7.8. Anti-particles Consider the free theory of a complex field (x). In the Minkowski space the action

7.8. Anti-particles Consider the free theory of a complex field φ(x). In the Minkowski space the action is S =  ddx∂μφ

∗

∂μφ −m2φ

∗

φ.

a. Show that the Hamiltonian is given by H =  dd−1x(π

∗

π +∇φ

∗ · ∇φ +m2φ

∗

φ).

b. Prove that the system is invariant under the continuous symmetry

φ→eiαφ, φ

∗→e−iαφ.

Use the Noether theorem to derive the conserved charge Q = −i  dd−1x(π

∗

φ

∗ −πφ).

c. Diagonalize the Hamiltonian by introducing the creation and annihilation operators.

Show that the theory contains two sets of operators that can be distinguished by the different eigenvalues of the charge Q: the first set describes the creation and the annihilation of a particle A, while the second set describes the same processes for an anti-particle A¯.

d. Show that the propagation of a particle in a space-like interval is the same as the propagation of an anti-particle back in time.