An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 2.2

The Klein-Gordon Field — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · complex-scalar-field

Problem

The complex scalar field. Consider the field theory of a complex-valued scalar field obeying the Klein-Gordon equation. The action of this theory is S=∫d4x (∂μϕ∗∂μϕ−m2ϕ∗ϕ).S = \int d^4x\,(\partial_\mu\phi^*\partial^\mu\phi - m^2\phi^*\phi). It is easiest to analyze this theory by considering ϕ(x)\phi(x) and ϕ∗(x)\phi^*(x), rather than the real and imaginary parts of ϕ(x)\phi(x), as the basic dynamical variables. (a) Find the conjugate momenta to ϕ(x)\phi(x) and ϕ∗(x)\phi^*(x) and the canonical commutation relations. Show that the Hamiltonian is H=∫d3x (π∗π+∇ϕ∗⋅∇ϕ+m2ϕ∗ϕ).H = \int d^3x\,(\pi^*\pi + \nabla\phi^* \cdot \nabla\phi + m^2\phi^*\phi). Compute the Heisenberg equation of motion for ϕ(x)\phi(x) and show that it is indeed the Klein-Gordon equation. (b) Diagonalize HH by introducing creation and annihilation operators. Show that the theory contains two sets of particles of mass mm. (c) Rewrite the conserved charge Q=∫d3x i2(ϕ∗π∗−πϕ)Q = \int d^3x\,\frac{i}{2}(\phi^*\pi^* - \pi\phi) in terms of creation and annihilation operators, and evaluate the charge of the particles of each type. (d) Consider the case of two complex Klein-Gordon fields with the same mass. Label the fields as ϕa(x)\phi_a(x), where a=1,2a = 1, 2. Show that there are now four conserved charges, one given by the generalization of part (c), and the other three given by Qi=∫d3x i2(ϕa∗(σi)abπb∗−πa(σi)abϕb),Q^i = \int d^3x\,\frac{i}{2}(\phi^*_a(\sigma^i)_{ab}\pi^*_b - \pi_a(\sigma^i)_{ab}\phi_b), where σi\sigma^i are the Pauli sigma matrices. Show that these three charges have the commutation relations of angular momentum (SU(2)SU(2)). Generalize these results to the case of nn identical complex scalar fields.

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