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.
Only by separating out the positive and negative frequency parts does one obtain an equation describing a relativistic wavefunction.
The quantization process introduces then a quantum field whose quanta are spinless particles.
[2] The equation solutions include a scalar or pseudoscalar field[clarification needed].
Despite 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.
This restricts the momenta to those that lie on shell, giving positive and negative energy solutions
Note that because the initial Fourier transformation contained Lorentz invariant quantities like
The equation was named after the physicists Oskar Klein[9] and Walter Gordon,[10] who in 1926 proposed that it describes relativistic electrons.
On 4 July 2012, European Organization for Nuclear Research CERN announced the discovery of the Higgs boson.
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 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 4n/2n − 1 for the n-th energy level.
[12] 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.
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.
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 diag(−c2, 1, 1, 1), and writing the Einstein summation convention explicitly we get Thus the Klein–Gordon equation can be written in a covariant notation.
This observation is an important one in the theory of spontaneous symmetry breaking in the Standard model.
The form of the conserved current can be derived systematically by applying Noether's theorem to the
, written in covariant notation and mostly plus signature, and its complex conjugate Multiplying by the left respectively by
dependence), Subtracting the former from the latter, we obtain or in index notation, Applying this to the derivative of the current
It is and in natural units, By integration of the time–time component T00 over all space, one may show that both the positive- and negative-frequency plane-wave solutions can be physically associated with particles with positive energy.
[6] The stress energy tensor is the set of conserved currents corresponding to the invariance of the Klein–Gordon equation under space-time translations
, yields which (by dividing out the exponential and subtracting the mass term) simplifies to This is a classical Schrödinger field.
In the limit v ≪ c, the creation and annihilation operators decouple and behave as independent quantum Schrödinger fields.
A subtle point is that global transformations can arise as local ones, when the function
as With these definitions, the covariant derivative transforms as In natural units, the Klein–Gordon equation therefore becomes Since an ungauged
symmetry is only present in complex Klein–Gordon theory, this coupling and promotion to a gauged
In general relativity, we include the effect of gravity by replacing partial derivatives with covariant derivatives, and the Klein–Gordon equation becomes (in the mostly pluses signature)[15] or equivalently, where gαβ is the inverse of the metric tensor that is the gravitational potential field, g is the determinant of the metric tensor, ∇μ is the covariant derivative, and Γσμν is the Christoffel symbol that is the gravitational force field.