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In quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
, a bosonic field is a quantum field whose quanta are boson
Boson

In particle physics, bosons are subatomic particle which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein....
s; that is, they obey Bose-Einstein statistics. Bosonic fields obey canonical commutation relation
Canonical commutation relation

In physics, the canonical commutation relation is the relation between canonical conjugate quantities , for example:between the position and momentum in the direction of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Planck's constant ....
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
Higgs boson

In particle physics, the Higgs boson is a massive Scalar field theory elementary particle predicted to exist by the Standard Model.The Higgs boson is the only Standard Model particle that has not yet been observed....
, and gauge fields, describing spin 1 particles such as the photon.

Basic properties
Free (non-interacting) bosonic fields obey canonical commutation relation
Canonical commutation relation

In physics, the canonical commutation relation is the relation between canonical conjugate quantities , for example:between the position and momentum in the direction of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Planck's constant ....
s.






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In quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
, a bosonic field is a quantum field whose quanta are boson
Boson

In particle physics, bosons are subatomic particle which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein....
s; that is, they obey Bose-Einstein statistics. Bosonic fields obey canonical commutation relation
Canonical commutation relation

In physics, the canonical commutation relation is the relation between canonical conjugate quantities , for example:between the position and momentum in the direction of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Planck's constant ....
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
Higgs boson

In particle physics, the Higgs boson is a massive Scalar field theory elementary particle predicted to exist by the Standard Model.The Higgs boson is the only Standard Model particle that has not yet been observed....
, and gauge fields, describing spin 1 particles such as the photon.

Basic properties


Free (non-interacting) bosonic fields obey canonical commutation relation
Canonical commutation relation

In physics, the canonical commutation relation is the relation between canonical conjugate quantities , for example:between the position and momentum in the direction of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Planck's constant ....
s. 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
Symmetric tensor

In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments. Symmetric tensors of rank two are sometimes called quadratic forms....
 fields
Tensor field

In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. It is used in differential geometry and the theory of manifolds, in algebraic geometry, in general relativity, in the analysis of stress and strain tensor in materials, and in numerous applications in the physical sciences and en...
, which are characterized by their covariance
Covariance

In probability theory and statistics, covariance is a measure of how much two variables change together .If two variables tend to vary together , then the covariance between the two variables will be positive....
 under Lorentz transformation
Lorentz transformation

In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other....
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. 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, one of the most successful theories in physics. Quantization of gravity
Quantum gravity

Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the Fundamental interaction , with general relativity, the theory of the fourth fundamental force: Gravitation....
, on the other hand, is a long standing problem that has led to development of theories such as string theory
String theory

String theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum gravity. The String s of string theory are one-dimensional oscillating lines, but they are no longer considered fundamental to the theory, which can be formulated in terms of points or surfaces too....
 and loop quantum gravity
Loop quantum gravity

Loop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity....
.

Spin and statistics


The spin-statistics theorem
Spin-statistics theorem

In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics obeyed by it. The spin of a particle is its intrinsic angular momentum ....
 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
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
 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

  • Fermionic field
    Fermionic field

    In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi-Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields....
  • Spin-statistics theorem
    Spin-statistics theorem

    In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics obeyed by it. The spin of a particle is its intrinsic angular momentum ....