Bosonic field

In quantum field theory, bosonic fields are quantum fields whose quanta are bosons; that is, they obey Bose-Einstein statistics.

Examples

Examples of bosonic fields include scalar fields, vector fields, and tensor fields, as characterized by their properties under Lorentz transformations (or equivalently by their spin, 0, 1 and 2, respectively). Physical examples, in the same order, are: the Higgs field, the photon field and the graviton field. While the first one remains to be observed, it is widely believed to exist. Of the last two, only the photon field can be quantized using the conventional methods of canonical or path integral quantization. This has led to the theory of quantum electrodynamics, arguably the most successful theory in Physics. Quantization of gravity, on the other hand, is a long standing problem that has led to theories such as string theory.

Properties

By definition, free (non-interacting) bosonic fields obey canonical commutation relations. Those relations also hold for interacting bosonic fields in the interaction picture, where the fields evolve in time as if free and the effects of the interaction are encoded in the evolution of the states. It is these commutation relations that imply Bose-Einstein statistics for the field quanta.

As implied by the Spin-statistics theorem, quantization of local, relativistic field theories in 3+1 dimensions may lead to either bosonic or fermionic quantum fields, i.e., fields obeying commutation or anti-commutation relations, according to whether they have integral or half integral spin, respectively. In this sense, bosonic fields are one of the two theoretically possible types of quantum field, namely those with integral spin. In lower dimensions, e.g. 2+1, one may have other types of fields, such as anyons, that obey fractional statistics.

In non-relativistic many-body theory, the spin and the statistical properties of the quanta are not directly related. In fact, the commutation or anti-commutation relations are assumed based on whether the theory one intends to study corresponds to particles obeying Bose-Einstein or Fermi-Dirac statistics. In this context the spin remains an internal quantum number that is only phenomenologically related to the statistical properties of the quanta. It must be stressed that such non-relativistic fields arise merely as an extremely convenient 're-packaging' of the many-body wave function describing the state of the system. This is to be contrasted with their relativistic counterparts (described above): they are a necessary consequence of the consistent union of relativity and quantum mechanics. Examples of non-relativistic bosonic fields include those describing cold bosonic atoms, such as Helium-4.

See also

• Fermionic field
• Spin-statistics theorem