Quantum Field Theory

There are two problems with this equation. Firstly, it is not relativistic, reducing to classical mechanics rather than relativistic mechanics in the correspondence limit. To see this, we note that the first term on the left is only the classical kinetic energy p²/2m, with the rest energy mc² omitted. It is possible to modify the Schrödinger equation to include the rest energy, resulting in the Klein-Gordon equation or the Dirac equation. However, these equations have many unsatisfactory qualities; for instance, they possess energy spectra which extend to -∞, so that there is no ground state. Such inconsistencies occur because these equations neglect the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein's famous mass-energy relation predicts that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. For example, an electron and a positron can annihilate each other to create photons. Such processes must be accounted for in a truly relativistic quantum theory.

The second problem occurs when we seek to extend the equation to large numbers of particles. It was discovered that quantum mechanical particles of the same species are indistinguishable, in the sense that the wavefunction of the entire system must be symmetric (bosons) or antisymmetric (fermions) when the coordinates of its consituent particles are exchanged. This makes the wavefunction of systems of many particles extremely complicated. For example, the general wavefunction of a system of N bosons is written as

In second quantization, we make use of particle indistinguishability by specifying multi-particle wavefunctions in terms of single-particle occupation numbers. For example, suppose we have a system of N bosons which can occupy various single-particle states φ₁, φ₂, φ₃, and so on. The usual method of writing a multi-particle wavefunction is to assign a state to each particle and then impose exchange symmetry. As we have seen, the resulting wavefunction is an unwieldy sum of N! terms. In the second quantized approach, we simply list the number of particles in each of the single-particle states, with the understanding that the multi-particle wavefunction is symmetric. To be precise, suppose that N = 3, with one particle in state φ₁ and two in state φ₂. The normal way of writing the wavefunction is

Finally, we introduce field operators that define the probability of creating or destroying a particle at a particular point in space. It turns out that single-particle wavefunction are usually enumerated in terms of their momenta (as in the particle in a box problem), so field operators can be constructed by applying the Fourier transform to the creation and annihilation operators. For example, the bosonic field annihilation operator φ(r) (which is not to be confused with the wavefunction) is

See Wightman axioms.