section_path: Quantum field theory/Quantization of the electromagnetic field
section_name: Quantization of the electromagnetic field
paragraph_text: Quantization of the electromagnetic field
  
  In 1910, Peter Debye derived Planck's law of black-body radiation from a relatively simple assumption. He decomposed the electromagnetic field in a cavity into its Fourier modes, and assumed that the energy in any mode was an integer multiple of h ν {\displaystyle h\nu }, where ν {\displaystyle \nu } is the frequency of the electromagnetic mode. Planck's law of black-body radiation follows immediately as a geometric sum. However, Debye's approach failed to give the correct formula for the energy fluctuations of black-body radiation, which were derived by Einstein in 1909.
  
  In 1925, Born, Heisenberg and Jordan reinterpreted Debye's concept in a key way. As may be shown classically, the Fourier modes of the electromagnetic field —a complete set of electromagnetic plane waves indexed by their wave vector k and polarization state—are equivalent to a set of uncoupled simple harmonic oscillators. Treated quantum mechanically, the energy levels of such oscillators are known to be E = n h ν {\displaystyle E=nh\nu }, where ν {\displaystyle \nu } is the oscillator frequency. The key new step was to identify an electromagnetic mode with energy E = n h ν {\displaystyle E=nh\nu } as a state with n {\displaystyle n} photons, each of energy h ν {\displaystyle h\nu }. This approach gives the correct energy fluctuation formula.
  
  Dirac took this one step further. He treated the interaction between a charge and an electromagnetic field as a small perturbation that induces transitions in the photon states, changing the numbers of photons in the modes, while conserving energy and momentum overall. Dirac was able to derive Einstein's A i j {\displaystyle A_{ij}} and B i j {\displaystyle B_{ij}} coefficients from first principles, and showed that the Bose–Einstein statistics of photons is a natural consequence of quantizing the electromagnetic field correctly (Bose's reasoning went in the opposite direction; he derived Planck's law of black-body radiation by assuming B–E statistics). In Dirac's time, it was not yet known that all bosons, including photons, must obey Bose–Einstein statistics.
  
  Dirac's second-order perturbation theory can involve virtual photons, transient intermediate states of the electromagnetic field; the static electric and magnetic interactions are mediated by such virtual photons. In such quantum field theories, the probability amplitude of observable events is calculated by summing over all possible intermediate steps, even ones that are unphysical; hence, virtual photons are not constrained to satisfy E = p c {\displaystyle E=pc}, and may have extra polarization states; depending on the gauge used, virtual photons may have three or four polarization states, instead of the two states of real photons. Although these transient virtual photons can never be observed, they contribute measurably to the probabilities of observable events.
  
  Indeed, such second-order and higher-order perturbation calculations can give apparently infinite contributions to the sum. Such unphysical results are corrected for using the technique of renormalization.
  
  Other virtual particles may contribute to the summation as well; for example, two photons may interact indirectly through virtual electron – positron pairs. Such photon–photon scattering (see two-photon physics), as well as electron–photon scattering, is meant to be one of the modes of operations of the planned particle accelerator, the International Linear Collider.
  
  In modern physics notation, the quantum state of the electromagnetic field is written as a Fock state, a tensor product of the states for each electromagnetic mode
  
  where | n k i ⟩ {\displaystyle |n_{k_{i}}\rangle } represents the state in which n k i {\displaystyle \,n_{k_{i}}} photons are in the mode k i {\displaystyle k_{i}}. In this notation, the creation of a new photon in mode k i {\displaystyle k_{i}} (e.g., emitted from an atomic transition) is written as | n k i ⟩ → | n k i + 1 ⟩ {\displaystyle |n_{k_{i}}\rangle \rightarrow |n_{k_{i}}+1\rangle }. This notation merely expresses the concept of Born, Heisenberg and Jordan described above, and does not add any physics.