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64-628: 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   )   {\displaystyle \ \left(\ t,\mathbf {x} \ \right)\ } or by combining them into a four-vector   x μ = (   c   t , x   )   . {\displaystyle \ x^{\mu }=\left(\ c\ t,\mathbf {x} \ \right)~.} By Fourier transforming

128-565: 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. 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. The equation can be put into

192-424: A Lorentz transformation on a general contravariant four-vector X (like the examples above), regarded as a column vector with Cartesian coordinates with respect to an inertial frame in the entries, is given by X ′ = Λ X , {\displaystyle X'=\Lambda X,} (matrix multiplication) where the components of the primed object refer to the new frame. Related to

256-517: A circle of minima. This observation is an important one in the theory of spontaneous symmetry breaking in the Standard model. The Klein–Gordon equation (and action) for a complex field ψ {\displaystyle \psi } admits a U ( 1 ) {\displaystyle {\text{U}}(1)} symmetry. That is, under the transformations the Klein–Gordon equation

320-617: A consistent quantum relativistic one-particle theory, any relativistic theory implies creation and annihilation of particles beyond a certain energy threshold. Here, the Klein–Gordon equation in natural units, ( ◻ + m 2 ) ψ ( x ) = 0 {\displaystyle (\Box +m^{2})\psi (x)=0} , with the metric signature η μ ν = diag ( + 1 , − 1 , − 1 , − 1 ) {\displaystyle \eta _{\mu \nu }={\text{diag}}(+1,-1,-1,-1)}

384-506: A constant velocity to another inertial reference frame ). Four-vectors describe, for instance, position x in spacetime modeled as Minkowski space , a particle's four-momentum p , the amplitude of the electromagnetic four-potential A ( x ) at a point x in spacetime, and the elements of the subspace spanned by the gamma matrices inside the Dirac algebra . The Lorentz group may be represented by 4×4 matrices Λ . The action of

448-399: A field in some potential V ( ψ ) {\displaystyle V(\psi )} as Then the Klein–Gordon equation is the case V ( ψ ) = M 2 ψ ¯ ψ {\displaystyle V(\psi )=M^{2}{\bar {\psi }}\psi } . Another common choice of potential which arises in interacting theories

512-412: A fixed angle θ about an axis defined by the unit vector : n ^ = ( n ^ 1 , n ^ 2 , n ^ 3 ) , {\displaystyle {\hat {\mathbf {n} }}=\left({\hat {n}}_{1},{\hat {n}}_{2},{\hat {n}}_{3}\right)\,,} without any boosts,

576-563: A four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group , the ( ⁠ 1 / 2 ⁠ , ⁠ 1 / 2 ⁠ ) representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts (a change by

640-576: A new set of constants C ( p ) {\displaystyle C(p)} , the solution then becomes ψ ( x ) = ∫ d 4 p ( 2 π ) 4 e i p ⋅ x C ( p ) δ ( ( p 0 ) 2 − E ( p ) 2 ) . {\displaystyle \psi (x)=\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}e^{ip\cdot x}C(p)\delta ((p^{0})^{2}-E(\mathbf {p} )^{2}).} It

704-601: 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. which, when quantized, gives which simplifies to Rearranging terms yields 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

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768-529: Is not a four-vector, see bispinor . It is similarly defined, the difference being that the transformation rule under Lorentz transformations is given by a representation other than the standard representation. In this case, the rule reads X ′ = Π(Λ) X , where Π(Λ) is a 4×4 matrix other than Λ . Similar remarks apply to objects with fewer or more components that are well-behaved under Lorentz transformations. These include scalars , spinors , tensors and spinor-tensors. The article considers four-vectors in

832-978: Is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notations: A = ( A 0 , A 1 , A 2 , A 3 ) = A 0 E 0 + A 1 E 1 + A 2 E 2 + A 3 E 3 = A 0 E 0 + A i E i = A α E α {\displaystyle {\begin{aligned}\mathbf {A} &=\left(A^{0},\,A^{1},\,A^{2},\,A^{3}\right)\\&=A^{0}\mathbf {E} _{0}+A^{1}\mathbf {E} _{1}+A^{2}\mathbf {E} _{2}+A^{3}\mathbf {E} _{3}\\&=A^{0}\mathbf {E} _{0}+A^{i}\mathbf {E} _{i}\\&=A^{\alpha }\mathbf {E} _{\alpha }\end{aligned}}} where A

