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. It is second order in space and time an 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 the same as that of the Dirac equation,[1] Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but due to that common spinless particles like the pi-mesons are unstable and also experience the strong interaction (with unknown Hamiltonian), the practical utility is limited.

The equation can be put into the form of a Schrödinger equation. In this form it is two coupled differential equations, each of first order in time. The solutions have two components, reflecting the charge degree of freedom in relativity.[2][3] 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 as well as zero charge. Within the Feynman–Stueckelberg interpretation, particles and antiparticles are treated mathematically as if' they propagate forward and backward in time respectively, the advanced propagator (as opposed to the retarded propagator) is employed for antiparticles. Physically, all particles move forward in time. Any solution of the free Dirac equation is componentwise a solution of the free Klein–Gordon equation.

The equation does not form the basis for 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.[nb 1] 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 field by using complete sets (spanning sets of Hilbert space) of wave functions.

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 is replaced by total angular momentum quantum number j.[4] 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).

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.[5] Furthermore, any component of any solution to the free Dirac equation (for a spin-one-half particle) is automatically a solution to the free Klein–Gordon equation. This generalizes to particles of any spin and the relevant equation, see Bargmann–Wigner equations. More is true. In quantum field theory, every component of every quantum field (an operator) satisfies the free Klein–Gordon equation.[6]

Klein–Gordon equation in a potential

The Klein–Gordon equation can be generalized to describe a field in some potential V(ψ) as:[7]

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

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.

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

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

By integration of the time-time component T00 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]

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)[9]

\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 gαβ 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 Γσμν is the Christoffel symbol that is the gravitational force field.

Remarks

1. ^ Stephen Weinberg makes a point about this. He leaves out the treatment of relativistic wave mechanics all together in his otherwise complete introduction to modern applications of quantum mechanics. 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.

Notes

1. ^ Gross 1993
2. ^ Feshbach & Villars 1958
3. ^ Greiner 2000, Ch. 1
4. ^ See C Itzykson and J-B Zuber, Quantum Field Theory, McGraw-Hill Co., 1985, pp. 73–74. Eq. 2.87 is identical to eq. 2.86 except that it features j instead of .
5. ^ Greiner 2000, Ch. 1
6. ^ Weinberg 2002, Ch. 5
7. ^ David Tong, Lectures on Quantum Field Theory, Lecture 1, Section 1.1.1
8. ^ Greiner 2000, Ch. 1
9. ^ S.A. Fulling, Aspects of Quantum Field Theory in Curved Space–Time, Cambridge University Press, 1996, p. 117

References

• Feshbach, H.; Villars, F. (1958). "Elementary relativistic wave mechanics of spin 0 and spin 1/2 particles". Rev. Mod. Phys. 30 (1). doi:10.1103/RevModPhys.30.24.