E2=(pc)2+
2\right) | |
\left(m | |
0c |
2
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
(t,x)
x\mu=(ct,x)
η\mu=diag(\pm1,\mp1,\mp1,\mp1)
Separated time and space | \left(
-\nabla2+
\right)\psi(t,x)=0 | \psi(t,x)=\int
\int
e\mp\psi(\omega,k) | E2=p2c2+m2c4 | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Four-vector form | (\Box+\mu2)\psi=0 , \mu=mc/\hbar | \psi(x\mu)=\int
\psi(p\mu) | p\mup\mu=\pmm2c2 |
\Box=\pmη\mu\partial\mu\partial\nu
\nabla2
c
\hbar
c=\hbar=1
Separated time and space | \left(
-\nabla2+m2\right)\psi(t,x)=0 | \psi(t,x)=\int
\int
e\mp\psi(\omega,k) | E2=p2+m2 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Four-vector form | (\Box+m2)\psi=0 | \psi(x\mu)=\int
\psi(p\mu) | p\mup\mu=\pmm2 |
\left[\nabla2-
m2c2 | |
\hbar2 |
\right]\psi(r)=0,
\hat{p}\mu\hat{p}\mu\psi=m2c2\psi
where, the momentum operator is given as: .
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.[2] 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.[3] [4] 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.[5] [6] [7]
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.[8] [9] 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. 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. [10] [11]
Here, the Klein–Gordon equation in natural units,
(\Box+m2)\psi(x)=0
η\mu=diag(+1,-1,-1,-1)
C(p)
p0
B(p) → B(-p)
p0
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
p ⋅ x=p\mux\mu
1/2E(p)
A(p)
B(p)
The equation was named after the physicists Oskar Klein[12] and Walter Gordon,[13] who in 1926 proposed that it describes relativistic electrons. Vladimir Fock also discovered the equation independently in 1926 slightly after Klein's work,[14] 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 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 .[15] 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 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.
The non-relativistic equation for the energy of a free particle is
p2 | |
2m |
=E.
By quantizing this, we get the non-relativistic Schrödinger equation for a free particle:
\hat{p | |
2}{2m} |
\psi=\hat{E}\psi,
\hat{p
\hat{E}=i\hbar
\partial | |
\partialt |
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:
\sqrt{p2c2+m2c4}=E.
Then, just inserting the quantum-mechanical operators for momentum and energy yields the equation
\sqrt{(-i\hbar\nabla)2c2+m2c4}\psi=i\hbar
\partial | |
\partialt |
\psi.
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 also Introduction to nonlocal equations).
Klein and Gordon instead began with the square of the above identity, i.e.
p2c2+m2c4=E2,
\left((-i\hbar\nabla)2c2+m2c4\right)\psi=\left(i\hbar
\partial | |
\partialt |
\right)2\psi,
which simplifies to
-\hbar2c2\nabla2\psi+m2c4\psi=-\hbar2
\partial2 | |
\partialt2 |
\psi.
Rearranging terms yields
1 | |
c2 |
\partial2 | |
\partialt2 |
\psi-\nabla2\psi+
m2c2 | |
\hbar2 |
\psi=0.
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
-η\mu\partial\mu\partial\nu\psi\equiv
3 | |
\sum | |
\mu=0 |
3 | |
\sum | |
\nu=0 |
-η\mu\nu\partial\mu\partial\nu\psi=
1 | |
c2 |
2 | |
\partial | |
0 |
\psi-
3 | |
\sum | |
\nu=1 |
\partial\nu\partial\nu\psi=
1 | |
c2 |
\partial2 | |
\partialt2 |
\psi-\nabla2\psi.
Thus the Klein–Gordon equation can be written in a covariant notation. This often means an abbreviation in the form of
(\Box+\mu2)\psi=0,
\mu=
mc | |
\hbar |
,
\Box=
1 | |
c2 |
\partial2 | |
\partialt2 |
-\nabla2.
This operator is called the wave operator.
