The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a
relativistic wave equation
In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the con ...
, related to the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
. It is second-order in space and time and manifestly
Lorentz-covariant
In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same ...
. It is a quantized version of the relativistic
energy–momentum relation
In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also called relativistic energy) to invariant mass (which is also called rest mass) and momentum. It is t ...
. Its solutions include a
quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
. Electromagnetic interactions can be incorporated, forming the topic of
scalar electrodynamics, but because common spinless particles like the
pion
In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
s are unstable and also experience the strong interaction (with unknown interaction term in the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
,) the practical utility is limited.
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
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density.
Probability amplitudes provide a relationship between the qu ...
. The conserved quantity is instead interpreted as
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
, and the norm squared of the wave function is interpreted as a
charge density
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in ...
. 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. The Klein–Gordon equation does not form the basis of a consistent quantum relativistic ''one-particle'' theory. There is no known such theory for particles of any spin. For full reconciliation of quantum mechanics with special relativity,
quantum field theory is needed, in which the Klein–Gordon equation reemerges as the equation obeyed by the components of all free quantum fields.
Steven Weinberg
Steven Weinberg (; May 3, 1933 – July 23, 2021) was an American theoretical physicist and Nobel laureate in physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic interac ...
makes a point about this. He leaves out the treatment of relativistic wave mechanics altogether in his otherwise complete introduction to modern applications of quantum mechanics, explaining: "It seems to me that the way this is usually presented in books on quantum mechanics is profoundly misleading." (From the preface in ''Lectures on Quantum Mechanics'', referring to treatments of the Dirac equation in its original flavor.)
Others, like Walter Greiner
Walter Greiner (29 October 1935 – 6 October 2016) was a German theoretical physicist. His research interests lay in atomic physics, heavy ion physics, nuclear physics, elementary particle physics (particularly in quantum electrodynamics and qu ...
does in his series on theoretical physics, give a full account of the historical development and view of relativistic quantum mechanics
In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light  ...
before they get to the modern interpretation, with the rationale that it is highly desirable or even necessary from a pedagogical point of view to take the long route. In quantum field theory, the solutions of the free (noninteracting) versions of the original equations still play a role. They are needed to build the Hilbert space (
Fock space
The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first intr ...
) and to express quantum fields by using complete sets (spanning sets of Hilbert space) of wave functions.
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
In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
. By
Fourier transforming the field into momentum space, the solution is usually written in terms of a superposition of
plane waves
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, t ...
whose energy and momentum obey the energy-momentum
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given t ...
from
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
. Here, the Klein–Gordon equation is given for both of the two common
metric signature
In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
conventions
.
Here,
is the
wave operator
In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of Mi ...
and
is the
Laplace operator. The
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
and
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
are often seen to clutter the equations, so they are therefore often expressed in
natural units
In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
where
.
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 physics, the screened Poisson equation is a Poisson equation, which arises in (for example) the Klein–Gordon equation, electric field screening in plasmas, and nonlocal granular fluidity in granular flow.
Statement of the equation
The equat ...
.
Solution for free particle
Here, the Klein–Gordon equation in natural units,
, with the metric signature
is solved by Fourier transformation. Inserting the Fourier transformation
and using
orthogonality of the complex exponentials gives the dispersion relation
This restricts the momenta to those that lie
on shell
In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called "on the mass shell" or simply more often on shell; while those that do not are called "off the mass shell", ...
, giving positive and negative energy solutions
For a new set of constants
, the solution then becomes
It 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 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
Vladimir Aleksandrovich Fock (or Fok; russian: Влади́мир Алекса́ндрович Фок) (December 22, 1898 – December 27, 1974) was a Soviet physicist, who did foundational work on quantum mechanics and quantum electrodynamic ...
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 Johann Kudar,
Théophile de Donder
Théophile Ernest de Donder (; 19 August 1872 – 11 May 1957) was a Belgian mathematician and physicist famous for his work (published in 1923) in developing correlations between the Newtonian concept of chemical affinity and the Gibbsian concep ...
and
Frans-H. van den Dungen, and
Louis de Broglie
Louis Victor Pierre Raymond, 7th Duc de Broglie (, also , or ; 15 August 1892 – 19 March 1987) was a French physicist and aristocrat who made groundbreaking contributions to Old quantum theory, quantum theory. In his 1924 PhD thesis, he pos ...
