In particle physics, the Klein–Gordon equation is a relativistic wave equation for spinless particles. It was discovered 1926 as the relativistic generalization of the Schrödinger equation, developed independently by numerous authors, among them Oskar Klein and Walter Gordon, after whom it is commonly named. Within relativistic quantum mechanics, it suffers from numerous conceptual problems that are only resolved in quantum field theory, where the equation describes the dynamics of spin-0 fields. Mathematically, it is a linear second-order hyperbolic partial differential equation that is manifestly Lorentz covariant and can be viewed as the wave equation form of the relativistic energy–momentum relation. It plays a fundamental role in many areas of modern physics, such as quantum field theory, particle physics, and cosmology.
History
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. His derivation was motivated by de Broglie's theory for matter waves, with Schrödinger attempting to find a wave equation describing their evolution. 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. In particular, he became convinced that his wave theory of atoms could be connected to a unification of Maxwellian electrodynamics and Einsteinian general relativity. This led him to the idea of unifying them in a five-dimensional theory of relativity (see Kaluza–Klein theory), where he interpreted the extra dimension to not be a real physical direction, rather a formal internal coordinate encoding the electric charge. 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. Klein later claimed to have discovered the equation originally in summer 1925, before Schrödinger, but there is no direct primary evidence for this. After Schrödinger's original publication of the non-relativistic equation, it was a straightforward matter to generalize the equation to its relativistic form. In this way Pauli got the relativistic equation in April 1926 but did not publish it. He lost confidence in the equation after he did not manage to use it to establish the equivalence between matrix mechanics and wave mechanics, like he could for the Schrödinger equation. Meanwhile, unaware of Kleins paper, Vladimir Fock likewise derived the equation and published it in June. 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. Finally, there were three other discoverers of the Klein–Gordon equation. They were Théophile De Donder and Frans-H. van den Dungen, who were treating general dynamical systems and were not explicitly focused on quantum theory at all. Their paper was published in July 1926. 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. More importantly, it was unappealing to the theorists as it failed to accommodate the principles of quantum mechanics as understood at the time, such as the Heisenberg uncertainty principle, Borns probabilistic interpretation, and the transformation theory of Paul Dirac and Pascual Jordan. 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. This allowed for the negative energy states that appeared in naive quantum mechanical interpretation to be instead reinterpreted as antiparticles. The first spin-0 particle to be predicted was the meson in 1935 by Hideki Yukawa, which utilized a form of the equation. 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 scalar field ϕ ( t , x ) {\displaystyle \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
( 1 c 2 ∂ 2 ∂ t 2 − ∇ 2 + c 2 m 2 ℏ 2 ) ϕ ( t , x ) = 0. {\displaystyle {\bigg (}{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}+{\frac {c^{2}m^{2}}{\hbar ^{2}}}{\bigg )}\phi (t,{\boldsymbol {x}})=0.}
Here ∇ 2 {\displaystyle \nabla ^{2}} is the Laplace operator, c {\displaystyle c} the speed of light, and ℏ {\displaystyle \hbar } is the reduced Planck constant. The parameter m {\displaystyle m} plays the role of the mass of the scalar field. The time-independent case of the equation is equivalent to the screened Poisson equation. A more concise expression for the equation employs natural units, where both the speed of light and the reduced Planck's constant are set to unity c = ℏ = 1 {\displaystyle c=\hbar =1} . The differential operator takes the form of the d'Alembert operator ◻ = η μ ν ∂ μ ∂ ν {\displaystyle \square =\eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }} , where η μ ν {\displaystyle \eta ^{\mu \nu }} is the raised flat metric. The Klein–Gordon equation then takes the form
The Klein–Gordon equation can also act on a complex scalar field χ ( x ) {\displaystyle \chi (x)} , which is a field that assigns to each point in spacetime a complex number. In this case the equation can be decomposed into a pair of Klein–Gordon equations acting independently on the two degrees of freedom of the complex scalar field. These can be the real and imaginary components, or alternatively the field and its complex conjugate. This decomposition holds because a non-interacting theory of a single complex scalar field is equivalent to a theory of two decoupled real scalar fields. In the case of a complex scalar field, electromagnetic interactions can be introduced by minimally coupling the fields through the introduction of a gauge covariant derivative
D μ χ = ( ∂ μ − i e A μ ) χ ( x ) , {\displaystyle D_{\mu }\chi =(\partial _{\mu }-ieA_{\mu })\chi (x),}
where A μ {\displaystyle A_{\mu }} is the gauge field and e {\displaystyle e} is the electric charge of the scalar field. The covariant derivative is introduced to ensure that the resulting theory is gauge invariant under gauge transformations where the gauge field and scalar field transform as
A μ ( x ) → A μ ( x ) + ∂ μ λ ( x ) , χ ( x ) → e i e λ ( x ) χ ( x ) , {\displaystyle A_{\mu }(x)\rightarrow A_{\mu }(x)+\partial _{\mu }\lambda (x),\ \ \ \ \ \ \ \ \chi (x)\rightarrow e^{ie\lambda (x)}\chi (x),}
for a gauge parameter λ ( x ) {\displaystyle \lambda (x)} . The Klein–Gordon equation minimally coupled to electromagnetism then takes the form
( D μ D μ + m 2 ) χ ( x ) = 0. {\displaystyle (D_{\mu }D^{\mu }+m^{2})\chi (x)=0.}
This equation describes the evolution of a complex scalar fields in scalar electrodynamics.
