Scalar field (quantum field theory)

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In quantum field theory, a scalar field is a quantum field whose quanta are spin-zero particles. As with any particles of integer spin, scalar particles obey Bose-Einstein statistics; they are bosons. The field transforms as a Lorentz scalar under general Lorentz transforms, and is thus an example of a general scalar field. The most common example of a scalar field is the quantization of the Klein-Gordon equation which is a free field, sometimes called a Klein-Gordon field.

Although scalar fields are Lorentz scalars, they may (and usually do) transform nontrivially under other symmetries, such as flavour or isospin. For example, the pion is invariant under the restricted Lorentz group, but is an isospin triplet furthermore (meaning it transforms like a three component vector under the SU(2) isospin symmetry). Furthermore, it picks up a negative phase under parity inversion, so it transforms nontrivially under the full Lorentz group; such particles are called pseudoscalar rather than scalar. Most mesons are only pseudoscalar particles.

Because of the relative simplicity of the mathematics involved, quantum scalar fields are often the first field introduced to a student of quantum field theory. The simplest example of an interacting field theory is the φ⁴ theory. The Yukawa potential describes how a scalar field can generate a force between spin-1/2 particles, and in particular, how pions mediate the strong interaction between nucleons. The Yukawa interaction is the most commmon way of coupling a scalar field to a spinor field.

Another phenomenon commonly studied in the context of the scalar field is in the Higgs mechanism. In supersymmetry, the auxiliary fields are commonly scalar fields; note however, unlike that described below, they do not have a kinetic term(!).

Contents

1 Free field Lagrangian
2 Equations of motion
3 Propagator
4 The action
5 References

Free field Lagrangian

The generic free complex scalar field Lagrangian is given by

[\mathcal(\phi(x)) =\frac \left[ partial^mu overline^a(x) partial_mu phi^b (x) - m^overline^a(x) phi^b(x) right] \delta_]

Natural units are assumed, that is, [\hbar=c=1].

The argument x indicates tha the field [\phi] is a function of the position x, usually taken to be four-dimensional spacetime, although it can be two-dimensional in conformal field theory or five or more dimensional in Kaluza-Klein theory. The [\partial_\mu] indicates the partial derivative with respect to the space-time coordinate. The greek-letter index [\mu] indicates the spatial direction of the derivative.

The repeated greek-index notation indicates the Einstein convention for summation over repeated indices. The upper and lower indices indicate a contraction or inner product

[\partial^\mu \overline \partial_\mu \phi = g_ \partial^\mu \overline \partial^\nu \phi ]

where [g_] is the metric tensor. In flat Minkowski space, it has a particularly simple form: it is diagonal, with entries +--- or -+++ along the diagonal, depending on one's conventions.

By definition, a scalar field transforms trivial under the Lorentz group. However if there are many particles, there may be an internal symmetry among them, in which case the scalar field may transform nontrivially under the symmetry group. Examples include the flavour symmetry among quarks or the isospin symmetry among nucleons, which are described by the symmetry groups SU(3) and SU(3) respectively. The roman-alphabet indices denote components of the vectors in the representation of the group. Thus for example, in the case of an isospin triplet, the indices a and b would run over the values 1,2,3, while for mesons belonging to the eightfold way, these indexes would run over the values 1,...,8 for the adjoint representation of SU(3). In the simplest case, of no flavour symmetry, these indices can be entirely dropped and ignored.

Any electrically charged field should be described by a complex scalar field. The overline [\overline] denotes the complex conjugate field, whose particle excitations are the antiparticles. On the other hand, an electrically neutral field is described by a real scalar field, that is, [\overline=\phi.]

Equations of motion

The equations of motion for the classical Klein-Gordon field (that is, the unquantized field) are the Klein-Gordon equations. These may be obtained by applying Hamilton's variational principle to the Lagrangian, to obtain the Euler-Lagrange equations, which are seen to be the Klein-Gordon equation. This equation is, by construction, relativistically covariant.

Propagator

The propagator of the free relativistic scalar boson is (in natural units):

[ D(k) = ]

where k is wave number and m is the scalar boson's mass.

The action

The action of the second quantized field theory is written as

[Z[J, overline J] = \int D[phi] \exp\left(\frac \int d^4x \mathcal(\phi(x)) + \overline(x)\phi(x) + \overline(x) J(x) \right)]

where J is te interaction current, and [\mathcal(\phi(x))] is the Lagrangian given above. The integal [\int D[phi]] is the Feynman path integral or functional integral of quantum field theory. Note that the inner integral over the Lagrangian corresponds to the classical action.

References

A. Zee, Quantum Field Theory in a Nutshell, Princeton: Princeton University Press, 2003. ISBN 0-691-01019-6. (Page 109 in Chapter II.2.)

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