896-1074: Is also customary to represent the bases by column vectors : E 0 = ( 1 0 0 0 ) , E 1 = ( 0 1 0 0 ) , E 2 = ( 0 0 1 0 ) , E 3 = ( 0 0 0 1 ) {\displaystyle \mathbf {E} _{0}={\begin{pmatrix}1\\0\\0\\0\end{pmatrix}}\,,\quad \mathbf {E} _{1}={\begin{pmatrix}0\\1\\0\\0\end{pmatrix}}\,,\quad \mathbf {E} _{2}={\begin{pmatrix}0\\0\\1\\0\end{pmatrix}}\,,\quad \mathbf {E} _{3}={\begin{pmatrix}0\\0\\0\\1\end{pmatrix}}} so that: A = ( A 0 A 1 A 2 A 3 ) {\displaystyle \mathbf {A} ={\begin{pmatrix}A^{0}\\A^{1}\\A^{2}\\A^{3}\end{pmatrix}}} The relation between

960-2421: Is common to handle the positive and negative energy solutions by separating out the negative energies and work only with positive p 0 {\displaystyle p^{0}} : ψ ( x ) = ∫ d 4 p ( 2 π ) 4 δ ( ( p 0 ) 2 − E ( p ) 2 ) ( A ( p ) e − i p 0 x 0 + i p i x i + B ( p ) e + i p 0 x 0 + i p i x i ) θ ( p 0 ) = ∫ d 4 p ( 2 π ) 4 δ ( ( p 0 ) 2 − E ( p ) 2 ) ( A ( p ) e − i p 0 x 0 + i p i x i + B ( − p ) e + i p 0 x 0 − i p i x i ) θ ( p 0 ) → ∫ d 4 p ( 2 π ) 4 δ ( ( p 0 ) 2 − E ( p ) 2 ) ( A ( p ) e − i p ⋅ x + B ( p ) e + i p ⋅ x ) θ ( p 0 ) {\displaystyle {\begin{aligned}\psi (x)=&\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}\delta ((p^{0})^{2}-E(\mathbf {p} )^{2})\left(A(p)e^{-ip^{0}x^{0}+ip^{i}x^{i}}+B(p)e^{+ip^{0}x^{0}+ip^{i}x^{i}}\right)\theta (p^{0})\\=&\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}\delta ((p^{0})^{2}-E(\mathbf {p} )^{2})\left(A(p)e^{-ip^{0}x^{0}+ip^{i}x^{i}}+B(-p)e^{+ip^{0}x^{0}-ip^{i}x^{i}}\right)\theta (p^{0})\\\rightarrow &\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}\delta ((p^{0})^{2}-E(\mathbf {p} )^{2})\left(A(p)e^{-ip\cdot x}+B(p)e^{+ip\cdot x}\right)\theta (p^{0})\\\end{aligned}}} In

1024-468: 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 μ x μ {\displaystyle p\cdot x=p_{\mu }x^{\mu }} only, the last expression is also a Lorentz invariant solution to the Klein–Gordon equation. If one does not require Lorentz invariance, one can absorb

1088-537: Is convenient to denote and define the relative velocity in units of c by: β = ( β 1 , β 2 , β 3 ) = 1 c ( v 1 , v 2 , v 3 ) = 1 c v . {\displaystyle {\boldsymbol {\beta }}=(\beta _{1},\,\beta _{2},\,\beta _{3})={\frac {1}{c}}(v_{1},\,v_{2},\,v_{3})={\frac {1}{c}}\mathbf {v} \,.} Then without rotations,

1152-464: Is inconsistent with special relativity . It is natural to try to use the identity from special relativity describing the energy: Then, just inserting the quantum-mechanical operators for momentum and energy yields the equation 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

1216-442: Is invariant, as is the action (see below). By Noether's theorem for fields, corresponding to this symmetry there is a current J μ {\displaystyle J^{\mu }} defined as Four-vector In special relativity , a four-vector (or 4-vector , sometimes Lorentz vector ) is an object with four components, which transform in a specific way under Lorentz transformations . Specifically,

