The
Klein–Gordon equation (
Klein–Fock–Gordon equation or sometimes
Klein–Gordon–Fock equation) is a
relativisticSpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
version of the
Schrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
.
It is the equation of motion of a
quantum scalar or pseudoscalar fieldQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and manybody systems. It is the natural and quantitative language of particle physics and...
, a field whose quanta are spinless particles. It cannot be straightforwardly interpreted as a
Schrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
for a quantum state, because it is second order in time and because it does not admit a positive definite conserved probability density. Still, with the appropriate interpretation, it does describe the quantum amplitude for finding a point particle in various places, the relativistic wavefunction, but the particle propagates both forwards and backwards in time. Any solution to the
Dirac equationThe Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
is automatically a solution to the Klein–Gordon equation, but the converse is not true.
Statement
The Klein–Gordon equation is

It is most often written in
natural unitsIn physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...
:

The form is determined by requiring that
plane waveIn the physics of wave propagation, a plane wave is a constantfrequency wave whose wavefronts are infinite parallel planes of constant peaktopeak amplitude normal to the phase velocity vector....
solutions of the equation:

obey the energy momentum relation of special relativity:

Unlike the Schrödinger equation, there are two values of for each k, one positive and one negative. Only by separating out the positive and negative frequency parts does the equation describe a relativistic wavefunction. For the timeindependent case, the Klein–Gordon equation becomes
which is the homogeneous screened Poisson equationIn Physics, the screened Poisson equation is the following partial differential equation:\left[ \Delta  \lambda^2 \right] u =  f...
.
History
The equation was named after the physicists Oskar KleinOskar Benjamin Klein was a Swedish theoretical physicist.Klein was born in Danderyd outside Stockholm, son of the chief rabbi of Stockholm, Dr. Gottlieb Klein from Homonna in Hungary and Antonie Levy...
and Walter GordonWalter Gordon was a German theoretical physicist.Life:Walter Gordon was the son of businessman Arnold Gordon and his wife Bianca Gordon . The family moved to Switzerland in his early years. In 1900 he attended school in St. Gallen and in 1915 he began his studies of mathematics and physics at...
, who in 1927 proposed that it describes relativistic electrons. Although it turned out that the Dirac equationThe Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
describes the spinning electron, the Klein–Gordon equation correctly describes the spinless pionIn particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the lowenergy properties of the strong nuclear force....
. The pion is a composite particle; no spinless elementary particles have yet been found, although the Higgs bosonThe Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...
is theorized to exist as a spinzero boson, according to the Standard ModelThe Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
.
The Klein–Gordon equation was first considered as a quantum wave equation by SchrödingerErwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...
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, without taking into account the electron's spin, the Klein–Gordon equation predicts the hydrogen atom's fine structure incorrectly, including overestimating the overall magnitude of the splitting pattern by a factor of for the nth energy level. The Dirac result 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 nonrelativistic approximation that predicts the Bohr energy levels of hydrogen without fine structure.
In 1927, soon after the Schrödinger equation was introduced, Vladimir Fock wrote an article about its generalization for the case of magnetic fieldA magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
s, where forceIn physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
s were dependent on velocityIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
, and independently derived this equation. Both Klein and Fock used Kaluza and Klein's method. Fock also determined the gauge theoryIn physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
for the wave equationThe wave equation is an important secondorder linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
. The Klein–Gordon equation for a free particleIn physics, a free particle is a particle that, in some sense, is not bound. In classical physics, this means the particle is present in a "fieldfree" space.Classical Free Particle:The classical free particle is characterized simply by a fixed velocity...
has a simple plane waveIn the physics of wave propagation, a plane wave is a constantfrequency wave whose wavefronts are infinite parallel planes of constant peaktopeak amplitude normal to the phase velocity vector....
solution.
Derivation
The nonrelativistic equation for the energy of a free particle is
By quantizing this, we get the nonrelativistic Schrödinger equation for a free particle,
where
is the momentum operatorIn quantum mechanics, momentum is defined as an operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once...
