Quantum Field Theory

We interpret the state |⃗p〉 as the momentum eigenstate of a single particle of mass m.To stress this, from now on we’ll write E ⃗p everywhere instead of ω ⃗p . Let’s check thisparticle interpretation by studying the other quantum numbers of |⃗p〉. We may takethe classical total momentum P ⃗ given in (1.46) and turn it into an operator. Afternormal ordering, it becomes∫⃗P = −∫d 3 x π ∇φ ⃗ =d 3 p(2π) 3 ⃗p a† ⃗p a ⃗p (2.45)Acting on our state |⃗p〉 with ⃗ P, we learn that it is indeed an eigenstate,⃗P |⃗p〉 = ⃗p |⃗p〉 (2.46)telling us that the state |⃗p〉 has momentum ⃗p. Another property of |⃗p〉 that we canstudy is its angular momentum. Once again, we may take the classical expression forthe total angular momentum of the field (1.55) and turn it into an operator,J i = ǫ ijk ∫d 3 x (J 0 ) jk (2.47)It’s not hard to show that acting on the one-particle state with zero momentum,J i |⃗p = 0〉 = 0, which we interpret as telling us that the particle carries no internalangular momentum. In other words, quantizing a scalar field gives rise to a spin 0particle.Multi-Particle States, Bosonic Statistics and Fock SpaceWe can create multi-particle states by acting multiple times with a † ’s. We interpretthe state in which n a † ’s act on the vacuum as an n-particle state,|⃗p 1 , . . .,⃗p n 〉 = a † ⃗p 1. . .a † ⃗p n|0〉 (2.48)Because all the a † ’s commute among themselves, the state is symmetric under exchangeof any two particles. For example,This means that the particles are bosons.|⃗p,⃗q〉 = |⃗q, ⃗p〉 (2.49)The full Hilbert space of our theory is spanned by acting on the vacuum with allpossible combinations of a † ’s,|0〉 , a † ⃗p |0〉 , a† ⃗p a† ⃗q |0〉 , a† ⃗p a† ⃗q a† ⃗r|0〉... (2.50)– 30 –