1280-672: 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

1344-521: 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

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1408-483: Is solved by Fourier transformation. Inserting the Fourier transformation ψ ( x ) = ∫ d 4 p ( 2 π ) 4 e − i p ⋅ x ψ ( p ) {\displaystyle \psi (x)=\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}e^{-ip\cdot x}\psi (p)} and using orthogonality of

1472-417: Is still referred to as a scalar field, as scalar describes its transformation (formally, representation) under the Lorentz group. This is also discussed below in the scalar chromodynamics section. The Higgs field is modelled by a potential which can be viewed as a generalization of the ϕ 4 {\displaystyle \phi ^{4}} potential, but has an important difference: it has

1536-407: Is the ϕ 4 {\displaystyle \phi ^{4}} potential for a real scalar field ϕ , {\displaystyle \phi ,} The pure Higgs boson sector of the Standard model is modelled by a Klein–Gordon field with a potential, denoted H {\displaystyle H} for this section. The Standard model is a gauge theory and so while

1600-1483: Is the Kronecker delta , and ε ijk is the three-dimensional Levi-Civita symbol . The spacelike components of four-vectors are rotated, while the timelike components remain unchanged. For the case of rotations about the z -axis only, the spacelike part of the Lorentz matrix reduces to the rotation matrix about the z -axis: ( A ′ 0 A ′ 1 A ′ 2 A ′ 3 ) = ( 1 0 0 0 0 cos ⁡ θ − sin ⁡ θ 0 0 sin ⁡ θ cos ⁡ θ 0 0 0 0 1 ) ( A 0 A 1 A 2 A 3 )   . {\displaystyle {\begin{pmatrix}{A'}^{0}\\{A'}^{1}\\{A'}^{2}\\{A'}^{3}\end{pmatrix}}={\begin{pmatrix}1&0&0&0\\0&\cos \theta &-\sin \theta &0\\0&\sin \theta &\cos \theta &0\\0&0&0&1\\\end{pmatrix}}{\begin{pmatrix}A^{0}\\A^{1}\\A^{2}\\A^{3}\end{pmatrix}}\ .} For two frames moving at constant relative three-velocity v (not four-velocity, see below ), it

1664-1170: Is the wave operator and ∇ 2 {\displaystyle \nabla ^{2}} is the Laplace operator . The speed of light   c   {\displaystyle \ c\ } and Planck constant   ℏ   {\displaystyle \ \hbar \ } are often seen to clutter the equations, so they are therefore often expressed in natural units where   c = ℏ = 1   . {\displaystyle \ c=\hbar =1~.}   x μ = (   t , x   )   {\displaystyle \ x^{\mu }=\left(\ t,\mathbf {x} \ \right)\ }   ω = E , k = p   {\displaystyle \ \omega =E,\quad \mathbf {k} =\mathbf {p} \ }   p μ = (   E , p   )   {\displaystyle \ p^{\mu }=\left(\ E,\mathbf {p} \ \right)\ } time and space Unlike

1728-551: Is the 3-dimensional Laplacian and η is the inverse Minkowski metric with Note that the μ and ν summation indices range from 0 to 3: see Einstein notation . (Some authors alternatively use the negative metric signature of (− + + +) , with η 00 = − 1 , η 11 = η 22 = η 33 = 1 {\displaystyle \eta _{00}=-1,\;\eta _{11}=\eta _{22}=\eta _{33}=1} .) Lorentz transformations leave

1792-470: Is the magnitude component and E α is the basis vector component; note that both are necessary to make a vector, and that when A is seen alone, it refers strictly to the components of the vector. The upper indices indicate contravariant components. Here the standard convention is that Latin indices take values for spatial components, so that i = 1, 2, 3, and Greek indices take values for space and time components, so α = 0, 1, 2, 3, used with

1856-652: Is then used to represent the space derivatives, but this is coordinate chart dependent. The wave equation for small vibrations is of the form where u ( x , t ) is the displacement. The wave equation for the electromagnetic field in vacuum is where A is the electromagnetic four-potential in Lorenz gauge . The Klein–Gordon equation has the form The Green's function , G ( x ~ − x ~ ′ ) {\displaystyle G\left({\tilde {x}}-{\tilde {x}}'\right)} , for

