that the lagrangian density [math]\displaystyle{ {\mathcal L}=(1/2)\sum_a(\partial_\mu \Phi_a)(\partial^\mu\Phi_a)-m^2\Phi_a^2 }[/math] has a continuous symmetry. Calculate the associated charge. Compare with the case of a complex scalar field.
3. In the case of two complex scalar fields [math]\displaystyle{ \Phi_a(x),\ a=1,2 }[/math] with the same mass, show that the lagrangian density [math]\displaystyle{ {\mathcal L}=\sum_a(\partial_\mu \Phi_a^*)(\partial^\mu\Phi_a)-m^2|\Phi_a|^2 }[/math] has several continuous symmetries. Calculate the corresponding charges.
4. Calculate the norm of a 1-particle state in the case of an inverted bosonic algebra [math]\displaystyle{ [a^\dagger,a]=1 }[/math].
5. In the case of a complex scalar field, show that the charge can be written [math]\displaystyle{ Q=\int\frac{d^3p}{(2\pi)^3}(a^\dagger_{\vec p} a_{\vec p}-b^\dagger_{\vec p} b_{\vec p}) }[/math].