Klein–Gordon equation

Discovery The Klein–Gordon equation was discovered independently in the mid-1920s by numerous physicists. As such, Wolfgang Pauli famously described it as "the equation with many fathers". It is sometimes referred to as Schrödinger's relativistic equation or the Klein–Gordon–Fock equation. The first to discover the equation in December 1925, but not publish it, was Erwin Schrödinger. He applied the Klein–Gordon equation to the hydrogen atom where he calculated the fine structure of its spectrum. Finding that this did not match the experimental results, it demotivated him from investigating the equation further. As a result, he only published the non-relativistic limit of the equation in early 1926, the Schrödinger equation. It was only in his fourth paper in July 1926 that he published the full Klein–Gordon equation. During the 1920s Oskar Klein was developing his own theory of waves, not dissimilar to de Broglie's theory. From this theory he derived the Klein–Gordon equation, but did not assign much importance to it as it only formed a small part of his April 1926 publication. This was the first time that the equation appeared in print. He focused on discussing the wave equation to study the Zeeman effect and Stark effect. He was also the first to publish the resulting fine structure for hydrogen using the equation, found earlier by Schrödinger. Shortly afterwards, Carl Eckart also calculated the Klein–Gordon hydrogen fine structure. Louis de Broglie likewise derived the equation and published it in July, motivated by Schrödinger’s non-relativistic formulation of his matter wave theory hypothesis. Walter Gordon also derived the equation and published it at the end of September, primarily focusing on applying it to the Compton effect, also deriving the current associated to the Klein–Gordon equation, a result found previously by Klein. Johann Kudar likewise published about the Klein–Gordon equation in 1926. Consequences The equation was used in the following years to investigate a handful of problems. For example, Victor Bursian used it to treat dispersion as a perturbation, finding only a small relativistic correction to the result found from the Schrödinger equation. Apart from its use in studying dispersion and Compton scattering, the equation was not very useful for treating most physical problems, primarily due to its inability to incorporate spin. As a result, the equation did not play an important role in the development of quantum mechanics. While the equation virtually disappeared from physics after the Dirac equation was discovered in 1928, it was revived in 1934 by Pauli and Victor Weisskopf who reinterpreted it in light of the theory of field quantization developed at the time, giving it a tenable quantum interpretation as describing spin-0 particles. The first mesons to be discovered were charged pions in 1947, whose kinematics are described by the Klein–Gordon equation. Many other mesons were discovered in the following decades. The first elementary spin-0 particle to be discovered was the Higgs boson in 2012. Since relativistic scalar fields play a prominent role in modern physics, the Klein–Gordon is likewise indispensable in many areas. For example, such scalar fields are found in cosmology and the theory of inflation, in the description of dark matter candidates such as axions and many other Beyond the Standard Model scenarios, in many areas of theoretical physics such as string theory in the form of moduli, and in the AdS/CFT correspondence. == Formulation ==

Definition The Klein–Gordon equation describes the time evolution of a real real scalar field \phi(t, \boldsymbol x), which is a field that assigns a real number to each point in spacetime. The equation is a second order hyperbolic partial differential equation given by The Klein–Gordon equation can also act on a complex scalar field \chi(x), which is a field that assigns to each point in spacetime a complex number. This Lagrangian corresponding to the Klein–Gordon equation is useful when calculating the currents associated with the symmetries using Noether's theorem. Correspondence principle derivation The equation can be derived analogously to how the Schrödinger equation is derived from the non-relativistic equation for the energy of a particle. In particular, in the non-relativistic limit, the energy E of a free particle with momentum \boldsymbol p is given by : E = \frac{1}{2m}\boldsymbol p^2. By elevating the momentum and energy to operators through the correspondence principle Additionally, one would have square roots of differential operators, which are usually handled by Taylor expanding the square root. This leads to an infinite series of higher derivative terms, making the theory difficult to work with. == Properties ==

