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In quantum field theory, a bosonic field is a quantum field whose quanta are
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
s; that is, they obey
Bose–Einstein statistics In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic ...
. Bosonic fields obey
canonical commutation relation In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p_ ...
s, as distinct from the canonical anticommutation relations obeyed by fermionic fields. Examples include scalar fields, describing spin-0 particles such as the Higgs boson, and gauge fields, describing spin-1 particles such as the
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they a ...
.


Basic properties

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.


Examples

Examples of bosonic fields include scalar fields, gauge fields, and symmetric 2-tensor
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
, which are characterized by their covariance under
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
s and have spins 0, 1 and 2, respectively. Physical examples, in the same order, are the Higgs field, the photon field, and the graviton field. 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 In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
, one of the most successful theories in physics. Quantization of gravity, on the other hand, is a long-standing problem that has led to development of theories such as string theory and loop quantum gravity.


Spin and statistics

The
spin–statistics theorem In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles tha ...
implies that quantization of local, relativistic field theories in 3+1 dimensions may lead either to bosonic or fermionic quantum fields, i.e., fields obeying commutation or anti-commutation relations, according to whether they have integer or half-integer spin, respectively. Thus bosonic fields are one of the two theoretically possible types of quantum field, namely those corresponding to particles with integer spin. In a 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. Examples of non-relativistic bosonic fields include those describing cold bosonic atoms, such as Helium-4. Such non-relativistic fields are not as fundamental as their relativistic counterparts: they provide a convenient 're-packaging' of the many-body wave function describing the state of the system, whereas the relativistic fields described above are a necessary consequence of the consistent union of relativity and quantum mechanics.


See also

* Quantum triviality


References

* * * Peskin, M and Schroeder, D. (1995). ''An Introduction to Quantum Field Theory'', Westview Press. * Srednicki, Mark (2007).
Quantum Field Theory
', Cambridge University Press, {{ISBN, 978-0-521-86449-7. * Weinberg, Steven (1995). ''The Quantum Theory of Fields'', (3 volumes) Cambridge University Press. Quantum field theory