Klein–Gordon equation

The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger equation.

Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. It cannot be straightforwardly interpreted as a Schrödinger equation 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 equation is automatically a solution to the Klein–Gordon equation, but the converse is not true.

Statement

The Klein–Gordon equation is

$\frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0.$

This is often abbreviated as

$(\Box + \mu^2) \psi = 0,$

where μ = mc/ħ and □ is the d'Alembert operator, defined by

$\Box = -\eta^{\mu\nu} \partial_\mu \partial_\nu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2.$

(We are using the (−, +, +, +) metric signature.)

The Klein-Gordon equation is most often written in natural units:

$- \partial_t^2 \psi + \nabla^2 \psi = m^2 \psi$

The form is determined by requiring that plane wave solutions of the equation:

$\psi = e^{-i\omega t + i k\cdot x } = e^{i k_\mu x^\mu}$

obey the energy momentum relation of special relativity:

$-p_\mu p^\mu = E^2 - P^2 = \omega^2 - k^2 = - k_\mu k^\mu = m^2$

Unlike the Schrödinger equation, the Klein–Gordon equation admits two values of ω for each k, 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

$\left[ \nabla^2 - \frac {m^2 c^2}{\hbar^2} \right] \psi(\mathbf{r}) = 0$

which is the homogeneous screened Poisson equation.

History

The equation was named after the physicists Oskar Klein and Walter Gordon, who in 1926 proposed that it describes relativistic electrons. Other authors making similar claims in that same year were Vladimir Fock, Johann Kudar, Théophile de Donder and Frans-H. van den Dungen, and Louis de Broglie. Although it turned out that the Dirac equation describes the spinning electron, the Klein–Gordon equation correctly describes the spinless pion, a composite particle. On July 4, 2012 CERN announced the discovery of the Higgs boson. Since the Higgs boson is a spin-zero particle, it is the first elementary particle that is described by the Klein-Gordon equation. Further experimentation and analysis is required to discern whether the Higgs boson found is that of the Standard Model, or a more exotic form.

The Klein–Gordon equation was first considered as a quantum wave equation by 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 4n/2n − 1 for the n-th energy level. The Dirac result is, however, easily recovered if the orbital momentum quantum number ℓ is replaced by total angular momentum quantum number j. 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.

Derivation

The non-relativistic equation for the energy of a free particle is

$\frac{\mathbf{p}^2}{2 m} = E.$

By quantizing this, we get the non-relativistic Schrödinger equation for a free particle,

$\frac{\mathbf{\hat{p}}^2}{2m} \psi = \hat{E}\psi$

where

$\mathbf{\hat{p}} =-i \hbar \mathbf{\nabla}$

is the momentum operator (∇ being the del operator), and

$\hat{E}=i \hbar \dfrac{\partial}{\partial t}$

is the energy operator.

The Schrödinger equation suffers from not being relativistically covariant, meaning it does not take into account Einstein's special relativity.

It is natural to try to use the identity from special relativity describing the energy:

$\sqrt{\mathbf{p}^2 c^2 + m^2 c^4} = E$

Then, just inserting the quantum mechanical operators for momentum and energy yields the equation

$\sqrt{(-i\hbar\mathbf{\nabla})^2 c^2 + m^2 c^4} \psi = i \hbar \frac{\partial}{\partial t}\psi.$

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 nonlocal (see also Introduction to nonlocal equations).

Despite the above mentioned difficulties, an Absolute Quantum Mechanics has been derived starting from the two equations immediately above. Absolute Quantum Mechanics dispenses with the problem of non-locality by returning to an Absolute theory of space and time wherein non-locality is not a problem. Also, the difficulty associated with the differential operator being inside of the radical is eliminated for the case of stationary states by rearranging terms and squaring the equation. As of the early 21st century, Absolute Quantum Mechanics is not considered a mainstream view, however.

Klein and Gordon instead began with the square of the above identity, i.e.

$\mathbf{p}^2 c^2 + m^2 c^4 = E^2$

which, when quantized, gives

$\left ((-i\hbar\mathbf{\nabla})^2 c^2 + m^2 c^4 \right ) \psi = \left(i \hbar \frac{\partial}{\partial t} \right)^2 \psi$

which simplifies to

$- \hbar^2 c^2 \mathbf{\nabla}^2 \psi + m^2 c^4 \psi = - \hbar^2 \frac{\partial^2}{\partial t^2} \psi.$

Rearranging terms yields

$\frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \mathbf{\nabla}^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0.$

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.

Using the inverse of the Minkowski metric diag(−c², 1, 1, 1), we get

$- \eta^{\mu \nu} \partial_{\mu} \partial_{\nu} \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0$

in covariant notation. This is often abbreviated as

$(\Box + \mu^2) \psi = 0,$

where

$\mu = \frac{mc}{\hbar}$

and

$\Box = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2.$

This operator is called the d'Alembert operator.

Today this form is interpreted as the relativistic field equation for spin-0 particles. Furthermore, any solution to the Dirac equation (for a spin-one-half 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 equation.

Klein–Gordon equation in a potential

The Klein–Gordon equation can be generalized to describe a field in some potential V(ψ) as:

$\Box \psi + \frac{\partial{}V}{\partial \psi} = 0$

Conserved current

The conserved current associated to the U(1) symmetry of a complex field $\phi(x) \in \mathbb{C}$ satisfying the Klein Gordon equation reads

$\partial_\mu J^\mu(x) = 0, \qquad J^\mu(x) \equiv \phi^*(x)\partial^\mu\phi(x) - \phi(x)\partial^\mu \phi^*(x).$

The form of the conserved current can be derived systematically by applying Noether's theorem to the U(1) symmetry. We will not do so here, but simply give a proof that this conserved current is correct.