Symmetries The equation transforms covariantly under spacetime translations and the Lorentz group. Together, these form the Poincare group which encodes the isometries of flat spacetime. Scalar fields transform as scalars under Lorentz transformations, meaning that under x'^\mu = \Lambda^\mu{}_\nu x^\nu, the scalar field transforms as \phi'(x') = \phi(x). Another common choice of boundary condition results in the Feynman propagator, which implements time-ordering and is given by}} : \phi(x) = \int \frac{d^3 \boldsymbol p}{(2\pi)^3} \frac{1}{2 E(\boldsymbol p)}(A(\boldsymbol p)e^{-ip_\mu x^\mu} + B(\boldsymbol p)e^{ip_\mu x^\mu}), where A(\boldsymbol p) and B(\boldsymbol p) are arbitrary functions of the 3-momentum, and E(\boldsymbol p) = \sqrt{\boldsymbol p^2+m^2} is the energy of the modes. This form makes explicit the inevitable presence of both positive and negative frequency solutions. Other solutions When the Klein–Gordon equation is minimally coupled to electromagnetism with a Coulomb potential, it is used to study exotic atoms whose nuclei are orbited by spinless bosonic particles. For example, in a pionic atom, the atomic nucleus is orbited by negatively charged pions \pi^- with mass m_\pi. The lifetime of pions is long enough for this system to form bound states before the pion decays. The case when only one pion orbits a nucleus of charge Ze can be solved directly. Stationary solutions for a Coulomb potential use separation of variables, where the field is written as \phi(x) = e^{-i\epsilon t}\Phi(\boldsymbol x), to express the Klein–Gordon equation as The solution breaks down when Z\alpha > l+1/2, where the square root becomes imaginary, which first happens for Z=69. Physically, the attractive Coulomb potential is too strong, with the kinetic energy of the pion being insufficient to stop the pion from spiralling into the nucleus. The properties of this solution are described by a reflection coefficient R and a transmission coefficient T, which satisfy R+T=1. They describe the probability of the incident wave being reflected off the barrier or passing through it, respectively. For large incoming energies E>V+m, both the reflection and transmission coefficients are non-zero, with there being some probability of the particle passing through the barrier and some probability of it being reflected. The paradox is resolved by reinterpreting the phenomenon as particle-antiparticle pair creation from the potential barrier itself. Antiparticles are then attracted to the potential while particles are repulsed, contributing to the reflected beam, increasing the amount of particles observed coming back from the barrier. A full analysis requires a relativistic multiparticle theory in the form of quantum field theory. == Quantum theory ==

Quantum mechanics The Klein–Gordon equation was originally devised in the context of relativistic quantum mechanics, where it serves the role of a relativistic generalization of the Schrödinger equation. There it describes the evolution of the complex-valued wavefunction \Psi(t,\boldsymbol x) through : \hbar^2 \frac{\partial^2}{\partial t^2}\Psi(t,\boldsymbol x) = \big(c^2 \hbar^2 \nabla^2 - c^4 m^2\big)\Psi(t,\boldsymbol x), where \nabla^2 is the Laplacian, \hbar is the reduced Planck constant, and c is the speed of light. If the Klein–Gordon equation is to describe a wavefunction, there needs to be a corresponding conserved probability density that can be built from the wavefunction. In the limit of large occupation numbers in a coherent or semiclassical state, the quantum field theory for a scalar field reduces to a classical field theory. The negative frequency states that cause issues in the quantum mechanical application of the Klein–Gordon equation are reinterpreted as positive energy antiparticles. For a real scalar field theory, particles are their own antiparticles, while for a complex scalar field theory the two have opposite charges. The candidate probability density for the quantum theory, which failed to be positive definite, is instead reinterpreted as describing the conserved charge density operator in the quantum field theory. == Related equations ==

