field is analogous to the promotion of a classical harmonic oscillator to a quantum harmonic oscillator. The displacement of a classical harmonic oscillator is described by where is a complex number (normalized by convention), and is the oscillator's frequency. Note that is the displacement of a particle in simple harmonic motion from the equilibrium position, not to be confused with the spatial label of a quantum field. For a quantum harmonic oscillator, is promoted to a linear operator : Complex numbers and are replaced by the annihilation operator and the creation operator , respectively, where denotes Hermitian conjugation. The commutation relation between the two is The Hamiltonian of the simple harmonic oscillator can be written as The vacuum state , which is the lowest energy state, is defined by and has energy One can easily check that which implies that increases the energy of the simple harmonic oscillator by . For example, the state is an eigenstate of energy . Any energy eigenstate state of a single harmonic oscillator can be obtained from by successively applying the creation operator : and any state of the system can be expressed as a linear combination of the states A similar procedure can be applied to the real scalar field , by promoting it to a quantum field operator , while the annihilation operator , the creation operator and the angular frequency are now for a particular : Their commutation relations are: where is the Dirac delta function. The vacuum state is defined by Any quantum state of the field can be obtained from by successively applying creation operators (or by a linear combination of such states), e.g. While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy levels of an arbitrary number of particles. The latter space is known as a Fock space, which can account for the fact that particle numbers are not fixed in relativistic quantum systems. The process of quantizing an arbitrary number of particles instead of a single particle is often also called second quantization. The foregoing procedure is a direct application of non-relativistic quantum mechanics and can be used to quantize (complex) scalar fields, Dirac fields, vector fields (e.g. the electromagnetic field), and even strings. However, creation and annihilation operators are only well defined in the