Klein–Gordon equation

The Klein–Gordon equation is a relativistic wave equation, related to the Schrödinger equation. It is named after Oskar Klein and Walter Gordon. It is second-order in space and time and manifestly Lorentz-covariant. It is a differential equation version of the relativistic energy–momentum relation.

Statement

The Klein–Gordon equation can be written in different ways. The equation itself usually refers to the position space form, where it can be written in terms of separated space and time components or by combining them into a four-vector By Fourier transforming the field into momentum space, the solution is usually written in terms of a superposition of plane waves whose energy and momentum obey the energy-momentum dispersion relation from special relativity. Here, the Klein–Gordon equation is given for both of the two common metric signature conventions

	Position space	Fourier transformation	Momentum space
Separated time and space
Four-vector form

Here, is the wave operator and is the Laplace operator. The speed of light and Planck constant are often seen to clutter the equations, so they are therefore often expressed in natural units where

	Position space	Fourier transformation	Momentum space
Separated time and space
Four-vector form

Unlike the Schrödinger equation, the Klein–Gordon equation admits two values of for each : One positive and one negative. Only by separating out the positive and negative frequency parts does one obtain an equation describing a relativistic wavefunction. For the time-independent case, the Klein–Gordon equation becomes
which is formally the same as the homogeneous screened Poisson equation. In addition, the Klein–Gordon equation can also be represented as:
where, the momentum operator is given as:

Relevance

The equation is to be understood first as a classical continuous scalar field equation that can be quantized. The quantization process introduces then a quantum field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation.
The equation solutions include a scalar or pseudoscalar field. In the realm of particle physics electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, the practical utility for particles like pions is limited. There is a second version of the equation for a complex scalar field that is theoretically important being the equation of the Higgs Boson. In the realm of condensed matter it can be used for many approximations of quasi-particles without spin.
The equation can be put into the form of a Schrödinger equation. In this form it is expressed as two coupled differential equations, each of first order in time. The solutions have two components, reflecting the charge degree of freedom in relativity. It admits a conserved quantity, but this is not positive definite. The wave function cannot therefore be interpreted as a probability amplitude. The conserved quantity is instead interpreted as electric charge, and the norm squared of the wave function is interpreted as a charge density. The equation describes all spinless particles with positive, negative, and zero charge.
Any solution of the free Dirac equation is, for each of its four components, a solution of the free Klein–Gordon equation. Although historically it was invented as a single particle equation, the Klein–Gordon equation cannot form the basis of a consistent quantum relativistic one-particle theory; any relativistic theory implies creation and annihilation of particles beyond a certain energy threshold.

Solution for free particle

Here, the Klein–Gordon equation in natural units,, with the metric signature is solved by Fourier transformation. Inserting the Fourier transformationand using orthogonality of the complex exponentials gives the dispersion relationThis restricts the momenta to those that lie on shell, giving positive and negative energy solutionsFor a new set of constants, the solution then becomesIt is common to handle the positive and negative energy solutions by separating out the negative energies and work only with positive :In the last step, was renamed. Now we can perform the -integration, picking up the positive frequency part from the delta function only:
This is commonly taken as a general solution to the free Klein–Gordon equation. Note that because the initial Fourier transformation contained Lorentz invariant quantities like only, the last expression is also a Lorentz invariant solution to the Klein–Gordon equation. If one does not require Lorentz invariance, one can absorb the -factor into the coefficients and.