. Although it turned out that modeling the electron's spin required the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
, the Klein–Gordon equation correctly describes the spinless relativistic
composite particle
This is a list of known and hypothesized particles.
Elementary particles
Elementary particles are particles with no measurable internal structure; that is, it is unknown whether they are composed of other particles. They are the fundamental ob ...
s, like the
pion
In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gene ...
. 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
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, a ...
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
Schrödinger in his search for an equation describing
de Broglie wave
Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave ...
s. 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 .
[See Eq. 2.87 is identical to eq. 2.86, except that it features instead of .] 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
Vladimir Aleksandrovich Fock (or Fok; russian: Влади́мир Алекса́ндрович Фок) (December 22, 1898 – December 27, 1974) was a Soviet physicist, who did foundational work on quantum mechanics and quantum electrodynamic ...
wrote an article about its generalization for the case of
magnetic fields, where
forces were dependent on
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
, and independently derived this equation. Both Klein and Fock used
Kaluza and Klein's method. Fock also determined the
gauge theory for the
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seism ...
. The Klein–Gordon equation for a
free particle
In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. I ...
has a simple
plane-wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, t ...
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 ( being the
del operator
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes t ...
), and
:
is the
energy operator
In quantum mechanics, energy is defined in terms of the energy operator, acting on the wave function of the system as a consequence of time translation symmetry.
Definition
It is given by:
\hat = i\hbar\frac
It acts on the wave function (the ...
.
The Schrödinger equation suffers from not being
relativistically invariant, meaning that it is inconsistent with
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
.
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 (see als
Introduction to nonlocal equations.
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
In mathematics, value may refer to several, strongly related notions.
In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an i ...
, as well as those that have
complex values.
Rewriting the first two terms using the inverse of the
Minkowski metric
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
, 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
In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of Mi ...
.
Today this form is interpreted as the relativistic
field equation
In theoretical physics and applied mathematics, a field equation is a partial differential equation which determines the dynamics of a physical field, specifically the time evolution and spatial distribution of the field. The solutions to the equ ...
for
spin-0 particles.
Furthermore, any ''component'' of any solution to the free
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
(for a
spin-1/2
In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of . The spin number describes how many symmetrical facets a particle has in one full ...
particle) 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 (formally, representation) 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
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
in the Standard model.
Conserved U(1) current
The Klein–Gordon equation (and action) for a complex field
admits a
symmetry. That is, under the transformations
:
:
the Klein–Gordon equation is invariant, as is the action (see below). By
Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...
for fields, corresponding to this symmetry there is a current
defined as
:
which satisfies the conservation equation
The form of the conserved current can be derived systematically by applying Noether's theorem to the
symmetry. We will not do so here, but simply verify that this current is conserved.
From the Klein–Gordon equation for a complex field
of mass
, written in covariant notation and ''mostly plus'' signature,
:
and its complex conjugate
:
Multiplying by the left respectively by
and
(and omitting for brevity the explicit
dependence),
:
:
Subtracting the former from the latter, we obtain
:
or in index notation,
:
Applying this to the derivative of the current
one finds
:
This
symmetry is a global symmetry, but it can also be gauged to create a local or gauge symmetry: see below scalar QED. The name of gauge symmetry is somewhat misleading: it is really a redundancy, while the global symmetry is a genuine symmetry.
Lagrangian formulation
The Klein–Gordon equation can also be derived by a
variational method, arising as the Euler–Lagrange equation of the action
:
In natural units, with signature ''mostly minus'', the actions take the simple form
for a real scalar field of mass
, and
for a complex scalar field of mass
.
Applying the formula for the
stress–energy tensor to the Lagrangian density (the quantity inside the integral), we can derive the stress–energy tensor of the scalar field. It is
:
and in natural units,
:
By integration of the time–time component 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. This is not the case for the Dirac equation and its energy–momentum tensor.
The stress energy tensor is the set of conserved currents corresponding to the invariance of the Klein–Gordon equation under space-time translations
. Therefore each component is conserved, that is,
(this holds only on-shell, that is, when the Klein–Gordon equations are satisfied). It follows that the integral of
over space is a conserved quantity for each
. These have the physical interpretation of total energy for
and total momentum for
with
.