Today this form is interpreted as the relativistic field equation for spin-0 particles.[8] 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. Furthermore, in quantum field theory, every component of every quantum field must satisfy the free Klein–Gordon equation,[16] making the equation a generic expression of quantum fields.
The Klein–Gordon equation can be generalized to describe a field in some potential
V(\psi)
\Box\psi+
\partialV | |
\partial\psi |
=0.
V(\psi)=M2\bar\psi\psi
Another common choice of potential which arises in interacting theories is the
\phi4
\phi,
V(\phi)=
1 | |
2 |
m2\phi2+λ\phi4.
See also: Higgs sector. The pure Higgs boson sector of the Standard model is modelled by a Klein–Gordon field with a potential, denoted
H
C2
SU(2)
H:R1,3 → C2
The Higgs field is modelled by a potential
V(H)=-m2H\daggerH+λ(H\daggerH)2
\phi4
The Klein–Gordon equation (and action) for a complex field
\psi
U(1)
\psi(x)\mapstoei\theta\psi(x),
\bar\psi(x)\mapstoe-i\theta\bar\psi(x),
J\mu
J\mu(x)=
e | |
2m |
\left(\bar\psi(x)\partial\mu\psi(x)-\psi(x)\partial\mu\bar\psi(x)\right).
\partial\muJ\mu(x)=0.
U(1)
From the Klein–Gordon equation for a complex field
\psi(x)
M
(\square+m2)\psi(x)=0
(\square+m2)\bar\psi(x)=0.
Multiplying by the left respectively by
\bar\psi(x)
\psi(x)
x
\bar\psi(\square+m2)\psi=0,
\psi(\square+m2)\bar\psi=0.
Subtracting the former from the latter, we obtain
\bar\psi\square\psi-\psi\square\bar\psi=0,
\bar\psi\partial\mu\partial\mu\psi-\psi\partial\mu\partial\mu\bar\psi=0.
Applying this to the derivative of the current
J\mu(x)\equiv\psi*(x)\partial\mu\psi(x)-\psi(x)\partial\mu\psi*(x),
\partial\muJ\mu(x)=0.
This
U(1)
The Klein–Gordon equation can also be derived by a variational method, arising as the Euler–Lagrange equation of the action
l{S}=\int\left(-\hbar2η\mu\partial\mu\bar\psi\partial\nu\psi-M2c2\bar\psi\psi\right)d4x,
In natural units, with signature mostly minus, the actions take the simple formfor a real scalar field of mass
m
M
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
T\mu\nu=\hbar2\left(η\muη\nu+η\muη\nu-η\mu\nuη\alpha\right)\partial\alpha\bar\psi\partial\beta\psi-η\mu\nuM2c2\bar\psi\psi.
and in natural units,
T\mu\nu=2\partial\mu\bar\psi\partial\nu\psi-η\mu\nu
\rho\bar\psi\partial | |
(\partial | |
\rho\psi |
-M2\bar\psi\psi)
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.[8]
The stress energy tensor is the set of conserved currents corresponding to the invariance of the Klein–Gordon equation under space-time translations
x\mu\mapstox\mu+c\mu
\partial\muT\mu\nu=0
T0\nu
\nu
\nu=0
\nu=i
i\in\{1,2,3\}
Taking the non-relativistic limit of a classical Klein–Gordon field begins with the ansatz factoring the oscillatory rest mass energy term,
\psi(x,t)=
| |||||
\phi(x,t)e |
rm{where}
| |||||
\phi(x,t)=u | |||||
E(x)e |
.
Defining the kinetic energy
E'=E-mc2=\sqrt{m2c4+c2p2}-mc
| ||||
E'\llmc2
v=p/m\llc
i\hbar
\partial\phi | |
\partialt |
=E'\phi\llmc2\phi rm{and} (i\hbar)2
\partial2\phi | |
\partialt2 |
=(E')2\phi\ll(mc2)2\phi.
