Consider a complex (that is, nonhermitian) scalar field φ with lagrangian density
L=−∂μφ†∂μφ−m2φ†φ+Ω0.(3.37)
a) Show that φ obeys the Klein-Gordon equation.
b) Treat φ and φ† as independent fields, and find the conjugate momentum for each. Compute the hamiltonian density in terms of these conjugate momenta and the fields themselves (but not their time derivatives).
c) Write the mode expansion of φ as
φ(x)=∫dk[a(k)eikx+b†(k)e−ikx].(3.38)
Express a(k) and b(k) in terms of φ and φ† and their time derivatives.
d) Assuming canonical commutation relations for the fields and their conjugate momenta, find the commutation relations obeyed by a(k) and b(k) and their hermitian conjugates.
e) Express the hamiltonian in terms of a(k) and b(k) and their hermitian conjugates. What value must Ω0 have in order for the ground state to have zero energy?