Lagrangian formulation In a Lagrangian formulation, the Klein–Gordon equation for a real scalar field can be obtained as the Euler–Lagrange equation of the Lagrangian
L = 1 2 ∂ μ ϕ ∂ μ ϕ − 1 2 m 2 ϕ 2 . {\displaystyle {\mathcal {L}}={\frac {1}{2}}\partial ^{\mu }\phi \partial _{\mu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}.}
In quantum field theory this Lagrangian describes a massive non-interacting real scalar field whose excitations are scalar bosons of mass m {\displaystyle m} . In the case of a complex scalar field χ ( x ) {\displaystyle \chi (x)} , its Lagrangian is given by
L = ∂ μ χ † ∂ μ χ − m 2 χ † χ . {\displaystyle {\mathcal {L}}=\partial ^{\mu }\chi ^{\dagger }\partial _{\mu }\chi -m^{2}\chi ^{\dagger }\chi .}
The Klein–Gordon equation for χ {\displaystyle \chi } and χ † {\displaystyle \chi ^{\dagger }} is found by varying the Lagrangian with respect to χ † {\displaystyle \chi ^{\dagger }} and χ {\displaystyle \chi } , respectively. This Lagrangian is manifestly invariant under a global U ( 1 ) {\displaystyle {\text{U}}(1)} symmetry, which when gauged results in scalar electrodynamics. The Lagrangian formulation 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 {\displaystyle E} of a free particle with momentum p {\displaystyle {\boldsymbol {p}}} is given by
E = 1 2 m p 2 . {\displaystyle E={\frac {1}{2m}}{\boldsymbol {p}}^{2}.}
By elevating the momentum and energy to operators through the correspondence principle
p → p ^ = − i ℏ ∇ , E → E ^ = i ℏ ∂ ∂ t , {\displaystyle {\boldsymbol {p}}\rightarrow {\hat {\boldsymbol {p}}}=-i\hbar \nabla ,\ \ \ \ \ \ \ \ E\rightarrow {\hat {E}}=i\hbar {\frac {\partial }{\partial t}},}
this gives the Schrödinger equation. The Klein–Gordon equation is similarly acquired by analogously replacing the energy and momentum by the corresponding operators in the relativistic energy-momentum relation
E 2 = p 2 + m 2 . {\displaystyle E^{2}={\boldsymbol {p}}^{2}+m^{2}.}
Attempting to acquire the equation for the square root of the above expression is on the other hand problematic since time and space do not appear on an equal footing. 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 Poincaré group which encodes the isometries of flat spacetime. Scalar fields transform as scalars under Lorentz transformations, meaning that under x ′ μ = Λ μ
ν x ν {\displaystyle x'^{\mu }=\Lambda ^{\mu }{}_{\nu }x^{\nu }} , the scalar field transforms as ϕ ′ ( x ′ ) = ϕ ( x ) {\displaystyle \phi '(x')=\phi (x)} . Global symmetries give rise to currents and charges through Noether's theorem. The four currents corresponding to the four spacetime translations are given by the stress-energy tensor, which for a real scalar field theory is given by
T μ ν = ∂ μ ϕ ∂ ν ϕ − g μ ν L , {\displaystyle T^{\mu \nu }=\partial ^{\mu }\phi \partial ^{\nu }\phi -g^{\mu \nu }{\mathcal {L}},}
where L {\displaystyle {\mathcal {L}}} is the Lagrangian. The conserved charges are given by the spatial integral of the zeroth component of the current. In this case the conserved charges are the total energy and momentum of the field. For Lorentz transformations, the six currents are expressed in terms of the stress-energy tensor as
M μ ν ρ = x ν T μ ρ − x ρ T μ ν . {\displaystyle {\mathcal {M}}^{\mu \nu \rho }=x^{\nu }T^{\mu \rho }-x^{\rho }T^{\mu \nu }.}
