This chapter deals with the quantum theory of a free scalar field. We will see how it corresponds to the existence of particles.

We must first recall the quantum theory of a harmonic oscillator, with the Hamiltonian (16.9). In the quantum theory q and p correspond to operators qˆ and pˆ, which satisfy the commutation relations

[qˆ, pˆ] = i. (17.1)

The Hamiltonian operator is

Hˆ = 1

2 qˆ2 +

2 ω2pˆ2. (17.2)

In the usual description of the quantum theory, qˆ and pˆ are constant operators, and the state vector has a time dependence given by the Schro¨dinger equation. That is called the Schro¨dinger picture and is usually the most convenient. But for the particular case of the harmonic oscillator, it is easier to go to what is called the Heisenberg picture, in which the state vector is constant while qˆ and pˆ satisfy the classical equations of motion ˆ¨q = −ω2qˆ. In the Heisenberg picture we can write

qˆ(t) = 1√ 2ω

[ e−iωtaˆ+ eiωtaˆ†

] . (17.3)

the The commutator (17.1) is equivalent to[

aˆ, aˆ† ] = 1. (17.4)

Using Eq. (17.3) the Hamiltonian is

Hˆ =

( aˆ†aˆ+

) ω. (17.5)

Eqs. (17.1) and (17.5) give the commutator

[Hˆ, aˆ†] = aˆ†ω, (17.6)

which is equivalent to Hˆaˆ† = (1 + ω) aˆ†Hˆ. (17.7)

This means that aˆ†, acting on a state with definite energy, increases the energy by an amount ω. Taking the Hermitian conjugates1 of Eqs. (17.6) and (17.7), we learn that aˆ decreases the energy by the same amount. The state |0〉 satisfying aˆ|0〉 = |0〉 is the ground state, and from Eq. (17.5) it’s energy is 12ω. The excited states obtained by the action of aˆ† have energies 32ω,

5 2ω · · · .