Proof using algebraic manipulations from the KG equation

From the Klein Gordon equation for a complex field $\phi(x)$ of mass $m$ written in covariant notation

$(\square + m^2) \phi(x) = 0,$

and its complex conjugate

$(\square + m^2) \phi^*(x) = 0,$

we have, multiplying by the left respectively by $\phi^*(x)$ and $\phi(x)$ (and omitting for brevity the explicit $x$ dependence),

$\phi^* (\square + m^2) \phi = 0,$

$\phi (\square + m^2) \phi^* = 0 .$

Subtracting the former from the latter we obtain

$\phi^* \square \phi - \phi \square \phi^* = 0$

from which we obtain the conservation law for the Klein Gordon field:

$\partial_\mu J^\mu(x) = 0, \qquad J^\mu(x) \equiv \phi^*(x)\partial^\mu\phi(x) - \phi(x)\partial^\mu \phi^*(x).$

Relativistic free particle solution

The Klein–Gordon equation for a free particle can be written as

$\mathbf{\nabla}^2\psi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi = \frac{m^2c^2}{\hbar^2}\psi$

We look for plane wave solutions of the form

$\psi(\mathbf{r}, t) = e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$

for some constant angular frequency ω ∈ ℝ and wave number k ∈ ℝ³. Substitution gives the dispersion relation:

$-|\mathbf{k}|^2+\frac{\omega^2}{c^2}=\frac{m^2c^2}{\hbar^2}.$

Energy and momentum are seen to be proportional to ω and k:

$\langle\mathbf{p}\rangle=\left\langle \psi \left|-i\hbar\mathbf{\nabla}\right|\psi\right\rangle = \hbar\mathbf{k},$

$\langle E\rangle=\left\langle \psi \left|i\hbar\frac{\partial}{\partial t}\right|\psi\right\rangle = \hbar\omega.$

So the dispersion relation is just the classic relativistic equation:

$\langle E \rangle^2=m^2c^4+\langle \mathbf{p} \rangle^2c^2.$

For massless particles, we may set m = 0, recovering the relationship between energy and momentum for massless particles:

$\langle E \rangle=\langle |\mathbf{p}| \rangle c.$

Action

The Klein–Gordon equation can also be derived via a variational method by considering the action:

$\mathcal{S} = \int \left( - \frac{\hbar^2}{m} \eta^{\mu \nu} \partial_{\mu}\bar\psi \partial_{\nu}\psi - m c^2 \bar\psi \psi \right) \mathrm{d}^4 x$

where ψ is the Klein–Gordon field and m is its mass. The complex conjugate of ψ is written ψ. If the scalar field is taken to be real-valued, then ψ = ψ.

Applying the formula for the Hilbert 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

$T^{\mu\nu} = \frac{\hbar^2}{m} \left (\eta^{\mu \alpha} \eta^{\nu \beta} + \eta^{\mu \beta} \eta^{\nu \alpha} - \eta^{\mu\nu} \eta^{\alpha \beta} \right ) \partial_{\alpha}\bar\psi \partial_{\beta}\psi - \eta^{\mu\nu} m c^2 \bar\psi \psi .$

Electromagnetic interaction

There is a simple way to make any field interact with electromagnetism in a gauge invariant way: replace the derivative operators with the gauge covariant derivative operators. The Klein Gordon equation becomes:

$D_\mu D^\mu \phi = -(\partial_t - ie A_0)^2 \phi + (\partial_i - ie A_i)^2 \phi = m^2 \phi$

in natural units, where A is the vector potential. While it is possible to add many higher order terms, for example,

$D_\mu D^\mu\phi + A F^{\mu\nu} D_\mu \phi D_\nu (D_\alpha D^\alpha \phi) =0$

these terms are not renormalizable 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:

$S= \int_x \left (\partial_\mu \phi^* + ie A_\mu \phi^* \right ) \left (\partial_\nu \phi - ie A_\nu\phi \right )\eta^{\mu\nu} = \int_x |D \phi|^2$

Gravitational interaction

In general relativity, we include the effect of gravity and the Klein–Gordon equation becomes (in the mostly pluses signature)

$\begin{align} 0 & = - g^{\mu \nu} \nabla_{\mu} \nabla_{\nu} \psi + \dfrac {m^2 c^2}{\hbar^2} \psi = - g^{\mu \nu} \nabla_{\mu} (\partial_{\nu} \psi) + \dfrac {m^2 c^2}{\hbar^2} \psi \\ & = - g^{\mu \nu} \partial_{\mu} \partial_{\nu} \psi + g^{\mu \nu} \Gamma^{\sigma}{}_{\mu \nu} \partial_{\sigma} \psi + \dfrac {m^2 c^2}{\hbar^2} \psi \end{align}$

or equivalently

$\frac{-1}{\sqrt{-g}} \partial_{\mu} \left ( g^{\mu \nu} \sqrt{-g} \partial_{\nu} \psi \right ) + \frac {m^2 c^2}{\hbar^2} \psi = 0$

where g^αβ is the inverse of the metric tensor that is the gravitational potential field, g is the determinant of the metric tensor, ∇_μ is the covariant derivative and Γ^σ_μν is the Christoffel symbol that is the gravitational force field.

See also

• Dirac equation
• Rarita–Schwinger equation
• Quantum field theory
• Scalar field theory
• Sine–Gordon equation

External links

• Weisstein, Eric W., "Klein–Gordon equation", MathWorld.
• Linear Klein–Gordon Equation at EqWorld: The World of Mathematical Equations.
• Nonlinear Klein–Gordon Equation at EqWorld: The World of Mathematical Equations.
• Introduction to nonlocal equations.