In
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, a bosonic field is a
quantum field
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct physical models of subatom ...
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 half odd-intege ...
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 identical particles may occupy a set of available discrete energy states at thermodynamic equilibri ...
. 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 relation
In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from the quantum mechanical study of bosons and fermions, respectively. They play a prominent role in ...
s obeyed by
fermionic fields.
Examples include
scalar fields
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
, describing spin-0 particles such as the
Higgs boson
The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field,
one of the field (physics), fields in particl ...
, 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 particles that can ...
.
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 field
In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
s,
gauge field
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
s, 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 by ...
, which are characterized by their
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.
The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one ...
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 vel ...
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 Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
, 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
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
and
loop quantum gravity
Loop quantum gravity (LQG) is a theory of quantum gravity that incorporates matter of the Standard Model into the framework established for the intrinsic quantum gravity case. It is an attempt to develop a quantum theory of gravity based direc ...
.
Spin and statistics
The
spin–statistics theorem
The spin–statistics theorem proves that the observed relationship between the intrinsic spin of a particle (angular momentum not due to the orbital motion) and the quantum particle statistics of collections of such particles is a consequence of ...
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
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
or
half-integer
In mathematics, a half-integer is a number of the form
n + \tfrac,
where n is an integer. For example,
4\tfrac12,\quad 7/2,\quad -\tfrac,\quad 8.5
are all ''half-integers''. The name "half-integer" is perhaps misleading, as each integer n is its ...
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
In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. ...
*
Composite field
In quantum field theory, a composite field is a field defined in terms of other more "elementary" fields. It might describe a composite particle (bound state) or it might not.
It might be local, or it might be nonlocal. However, "quantum field ...
*
Auxiliary field In physics, and especially quantum field theory, an auxiliary field is one whose equations of motion admit a single solution. Therefore, the Lagrangian (field theory), Lagrangian describing such a Field (physics), field A contains an algebraic quadr ...
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