Non-relativistic limit The non-relativistic limit v\ll c of the complex Klein–Gordon equation yields the Schrödinger equation at leading order. This can be derived by factoring out the oscillating rest mass energy through the field redefinition When interactions are included, these symmetries can break down to a single global \text{U}(2) \rightarrow \text{U}(1) corresponding to charge conservation inherited from the full relativistic theory. Additionally, the non-relativistic effective field theory may have a limited range of viability, such as only holding in the case when particles and antiparticles do not meet and annihilate into relativistic states. If such events are allowed to occur and the effective field theory does not include the relativistic degrees of freedom, then the non-relativistic effective field theory is no longer a closed unitary theory. The non-relativistic limit of the real Klein–Gordon equation is more subtle due to the particle being its own antiparticle. It can however be defined, with a single relativistic real scalar field resulting in a theory for a single complex scalar field in the non-relativistic limit. Despite being a complex field, the degrees of freedom match since the relativistic theory is a second derivative theory in time while the non-relativistic theory is a first order theory, thus each non-relativistic field has half as many degrees of freedom as in the relativistic theory. The naive definition of the non-relativistic complex scalar field \psi(t, \boldsymbol x) from the relativistic field \phi(t,\boldsymbol x) is : \phi(t, \boldsymbol x) = \frac{1}{\sqrt{2m}}[\psi(t, \boldsymbol x)e^{-imt}+\psi^*(t,\boldsymbol x)e^{imt}]. This definition works well for a free theory where the non-relativistic theory for \psi(t,\boldsymbol x) is obtained by dropping high frequency terms of order m, giving the Schrödinger equation to leading order. However, when interactions are taken into account, a modified non-local definition of \psi(t,\boldsymbol x) is usually used to more easily arrive at an effective non-relativistic theory. The non-relativistic theory also has an emergent \text{U}(1) symmetry, which is exact in the non-interacting case. This symmetry implies particle number conservation with a number charge of : N = \int d^3 \boldsymbol x \ |\psi|^2. In a self-interacting theory, this symmetry holds to all orders in perturbation theory in the limit of sufficiently low energies E\ll m and for low occupation numbers that kinematically disallow number violating processes. By constructing the field theory Hamiltonian, one can then show that this implies the Schrödinger equation for the wavefunction : i\frac{\partial \varphi}{\partial t} = -\frac{1}{2m}\nabla^2 \varphi. This procedure is rather technically imprecise, and does not easily generalize to include interactions. It also does not address what happened to the antiparticle degrees of freedom. Instead, a more careful analysis shows that the non-relativistic limit gives rise to the Schrödinger equation for both the particle and antiparticle. This procedure can also be used in the case of a real Klein–Gordon field to get a non-relativistic quantum theory. Klein–Gordon equation in curved spacetime In curved spacetime, the flat metric has to be elevated to a curved metric, and the derivatives need to be modified to account for the spacetime curvature through the introduction of a covariant derivative \nabla_\mu. In the mostly negative metric signature, the Klein–Gordon equation in curved spacetime takes the form : (g^{\mu\nu}\nabla_\mu\nabla_\nu + m^2) \phi = 0. An alternative form that replaces the covariant derivative with regular partial derivatives, avoiding having to explicitly calculate the Christoffel symbols, is given by : \frac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g} g^{\mu\nu}\partial_\nu \phi) + m^2 \phi = 0. Here g^{\mu\nu} is the inverse metric tensor and g is the determinant of the metric. A more general form of the Klein–Gordon equation that allows for non-minimal coupling to gravity is given by : (\nabla^\mu\nabla_\mu +m^2 +\xi R)\phi = 0, where R is the Ricci scalar. The case of \xi=0 is known as minimal coupling, with the equation reducing to the aforementioned form. Meanwhile, in d dimensional spacetime, when : \xi = \frac{1}{4}\frac{d-2}{d-1}, the Klein–Gordon field is said to be conformally coupled to gravity. When the conformally coupled field is also massless, the equation is conformally invariant. This case is often used as a particularly tractable model for a quantum field theory in curved spacetime calculations. In particular, for conformally flat spacetimes these fields do not experience any particle production due to the system being equivalent to a flat space theory. : (D_\mu D^\mu +m^2)\Phi(x)=0, where \Phi(x) is a multicomponent scalar field belonging to some representation of the Lie group. This means that it transforms as \Phi(x)\rightarrow \rho(g(x))\Phi(x), where \rho(g) is a matrix representation of the group element g \in G. Meanwhile, gauge covariant derivative is defined as This is a group theoretic consequence, with all unitary irreducible representations of the Poincare group satisfying the equation. A similar result holds for the Maxwell equation and the Proca equation, describing spin-1 massless and massive fields A^\mu, respectively. Their component equations both reduce to the massless and massive Klein–Gordon equation once the constraint \partial_\mu A^\mu = 0 is imposed, which is a necessary constraint for the Proca equation while only a gauge choice of the Lorenz gauge for the Maxwell equation. ==See also==