The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a
relativistic wave equation, 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. It is a quantized version of the relativistic
energy–momentum relation . 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. Electromagnetic interactions can be incorporated, forming the topic of
scalar electrodynamics
In theoretical physics, scalar electrodynamics is a theory of a U(1) gauge field coupled to a charged spin 0 scalar field that takes the place of the Dirac fermions in "ordinary" quantum electrodynamics. The scalar field is charged, and with an ap ...
, but because common spinless particles like the
pions are unstable and also experience the strong interaction (with unknown interaction term in the
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. 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 res ...
, 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 i ...
. 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
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
is needed, in which the Klein–Gordon equation reemerges as the equation obeyed by the components of all free quantum fields.
[ Steven Weinberg 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 does in his series on theoretical physics, give a full account of the historical development and view of relativistic quantum mechanics 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) 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 . 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, th ...
whose energy and momentum obey the energy-momentum
dispersion relation 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 law ...
. Here, the Klein–Gordon equation is given for both of the two common
metric signature conventions
.
Here,
is the
wave operator and
is the
Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
. 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 fo ...
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 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 equa ...
.
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
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
of the complex exponentials gives the dispersion relation
This restricts the momenta to those that lie
on 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
Oskar Benjamin Klein (; 15 September 1894 – 5 February 1977) was a Swedish theoretical physicist.
Biography
Klein was born in Danderyd outside Stockholm, son of the chief rabbi of Stockholm, Gottlieb Klein from Humenné in Kingdom of Hung ...
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 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
The European Organization for Nuclear Research, known as CERN (; ; ), is an intergovernmental organization that operates the largest particle physics laboratory in the world. Established in 1954, it is based in a northwestern suburb of Gen ...
announced the discovery of the
Higgs boson
The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field,
one of the fields in particle physics theory. In the Stan ...
. Since the
Higgs boson
The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field,
one of the fields in particle physics theory. In the Stan ...
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, ...
to be described by the Klein–Gordon equation. Further experimentation and analysis is required to discern whether the
Higgs boson
The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field,
one of the fields in particle physics theory. In the Stan ...
observed is that of the
Standard Model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces ( electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. It ...
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 atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom ...
.
In 1926, soon after the Schrödinger equation was introduced,
Vladimir Fock wrote an article about its generalization for the case of
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
s, where
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
s 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
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 grou ...
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 ...
. The Klein–Gordon equation for a
free particle 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, ...
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
In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimensi ...
( being the
del operator), 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 law ...
.
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 transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
ations, 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, 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
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
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally ...
-0 particles.
Furthermore, any ''component'' of any solution to the free
Dirac equation (for a
spin-1/2 particle) is automatically a solution to the free Klein–Gordon equation. This generalizes to particles of any spin due to the
Bargmann–Wigner equations
:''This article uses the Einstein summation convention for tensor/spinor indices, and uses hats for quantum operators.
In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles with non ...
. Furthermore, in
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, 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 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 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
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the str ...
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-bod ...
.
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 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-bod ...
s.
Scalar electrodynamics
There is a way to make the complex Klein–Gordon field
interact with electromagnetism in a
gauge-invariant 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 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 derivative
In 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 differe ...
s, and the Klein–Gordon equation becomes (in the
mostly pluses signature)
:
or equivalently,
:
where is the inverse of the
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allow ...
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
In 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 differe ...
, and is the
Christoffel symbol 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
*
Quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
*
Quartic interaction
*
Relativistic wave equations
*
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 Schwing ...
*
Scalar field theory
In theoretical physics, scalar field theory can refer to a relativistically invariant classical or quantum theory of scalar fields. A scalar field is invariant under any Lorentz transformation.
The only fundamental scalar quantum field that ha ...
*
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