1920-442: The 1 / 2 E ( p ) {\displaystyle 1/2E(\mathbf {p} )} -factor into the coefficients A ( p ) {\displaystyle A(p)} and B ( p ) {\displaystyle B(p)} . The equation was named after the physicists Oskar Klein and Walter Gordon , who in 1926 proposed that it describes relativistic electrons. Vladimir Fock also discovered

1984-425: The Lorentz factor is defined by: γ = 1 1 − β ⋅ β , {\displaystyle \gamma ={\frac {1}{\sqrt {1-{\boldsymbol {\beta }}\cdot {\boldsymbol {\beta }}}}}\,,} and δ ij is the Kronecker delta . Contrary to the case for pure rotations, the spacelike and timelike components are mixed together under boosts. For

Klein–Gordon equation - Misplaced Pages Continue

2048-469: The Minkowski metric invariant, so the d'Alembertian yields a Lorentz scalar . The above coordinate expressions remain valid for the standard coordinates in every inertial frame. There are a variety of notations for the d'Alembertian. The most common are the box symbol ◻ {\displaystyle \Box } ( Unicode : U+2610 ☐ BALLOT BOX ) whose four sides represent

2112-1256: The covariant and contravariant coordinates is through the Minkowski metric tensor (referred to as the metric), η which raises and lowers indices as follows: A μ = η μ ν A ν , {\displaystyle A_{\mu }=\eta _{\mu \nu }A^{\nu }\,,} and in various equivalent notations the covariant components are: A = ( A 0 , A 1 , A 2 , A 3 ) = A 0 E 0 + A 1 E 1 + A 2 E 2 + A 3 E 3 = A 0 E 0 + A i E i = A α E α {\displaystyle {\begin{aligned}\mathbf {A} &=(A_{0},\,A_{1},\,A_{2},\,A_{3})\\&=A_{0}\mathbf {E} ^{0}+A_{1}\mathbf {E} ^{1}+A_{2}\mathbf {E} ^{2}+A_{3}\mathbf {E} ^{3}\\&=A_{0}\mathbf {E} ^{0}+A_{i}\mathbf {E} ^{i}\\&=A_{\alpha }\mathbf {E} ^{\alpha }\\\end{aligned}}} where

2176-436: The d'Alembertian , wave operator , box operator or sometimes quabla operator ( cf . nabla symbol ) is the Laplace operator of Minkowski space . The operator is named after French mathematician and physicist Jean le Rond d'Alembert . In Minkowski space, in standard coordinates ( t , x , y , z ) , it has the form Here ∇ 2 := Δ {\displaystyle \nabla ^{2}:=\Delta }

2240-487: 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 By quantizing this, we get the non-relativistic Schrödinger equation for a free particle: where is the momentum operator ( ∇ being the del operator ), and is the energy operator . The Schrödinger equation suffers from not being relativistically invariant , meaning that it

2304-403: The matrix transpose . This rule is different from the above rule. It corresponds to the dual representation of the standard representation. However, for the Lorentz group the dual of any representation is equivalent to the original representation. Thus the objects with covariant indices are four-vectors as well. For an example of a well-behaved four-component object in special relativity that

2368-409: The rapidity ϕ expression has been used, written in terms of the hyperbolic functions : γ = cosh ⁡ ϕ {\displaystyle \gamma =\cosh \phi } Wave operator In special relativity , electromagnetism and wave theory , the d'Alembert operator (denoted by a box: ◻ {\displaystyle \Box } ), also called