History

The equation was named after the physicists Oskar Klein and Walter Gordon, who in 1926 proposed that it describes relativistic electrons. Vladimir Fock also discovered the equation independently in 1926 slightly after Klein's work, in that Klein's paper was received on 28 April 1926, Fock's paper was received on 30 July 1926 and Gordon's paper on 29 September 1926. Other authors making similar claims in that same year include Johann Kudar, Théophile de Donder and Frans-H. van den Dungen, and Louis de Broglie. Although it turned out that modeling the electron's spin required the Dirac equation, the Klein–Gordon equation correctly describes the spinless relativistic composite particles, like the pion. On 4 July 2012, European Organization for Nuclear Research CERN announced the discovery of the Higgs boson. Since the Higgs boson is a spin-zero particle, it is the first observed ostensibly elementary particle to be described by the Klein–Gordon equation. Further experimentation and analysis is required to discern whether the Higgs boson observed is that of the Standard Model or a more exotic, possibly composite, form.
The Klein–Gordon equation was first considered as a quantum wave equation by Erwin Schrödinger in his search for an equation describing de Broglie waves. The equation is found in his notebooks from late 1925, and he appears to have prepared a manuscript applying it to the hydrogen atom. Yet, because it fails to take into account the electron's spin, the equation predicts the hydrogen atom's fine structure incorrectly, including overestimating the overall magnitude of the splitting pattern by a factor of for the -th energy level. The Dirac equation relativistic spectrum is, however, easily recovered if the orbital-momentum quantum number is replaced by total angular-momentum quantum number. In January 1926, Schrödinger submitted for publication instead his equation, a non-relativistic approximation that predicts the Bohr energy levels of hydrogen without fine structure.
In 1926, soon after the Schrödinger equation was introduced, Vladimir Fock wrote an article about its generalization for the case of magnetic fields, where forces were dependent on velocity, and independently derived this equation. Both Klein and Fock used Theodor Kaluza and Klein's method. Fock also determined the gauge theory for the wave equation. The Klein–Gordon equation for a free particle has a simple plane-wave solution.

Derivation

The non-relativistic equation for the energy of a free particle is
By quantizing this, we get the non-relativistic Schrödinger equation for a free particle:
where
is the momentum operator, and
is the energy operator.
The Schrödinger equation suffers from not being relativistically invariant, meaning that it is inconsistent with special relativity.
It is natural to try to use the identity from special relativity describing the energy:
Then, just inserting the quantum-mechanical operators for momentum and energy yields the equation
The square root of a differential operator can be defined with the help of Fourier transformations, but due to the asymmetry of space and time derivatives, Dirac found it impossible to include external electromagnetic fields in a relativistically invariant way. So he looked for another equation that can be modified in order to describe the action of electromagnetic forces. In addition, this equation, as it stands, is nonlocal.
Klein and Gordon instead began with the square of the above identity, i.e.
which, when quantized, gives
which simplifies to
Rearranging terms yields
Since all reference to imaginary numbers has been eliminated from this equation, it can be applied to fields that are real-valued, as well as those that have complex values.
Rewriting the first two terms using the inverse of the Minkowski metric, and writing the Einstein summation convention explicitly we get
Thus the Klein–Gordon equation can be written in a covariant notation. This often means an abbreviation in the form of
where
and
This operator is called the wave operator.
Today this form is interpreted as the relativistic field equation for spin-0 particles. Furthermore, any component of any solution to the free Dirac equation is automatically a solution to the free Klein–Gordon equation. This generalizes to particles of any spin due to the Bargmann–Wigner equations. Furthermore, in quantum field theory, every component of every quantum field must satisfy the free Klein–Gordon equation, making the equation a generic expression of quantum fields.

Klein–Gordon equation in a potential

The Klein–Gordon equation can be generalized to describe a field in some potential as
Then the Klein–Gordon equation is the case.
Another common choice of potential which arises in interacting theories is the potential for a real scalar field

Higgs sector

The pure Higgs boson sector of the Standard model is modelled by a Klein–Gordon field with a potential, denoted for this section. The Standard model is a gauge theory and so while the field transforms trivially under the Lorentz group, it transforms as a -valued vector under the action of the part of the gauge group. Therefore, while it is a vector field, it is still referred to as a scalar field, as scalar describes its transformation under the Lorentz group. This is also discussed below in the scalar chromodynamics section.
The Higgs field is modelled by a potential
which can be viewed as a generalization of the potential, but has an important difference: it has a circle of minima. This observation is an important one in the theory of spontaneous symmetry breaking in the Standard model.