The conserved charges arising from the three rotational degrees of freedom are the angular momenta, while for the three boosts they correspond to the motion of the centre of energy. The real scalar field theory also has an internal discrete Z 2 {\displaystyle \mathbb {Z} _{2}} symmetry ϕ ( x ) → − ϕ ( x ) {\displaystyle \phi (x)\rightarrow -\phi (x)} , although this has no dynamical consequences in the free theory. In the case of a complex scalar field, there is a continuous internal U ( 1 ) {\displaystyle {\text{U}}(1)} phase symmetry
χ ( x ) → e i α χ ( x ) , {\displaystyle \chi (x)\rightarrow e^{i\alpha }\chi (x),}
with an associated Noether current given by
j μ = i ( χ † ∂ μ χ − χ ∂ μ χ † ) . {\displaystyle j^{\mu }=i(\chi ^{\dagger }\partial ^{\mu }\chi -\chi \partial ^{\mu }\chi ^{\dagger }).}
The conserved charge to this symmetry measures the difference in the number of particles to antiparticles. When coupled to electromagnetism, the charge is the electric charge, the conservation of which corresponds to charge conservation.
Green's functions The Klein–Gordon equation can be elevated to the inhomogeneous Klein–Gordon equation through the introduction of a source term J ( x ) {\displaystyle J(x)} as
( ◻ + m 2 ) ϕ ( x ) = J ( x ) . {\displaystyle (\square +m^{2})\phi (x)=J(x).}
This equation admits a general solution of the form
ϕ ( x ) = ∫ d 4 y G ( x , y ) J ( y ) , {\displaystyle \phi (x)=\int d^{4}y\ G(x,y)J(y),}
where G ( x , y ) {\displaystyle G(x,y)} is known as the Green's function, which is the formal solution to the equation
( ◻ + m 2 ) G ( x , y ) = − i δ 4 ( x − y ) . {\displaystyle (\square +m^{2})G(x,y)=-i\delta ^{4}(x-y).}
The principle behind the Green's function G ( x , y ) {\displaystyle G(x,y)} is to relate the solution at x {\displaystyle x} to some forcing term at y {\displaystyle y} . While the Green's function can be seen as the formal inverse of the Klein–Gordon operator ( ◻ + m 2 ) − 1 {\displaystyle (\square +m^{2})^{-1}} , this choice is not unique. A precise choice of the Green's function requires a choice of boundary conditions, specifying where it has support, such as vanishing for certain values of y {\displaystyle y} . For example, the retarded Green's function is defined as relating the solution at the present to impulses strictly from the past y 0 < x 0 {\displaystyle y^{0}<x^{0}} . Meanwhile, the advanced Green's function relates solutions in the present to impulses strictly from the future. These are given by
G r , a ( x , y ) = lim ϵ → 0 ∫ d 4 p ( 2 π ) 4 e i p ⋅ ( x − y ) ( p 0 ± i ϵ ) 2 − p 2 − m 2 , {\displaystyle G_{r,a}(x,y)=\lim _{\epsilon \rightarrow 0}\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {e^{ip\cdot (x-y)}}{(p_{0}\pm i\epsilon )^{2}-{\boldsymbol {p}}^{2}-m^{2}}},}
where the positive sign corresponds to the advanced Green's function. Another common choice of boundary condition results in the Feynman propagator, which implements time-ordering and is given by
G F ( x , y ) = lim ϵ → 0 ∫ d 4 p ( 2 π ) 4 e i p ⋅ ( x − y ) p 2 − m 2 + i ϵ . {\displaystyle G_{F}(x,y)=\lim _{\epsilon \rightarrow 0}\int {\frac {d^{4}p}{(2\pi )^{4}}}{\frac {e^{ip\cdot (x-y)}}{p^{2}-m^{2}+i\epsilon }}.}
In quantum field theory, this propagator is proportional to the time-ordered two-point correlation function for scalar fields.