2432-2517: The summation convention . The split between the time component and the spatial components is a useful one to make when determining contractions of one four vector with other tensor quantities, such as for calculating Lorentz invariants in inner products (examples are given below), or raising and lowering indices . In special relativity, the spacelike basis E 1 , E 2 , E 3 and components A , A , A are often Cartesian basis and components: A = ( A t , A x , A y , A z ) = A t E t + A x E x + A y E y + A z E z {\displaystyle {\begin{aligned}\mathbf {A} &=\left(A_{t},\,A_{x},\,A_{y},\,A_{z}\right)\\&=A_{t}\mathbf {E} _{t}+A_{x}\mathbf {E} _{x}+A_{y}\mathbf {E} _{y}+A_{z}\mathbf {E} _{z}\\\end{aligned}}} although, of course, any other basis and components may be used, such as spherical polar coordinates A = ( A t , A r , A θ , A ϕ ) = A t E t + A r E r + A θ E θ + A ϕ E ϕ {\displaystyle {\begin{aligned}\mathbf {A} &=\left(A_{t},\,A_{r},\,A_{\theta },\,A_{\phi }\right)\\&=A_{t}\mathbf {E} _{t}+A_{r}\mathbf {E} _{r}+A_{\theta }\mathbf {E} _{\theta }+A_{\phi }\mathbf {E} _{\phi }\\\end{aligned}}} or cylindrical polar coordinates , A = ( A t , A r , A θ , A z ) = A t E t + A r E r + A θ E θ + A z E z {\displaystyle {\begin{aligned}\mathbf {A} &=(A_{t},\,A_{r},\,A_{\theta },\,A_{z})\\&=A_{t}\mathbf {E} _{t}+A_{r}\mathbf {E} _{r}+A_{\theta }\mathbf {E} _{\theta }+A_{z}\mathbf {E} _{z}\\\end{aligned}}} or any other orthogonal coordinates , or even general curvilinear coordinates . Note

2496-535: 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

2560-408: The Klein–Gordon equation can also be represented as: 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 . The equation solutions include

2624-409: 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 which is formally the same as the homogeneous screened Poisson equation . In addition,

Klein–Gordon equation - Misplaced Pages Continue

2688-420: The above conventions are that the inner product is a scalar, see below for details. Given two inertial or rotated frames of reference , a four-vector is defined as a quantity which transforms according to the Lorentz transformation matrix  Λ : A ′ = Λ A {\displaystyle \mathbf {A} '={\boldsymbol {\Lambda }}\mathbf {A} } In index notation,

2752-1039: The case of a boost in the x -direction only, the matrix reduces to; ( A ′ 0 A ′ 1 A ′ 2 A ′ 3 ) = ( cosh ⁡ ϕ − sinh ⁡ ϕ 0 0 − sinh ⁡ ϕ cosh ⁡ ϕ 0 0 0 0 1 0 0 0 0 1 ) ( A 0 A 1 A 2 A 3 ) {\displaystyle {\begin{pmatrix}A'^{0}\\A'^{1}\\A'^{2}\\A'^{3}\end{pmatrix}}={\begin{pmatrix}\cosh \phi &-\sinh \phi &0&0\\-\sinh \phi &\cosh \phi &0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}{\begin{pmatrix}A^{0}\\A^{1}\\A^{2}\\A^{3}\end{pmatrix}}} Where

2816-663: The complex exponentials gives the dispersion relation p 2 = ( p 0 ) 2 − p 2 = m 2 {\displaystyle p^{2}=(p^{0})^{2}-\mathbf {p} ^{2}=m^{2}} This restricts the momenta to those that lie on shell , giving positive and negative energy solutions p 0 = ± E ( p ) where E ( p ) = p 2 + m 2 . {\displaystyle p^{0}=\pm E(\mathbf {p} )\quad {\text{where}}\quad E(\mathbf {p} )={\sqrt {\mathbf {p} ^{2}+m^{2}}}.} For

2880-450: The context of special relativity. Although the concept of four-vectors also extends to general relativity , some of the results stated in this article require modification in general relativity. The notations in this article are: lowercase bold for three-dimensional vectors, hats for three-dimensional unit vectors , capital bold for four dimensional vectors (except for the four-gradient), and tensor index notation . A four-vector A

2944-506: The contravariant and covariant components transform according to, respectively: A ′ μ = Λ μ ν A ν , A ′ μ = Λ μ ν A ν {\displaystyle {A'}^{\mu }=\Lambda ^{\mu }{}_{\nu }A^{\nu }\,,\quad {A'}_{\mu }=\Lambda _{\mu }{}^{\nu }A_{\nu }} in which