( being the del operatorIn vector calculus, del is a vector differential operator, usually represented by the nabla symbol \nabla . When applied to a function defined on a onedimensional domain, it denotes its standard derivative as defined in calculus...
).
The Schrödinger equation suffers from not being relativistically covariant, meaning it does not take into account Einstein's special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
.
It is natural to try to use the identity from special relativity
for the energy; then, just inserting the quantum mechanical momentum operator, yields the equation
This, however, is a cumbersome expression to work with because the differential operator cannot be evaluated while under the square root sign. In addition, this equation, as it stands, is nonlocalIn Classical physics, nonlocality is the direct influence of one object on another, distant object. In Quantum mechanics, nonlocality refers to the absence of a local, realist model in agreement with quantum mechanical predictions.Nonlocality may refer to:...
.
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 realIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
valued as well as those that have complexA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the onedimensional number line to the twodimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
values.
Using the reciprocal of the Minkowski metric , we get
in covariant notation. This is often abbreviated as
where
and
This operator is called the d'Alembert operatorIn special relativity, electromagnetism and wave theory, the d'Alembert operator , also called the d'Alembertian or the wave operator, is the Laplace operator of Minkowski space. The operator is named for French mathematician and physicist Jean le Rond d'Alembert...
. Today this form is interpreted as the relativistic field equationA field equation is an equation in a physical theory that describes how a fundamental force interacts with matter...
for a scalar (i.e. spinIn quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
0) particle. Furthermore, any solution to the Dirac equationThe Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
(for a spinonehalf particle) is automatically a solution to the Klein–Gordon equation, though not all solutions of the Klein–Gordon equation are solutions of the Dirac equation. It is noteworthy that the Klein–Gordon equation is very similar to the Proca equationIn physics, in the area of field theory, the Proca action describes a massive spin1 field of mass m in Minkowski spacetime. The field involved is a real vector field A...
.
Relativistic free particle solution
The Klein–Gordon equation for a free particle can be written as
with the same solution as in the nonrelativistic case:
except with the constraint
Just as with the nonrelativistic particle, we have for energy and momentum:
Except that now when we solve for k and ω and substitute into the constraint equation, we recover the relationship between energy and momentum for relativistic massive particles:
For massless particles, we may set m = 0 in the above equations. We then recover the relationship between energy and momentum for massless particles:
Action
The Klein–Gordon equation can also be derived from the following action
where is the Klein–Gordon field and is its mass. The complex conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
of is written If the scalar field is taken to be realvalued, then
From this we can derive the stressenergy tensorThe stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and nongravitational force fields...
of the scalar field. It is
Electromagnetic interaction
There is a simple way to make any field interact with electromagnetism in a gauge invariantIn physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
way: replace the derivative operators with the gauge covariant derivative operators. The Klein Gordon equation becomes:

in natural unitsIn physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...
, where A is the vector potential. While it is possible to add many higher order terms, for example,

these terms are not renormalizableIn quantum field theory, the statistical mechanics of fields, and the theory of selfsimilar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
in 3+1 dimensions.
The field equation for a charged scalar field multiplies by i, which means the field must be complex. In order for a field to be charged, it must have two components that can rotate into each other, the real and imaginary parts.
The action for a charged scalar is the covariant version of the uncharged action:

Gravitational interaction
In general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, we include the effect of gravity and the Klein–Gordon equation becomes
or equivalently
where is the reciprocal of the metric tensorIn the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
that is the gravitational potential field, is the determinantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the metric tensor,
is the covariant derivativeIn mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
and is the Christoffel symbol that is the gravitational force field.
See also
 Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
 Rarita–Schwinger equation
 Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and manybody systems. It is the natural and quantitative language of particle physics and...
 Scalar field theory
In theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a "scalar", in contrast to a vector or tensor field...
External links