Applying this yields the non-relativistic limit of the second time derivative of
\psi
\partial\psi | |
\partialt |
=\left(-i
mc2 | \phi+ | |
\hbar |
\partial\phi | |
\partialt |
| |||||
\right)e |
≈ -i
mc2 | |
\hbar |
| |||||
\phie |
\partial2\psi | |
\partialt2 |
=-\left(i
2mc2 | |
\hbar |
\partial\phi | |
\partialt |
+\left(
mc2 | |
\hbar |
\right)2\phi-
\partial2\phi | |
\partialt2 |
\right)
| |||||
e |
≈ -\left(i
2mc2 | |
\hbar |
\partial\phi | |
\partialt |
+\left(
mc2 | |
\hbar |
\right)2\phi\right)
| |||||
e |
Substituting into the free Klein–Gordon equation,
c-2
2 | |
\partial | |
t |
\psi=\nabla2\psi-m2\psi
- | 1 |
c2 |
\left(i
2mc2 | |
\hbar |
\partial\phi | |
\partialt |
+\left(
mc2 | |
\hbar |
\right)2\phi\right)
| |||||
e |
≈ \left(\nabla2-\left(
mc | |
\hbar |
\right)2\right)\phie
| |||||
which (by dividing out the exponential and subtracting the mass term) simplifies to
i\hbar | \partial\phi |
\partialt |
=-
\hbar2 | |
2m |
\nabla2\phi.
This is a classical Schrödinger 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 fields.
See also: Scalar electrodynamics. There is a way to make the complex Klein–Gordon field
\psi
U(1)
\psi\mapsto\psi'=ei\theta(x)\psi,
\bar\psi\mapsto\bar\psi'=e-i\theta(x)\bar\psi,
\theta(x)=\theta(t,bf{x})
U(1)
\theta(x)
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
\partial\mu
D\mu
D\mu\psi=(\partial\mu-ieA\mu)\psi
D\mu\bar\psi=(\partial\mu+ieA\mu)\bar\psi
A\mu
\theta
A\mu\mapstoA'\mu=A\mu+
1 | |
e |
\partial\mu\theta
D\mu\psi\mapstoei\thetaD\mu\psi
D\muD\mu\psi-M2\psi=0.
Since an ungauged
U(1)
U(1)
In natural units and mostly minus signature we have
where
F\mu\nu=\partial\muA\nu-\partial\nuA\mu
This theory is often known as scalar quantum electrodynamics or scalar QED, although all aspects we've discussed here are classical.
It is possible to extend this to a non-abelian gauge theory with a gauge group
G
For concreteness we fix
G
SU(N)
N\geq2
U(x)
U:R1,3 → SU(N),
\psi
CN
\psi(x)\mapstoU(x)\psi(x)
\psi\dagger(x)\mapsto\psi\dagger(x)U\dagger(x)
D\mu\psi=\partial\mu\psi-igA\mu\psi
\dagger | |
D | |
\mu\psi |
=\partial\mu\psi\dagger+ig\psi\dagger
\dagger | |
A | |
\mu |
A\mu\mapstoUA\muU-1-
i | |
g |
\partial\muUU-1.
CN
Finally defining the chromomagnetic field strength or curvature,
F\mu\nu=\partial\muA\nu-\partial\nuA\mu+g(A\muA\nu-A\nuA\mu),
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)[18]
\begin{align} 0&=-g\mu\nabla\mu\nabla\nu\psi+\dfrac{m2c2}{\hbar2}\psi=-g\mu\nabla\mu(\partial\nu\psi)+\dfrac{m2c2}{\hbar2}\psi\\ &=-g\mu\partial\mu\partial\nu\psi+g\mu\Gamma\sigma{}\mu\partial\sigma\psi+\dfrac{m2c2}{\hbar2}\psi, \end{align}
or equivalently,
-1 | |
\sqrt{-g |
where is the inverse of the metric tensor that is the gravitational potential field, g is the determinant of the metric tensor, is the covariant derivative, 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
M