3008-461: The coordinate labels are always subscripted as labels and are not indices taking numerical values. In general relativity, local curvilinear coordinates in a local basis must be used. Geometrically, a four-vector can still be interpreted as an arrow, but in spacetime - not just space. In relativity, the arrows are drawn as part of Minkowski diagram (also called spacetime diagram ). In this article, four-vectors will be referred to simply as vectors. It

3072-509: The d'Alembertian in flat standard coordinates is ∂ 2 {\displaystyle \partial ^{2}} . This notation is used extensively in quantum field theory , where partial derivatives are usually indexed, so the lack of an index with the squared partial derivative signals the presence of the d'Alembertian. Sometimes the box symbol is used to represent the four-dimensional Levi-Civita covariant derivative . The symbol ∇ {\displaystyle \nabla }

3136-609: The d'Alembertian is defined by the equation where δ ( x ~ − x ~ ′ ) {\displaystyle \delta \left({\tilde {x}}-{\tilde {x}}'\right)} is the multidimensional Dirac delta function and x ~ {\displaystyle {\tilde {x}}} and x ~ ′ {\displaystyle {\tilde {x}}'} are two points in Minkowski space. A special solution

3200-418: The equation independently in 1926 slightly after Klein's work, 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 include 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

3264-410: The examples above that are given as contravariant vectors, there are also the corresponding covariant vectors x μ , p μ and A μ ( x ) . These transform according to the rule X ′ = ( Λ − 1 ) T X , {\displaystyle X'=\left(\Lambda ^{-1}\right)^{\textrm {T}}X,} where denotes

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3328-1632: The field into momentum space, the solution is usually written in terms of a superposition of plane waves whose energy and momentum obey the energy-momentum dispersion relation from special relativity . Here, the Klein–Gordon equation is given for both of the two common metric signature conventions   η μ ν = diag (   ± 1 , ∓ 1 , ∓ 1 , ∓ 1   )   . {\displaystyle \ \eta _{\mu \nu }={\text{diag}}\left(\ \pm 1,\mp 1,\mp 1,\mp 1\ \right)~.}   x μ = (   c   t , x   )   {\displaystyle \ x^{\mu }=\left(\ c\ t,\mathbf {x} \ \right)\ }   ω =   E   ℏ , k =   p   ℏ   {\displaystyle \ \omega ={\frac {\ E\ }{\hbar }},\quad \mathbf {k} ={\frac {\ \mathbf {p} \ }{\hbar }}\ }   p μ = (   E   c , p )   {\displaystyle \ p^{\mu }=\left({\frac {\ E\ }{c}},\mathbf {p} \right)\ } Here,   ◻ = ± η μ ν ∂ μ ∂ ν   {\displaystyle \ \Box =\pm \eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }\ }

3392-536: The field transforms trivially under the Lorentz group, it transforms as a C 2 {\displaystyle \mathbb {C} ^{2}} -valued vector under the action of the SU ( 2 ) {\displaystyle {\text{SU}}(2)} part of the gauge group. Therefore, while it is a vector field H : R 1 , 3 → C 2 {\displaystyle H:\mathbb {R} ^{1,3}\rightarrow \mathbb {C} ^{2}} , it

3456-531: The first two terms using the inverse of the Minkowski metric diag(− c , 1, 1, 1) , and writing the Einstein summation convention explicitly we get Thus the Klein–Gordon equation can be written in a covariant notation. This often means an abbreviation in the form of where and This operator is called the wave operator . Today this form is interpreted as the relativistic field equation for spin -0 particles. Furthermore, any component of any solution to

3520-441: 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. 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

3584-465: The four dimensions of space-time and the box-squared symbol ◻ 2 {\displaystyle \Box ^{2}} which emphasizes the scalar property through the squared term (much like the Laplacian ). In keeping with the triangular notation for the Laplacian , sometimes Δ M {\displaystyle \Delta _{M}} is used. Another way to write

3648-494: 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, making the equation a generic expression of quantum fields. The Klein–Gordon equation can be generalized to describe

3712-421: 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 ⁠ 4 n / 2 n − 1 ⁠ for the n -th energy level. The Dirac equation relativistic spectrum is, however, easily recovered if the orbital-momentum quantum number l