Plane-wave solutions The Klein–Gordon equation can be solved directly by plugging in the Fourier transform of the scalar field
ϕ ( x ) = ∫ d 4 p ( 2 π ) 4 ϕ ( p ) e − i p μ x μ . {\displaystyle \phi (x)=\int {\frac {d^{4}p}{(2\pi )^{4}}}\phi (p)e^{-ip_{\mu }x^{\mu }}.}
The d'Alembert operator only acts on the 4-position term in the exponent, bringing down factors of momentum. The equation then reduces to a constraint on the 4-momentum of the plane waves
p μ p μ = m 2 . {\displaystyle p_{\mu }p^{\mu }=m^{2}.}
Writing this in terms of the energy and 3-momentum reveals that it is equivalent to the energy-momentum relation for a massive particle E 2 − p 2 = m 2 {\displaystyle E^{2}-{\boldsymbol {p}}^{2}=m^{2}} . Therefore, the most general solution to the Klein–Gordon equation is a superposition of plane wave solutions whose momenta satisfy the energy-momentum relation. By imposing the on-shell condition, such as through the insertion of delta functions into the Fourier transform solution, a more explicit form for the solutions is found
ϕ ( x ) = ∫ d 3 p ( 2 π ) 3 1 2 E ( p ) ( A ( p ) e − i p μ x μ + B ( p ) e i p μ x μ ) , {\displaystyle \phi (x)=\int {\frac {d^{3}{\boldsymbol {p}}}{(2\pi )^{3}}}{\frac {1}{2E({\boldsymbol {p}})}}(A({\boldsymbol {p}})e^{-ip_{\mu }x^{\mu }}+B({\boldsymbol {p}})e^{ip_{\mu }x^{\mu }}),}
where A ( p ) {\displaystyle A({\boldsymbol {p}})} and B ( p ) {\displaystyle B({\boldsymbol {p}})} are arbitrary functions of the 3-momentum, and E ( p ) = p 2 + m 2 {\displaystyle 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 π − {\displaystyle \pi ^{-}} with mass m π {\displaystyle 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 Z e {\displaystyle Ze} can be solved directly. Stationary solutions for a Coulomb potential use separation of variables, where the field is written as ϕ ( x ) = e − i ϵ t Φ ( x ) {\displaystyle \phi (x)=e^{-i\epsilon t}\Phi ({\boldsymbol {x}})} , to express the Klein–Gordon equation as
[ ( ϵ + Z α r ) 2 + ∇ 2 − m π 2 ] Φ ( x ) = 0 , {\displaystyle {\bigg [}{\bigg (}\epsilon +{\frac {Z\alpha }{r}}{\bigg )}^{2}+\nabla ^{2}-m_{\pi }^{2}{\bigg ]}\Phi ({\boldsymbol {x}})=0,}
where α {\displaystyle \alpha } is the fine-structure constant. The energy levels of this system are then given by
E KG = m π 1 + ( Z α ) 2 ( n − ( l + 1 2 ) + ( l + 1 2 ) 2 − ( Z α ) 2 ) 2 , {\displaystyle E_{\text{KG}}={\frac {m_{\pi }}{\sqrt {1+{\frac {(Z\alpha )^{2}}{{\Big (}n-{\big (}l+{\tfrac {1}{2}}{\big )}+{\sqrt {{\big (}l+{\tfrac {1}{2}}{\big )}^{2}-(Z\alpha )^{2}}}{\Big )}^{2}}}}}},}
where n {\displaystyle n} is the principal quantum number and l {\displaystyle l} is the orbital angular momentum quantum number. In contrast to the nonrelativistic case, relativistic effects lift the degeneracy in the l {\displaystyle l} states. These energy levels differ from traditional atoms because electrons are fermions with spin half, while the Klein–Gordon equation describes bosonic spin zero particles. However, the expression for the energy levels for regular atoms, known as the Sommerfeld formula, is very similar. It is directly acquired from the above through the replacement of the pion mass with the electron mass and by replacing l → j {\displaystyle l\rightarrow j} , the spin-orbital angular quantum number. The energy levels predicted by E KG {\displaystyle E_{\text{KG}}} have successfully been observed in pionic atoms. The solution b