Non-relativistic limit
Classical field
Taking the non-relativistic limit () of a classical Klein–Gordon field begins with the ansatz factoring the oscillatory
rest mass energy term,
:
Defining the kinetic energy
,
in the non-relativistic limit
, and hence
:
Applying this yields the non-relativistic limit of the second time derivative of
,
:
:
Substituting into the free Klein–Gordon equation,
, yields
:
which (by dividing out the exponential and subtracting the mass term) simplifies to
:
This is a ''classical''
Schrödinger field
In quantum mechanics and quantum field theory, a Schrödinger field, named after Erwin Schrödinger, is a quantum field which obeys the Schrödinger equation. While any situation described by a Schrödinger field can also be described by a many-bo ...
.
Quantum field
The analogous limit of a quantum Klein–Gordon field is complicated by the non-commutativity of the field operator. In the limit , the
creation and annihilation operators
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
decouple and behave as independent quantum
Schrödinger field
In quantum mechanics and quantum field theory, a Schrödinger field, named after Erwin Schrödinger, is a quantum field which obeys the Schrödinger equation. While any situation described by a Schrödinger field can also be described by a many-bo ...
s.
Scalar electrodynamics
There is a way to make the complex Klein–Gordon field
interact with electromagnetism in a
gauge-invariant
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 (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
way. We can replace the (partial) derivative with the gauge-covariant derivative. Under a local
gauge transformation, the fields transform as
:
:
where
is a function of spacetime, thus making it a local transformation, as opposed to a constant over all of spacetime, which would be a global
transformation. A subtle point is that global transformations can arise as local ones, when the function
is taken to be a constant function.
A well-formulated theory should be invariant under such transformations. Precisely, this means that the equations of motion and action (see below) are invariant. To achieve this, ordinary derivatives
must be replaced by gauge-covariant derivatives
, defined as
:
:
where the 4-potential or gauge field
transforms under a gauge transformation
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''
symmetry is compatible only with complex Klein–Gordon theory and not real Klein–Gordon theory.
In natural units and mostly minus signature we have
where
is known as the Maxwell tensor, field strength or curvature depending on viewpoint.
This theory is often known as scalar quantum electrodynamics or scalar QED, although all aspects we've discussed here are classical.
Scalar chromodynamics
It is possible to extend this to a non-abelian gauge theory with a gauge group
, where we couple the scalar Klein–Gordon action to a
Yang–Mills Lagrangian. Here, the field is actually vector-valued, but is still described as a scalar field: the scalar describes its transformation under
space-time transformations, but not its transformation under the action of the gauge group.
For concreteness we fix
to be
, the
special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
for some
. Under a gauge transformation
, which can be described as a function
the scalar field
transforms as a
vector
:
:
.
The covariant derivative is
:
:
where the gauge field or connection transforms as
:
This field can be seen as a matrix valued field which acts on the vector space
.
Finally defining the chromomagnetic field strength or curvature,
:
we can define the action.
Klein–Gordon on curved spacetime
In
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, we include the effect of gravity by replacing partial derivatives with
covariant derivatives, and the Klein–Gordon equation becomes (in the
mostly pluses signature)
:
or equivalently,
:
where is the inverse of the
metric tensor that is the gravitational potential field, ''g'' is the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the metric tensor, is the
covariant derivative, and is the
Christoffel symbol
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing dist ...
that is the gravitational
force field.
With natural units this becomes
This also admits an action formulation on a spacetime (Lorentzian) manifold
. Using
abstract index notation and in ''mostly plus'' signature this is
or
See also
*
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
*
Quantum field theory
*
Quartic interaction
In quantum field theory, a quartic interaction is a type of self-interaction in a scalar field. Other types of quartic interactions may be found under the topic of four-fermion interactions. A classical free scalar field \varphi satisfies the Kle ...
*
Relativistic wave equations
In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the con ...
*
Rarita–Schwinger equation
In theoretical physics, the Rarita–Schwinger equation is the
relativistic field equation of spin-3/2 fermions. It is similar to the Dirac equation for spin-1/2 fermions. This equation was first introduced by William Rarita and Julian Schwinge ...
*
Scalar field theory
*
Sine–Gordon equation
Remarks
Notes
References
*
*
*
*
*
*
*
*
*
External links
*
*
Linear Klein–Gordon Equationat EqWorld: The World of Mathematical Equations.
Nonlinear Klein–Gordon Equationat EqWorld: The World of Mathematical Equations.
Introduction to nonlocal equations
{{DEFAULTSORT:Klein-Gordon equation
Partial differential equations
Special relativity
Waves
Quantum field theory
Equations of physics
Mathematical physics