3776-1671: The last step, B ( p ) → B ( − p ) {\displaystyle B(p)\rightarrow B(-p)} was renamed. Now we can perform the p 0 {\displaystyle p^{0}} -integration, picking up the positive frequency part from the delta function only: ψ ( x ) = ∫ d 4 p ( 2 π ) 4 δ ( p 0 − E ( p ) ) 2 E ( p ) ( A ( p ) e − i p ⋅ x + B ( p ) e + i p ⋅ x ) θ ( p 0 ) = ∫ d 3 p ( 2 π ) 3 1 2 E ( p ) ( A ( p ) e − i p ⋅ x + B ( p ) e + i p ⋅ x ) | p 0 = + E ( p ) . {\displaystyle {\begin{aligned}\psi (x)&=\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}{\frac {\delta (p^{0}-E(\mathbf {p} ))}{2E(\mathbf {p} )}}\left(A(p)e^{-ip\cdot x}+B(p)e^{+ip\cdot x}\right)\theta (p^{0})\\&=\int \left.{\frac {\mathrm {d} ^{3}p}{(2\pi )^{3}}}{\frac {1}{2E(\mathbf {p} )}}\left(A(\mathbf {p} )e^{-ip\cdot x}+B(\mathbf {p} )e^{+ip\cdot x}\right)\right|_{p^{0}=+E(\mathbf {p} )}.\end{aligned}}} This

3840-1231: The lowered index indicates it to be covariant . Often the metric is diagonal, as is the case for orthogonal coordinates (see line element ), but not in general curvilinear coordinates . The bases can be represented by row vectors : E 0 = ( 1 0 0 0 ) , E 1 = ( 0 1 0 0 ) , E 2 = ( 0 0 1 0 ) , E 3 = ( 0 0 0 1 ) {\displaystyle \mathbf {E} ^{0}={\begin{pmatrix}1&0&0&0\end{pmatrix}}\,,\quad \mathbf {E} ^{1}={\begin{pmatrix}0&1&0&0\end{pmatrix}}\,,\quad \mathbf {E} ^{2}={\begin{pmatrix}0&0&1&0\end{pmatrix}}\,,\quad \mathbf {E} ^{3}={\begin{pmatrix}0&0&0&1\end{pmatrix}}} so that: A = ( A 0 A 1 A 2 A 3 ) {\displaystyle \mathbf {A} ={\begin{pmatrix}A_{0}&A_{1}&A_{2}&A_{3}\end{pmatrix}}} The motivation for

3904-425: The matrix Λ has components Λ ν in row  μ and column  ν , and the matrix ( Λ ) has components Λ μ in row  μ and column  ν . For background on the nature of this transformation definition, see tensor . All four-vectors transform in the same way, and this can be generalized to four-dimensional relativistic tensors; see special relativity . For two frames rotated by

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3968-948: The matrix Λ has components given by: Λ 00 = 1 Λ 0 i = Λ i 0 = 0 Λ i j = ( δ i j − n ^ i n ^ j ) cos ⁡ θ − ε i j k n ^ k sin ⁡ θ + n ^ i n ^ j {\displaystyle {\begin{aligned}\Lambda _{00}&=1\\\Lambda _{0i}=\Lambda _{i0}&=0\\\Lambda _{ij}&=\left(\delta _{ij}-{\hat {n}}_{i}{\hat {n}}_{j}\right)\cos \theta -\varepsilon _{ijk}{\hat {n}}_{k}\sin \theta +{\hat {n}}_{i}{\hat {n}}_{j}\end{aligned}}} where δ ij

4032-979: The matrix Λ has components given by: Λ 00 = γ , Λ 0 i = Λ i 0 = − γ β i , Λ i j = Λ j i = ( γ − 1 ) β i β j β 2 + δ i j = ( γ − 1 ) v i v j v 2 + δ i j , {\displaystyle {\begin{aligned}\Lambda _{00}&=\gamma ,\\\Lambda _{0i}=\Lambda _{i0}&=-\gamma \beta _{i},\\\Lambda _{ij}=\Lambda _{ji}&=(\gamma -1){\frac {\beta _{i}\beta _{j}}{\beta ^{2}}}+\delta _{ij}=(\gamma -1){\frac {v_{i}v_{j}}{v^{2}}}+\delta _{ij},\\\end{aligned}}} where

4096-406: 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

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