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Klein–Gordon equation

From Wikipedia, the free encyclopedia
Relativistic wave equation in quantum mechanics
Part of a series of articles about
Quantum mechanics
iℏddt|Ψ⟩=H^|Ψ⟩{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }

TheKlein–Gordon equation (Klein–Fock–Gordon equation or sometimesKlein–Gordon–Fock equation, in earlier publicationsSchrödinger–Gordon equation) is arelativistic wave equation, related to theSchrödinger equation. It is named afterOskar Klein andWalter Gordon. It is second-order in space and time and manifestlyLorentz-covariant. It is adifferential equation version of the relativisticenergy–momentum relationE2=(pc)2+(m0c2)2{\displaystyle E^{2}=(pc)^{2}+\left(m_{0}c^{2}\right)^{2}\,}.

Statement

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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 afour-vector xμ=( c t,x ) .{\displaystyle \ x^{\mu }=\left(\ c\ t,\mathbf {x} \ \right)~.} ByFourier transforming the field into momentum space, the solution is usually written in terms of a superposition ofplane waves whose energy and momentum obey the energy-momentumdispersion relation fromspecial relativity. Here, the Klein–Gordon equation is given for both of the two commonmetric signature conventions ημν=diag( ±1,∓1,∓1,∓1 ) .{\displaystyle \ \eta _{\mu \nu }={\text{diag}}\left(\ \pm 1,\mp 1,\mp 1,\mp 1\ \right)~.}

Klein–Gordon equation in normal units with metric signature ημν=diag(±1,∓1,∓1,∓1) {\displaystyle \ \eta _{\mu \nu }={\text{diag}}(\pm 1,\mp 1,\mp 1,\mp 1)\ }
Position space

 xμ=( c t,x ) {\displaystyle \ x^{\mu }=\left(\ c\ t,\mathbf {x} \ \right)\ }

Fourier transformation

 ω= E ℏ,k= p ℏ {\displaystyle \ \omega ={\frac {\ E\ }{\hbar }},\quad \mathbf {k} ={\frac {\ \mathbf {p} \ }{\hbar }}\ }

Momentum space

 pμ=( E c,p) {\displaystyle \ p^{\mu }=\left({\frac {\ E\ }{c}},\mathbf {p} \right)\ }

Separated time and space ( 1 c2 ∂2 ∂t2 −∇2+ m2c2 ℏ2 ) ψ( t,x )=0 {\displaystyle \ \left(\ {\frac {1}{\ c^{2}}}{\frac {\ \partial ^{2}}{\ \partial t^{2}\ }}-\nabla ^{2}+{\frac {\ m^{2}c^{2}\ }{\hbar ^{2}}}\ \right)\ \psi (\ t,\mathbf {x} \ )=0\ } ψ( t,x )=∫{ ∫e∓i( ωt−k⋅x) ψ( ω,k ) d3k  (2πℏ)3  } dω  2πℏ  {\displaystyle \ \psi (\ t,\mathbf {x} \ )=\int \left\{\ \int e^{\mp i\left(\ \omega t-\mathbf {k} \cdot \mathbf {x} \right)}~\psi (\ \omega ,\mathbf {k} \ )\;{\frac {\ \mathrm {d} ^{3}k\ }{~\left(2\pi \hbar \right)^{3}\ }}\ \right\}{\frac {\ \mathrm {d} \omega \ }{\ 2\pi \hbar \ }}\ } E2−p2c2=m2c4 {\displaystyle \ E^{2}-\mathbf {p} ^{2}c^{2}=m^{2}c^{4}\ }
Four-vector form ( ◻+μ2 )ψ=0,μ= m c ℏ {\displaystyle \ \left(\ \Box +\mu ^{2}\ \right)\psi =0,\quad \mu ={\frac {\ m\ c\ }{\hbar }}\ } ψ( xμ )=∫ e−i pμ xμ/ℏψ( pμ ) d4p  (2πℏ)4  {\displaystyle \ \psi (\ x^{\mu }\ )=\int \ e^{-i\ p_{\mu }\ x^{\mu }/\hbar }\;\psi (\ p^{\mu }\ )\;{\frac {\ \mathrm {d} ^{4}p\ }{~\left(2\pi \hbar \right)^{4}\ }}\ } pμ pμ=±m2 c2 {\displaystyle \ p^{\mu }\ p_{\mu }=\pm m^{2}\ c^{2}\ }

Here, ◻=±ημν∂μ∂ν {\displaystyle \ \Box =\pm \eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }\ } is thewave operator and∇2{\displaystyle \nabla ^{2}} is theLaplace operator. Thespeed of light c {\displaystyle \ c\ } andPlanck constant ℏ {\displaystyle \ \hbar \ } are often seen to clutter the equations, so they are therefore often expressed innatural units where c=ℏ=1 .{\displaystyle \ c=\hbar =1~.}

Klein–Gordon equation in natural units with metric signature ημν=diag( ±1,∓1,∓1,∓1 ) {\displaystyle \ \eta _{\mu \nu }={\text{diag}}\left(\ \pm 1,\mp 1,\mp 1,\mp 1\ \right)\ }
Position space

 xμ=( t,x ) {\displaystyle \ x^{\mu }=\left(\ t,\mathbf {x} \ \right)\ }

Fourier transformation

 ω=E,k=p {\displaystyle \ \omega =E,\quad \mathbf {k} =\mathbf {p} \ }

Momentum space

 pμ=( E,p ) {\displaystyle \ p^{\mu }=\left(\ E,\mathbf {p} \ \right)\ }

Separated

time and space

 ( ∂t2−∇2+m2) ψ( t,x )=0 {\displaystyle \ \left(\ \partial _{t}^{2}-\nabla ^{2}+m^{2}\right)\ \psi (\ t,\mathbf {x} \ )=0\ } ψ( t,x )=∫{ ∫e∓i ( ω t − k⋅x )ψ( ω,k ) d3k (2π)3 }dω 2π  {\displaystyle \ \psi (\ t,\mathbf {x} \ )=\int \left\{\ \int e^{\mp i\ \left(\ \omega \ t\ -\ \mathbf {k} \cdot \mathbf {x} \ \right)}\;\psi (\ \omega ,\mathbf {k} \ )\ {\frac {\mathrm {d} ^{3}k}{\ \left(2\pi \right)^{3}}}\ \right\}{\frac {\mathrm {d} \omega }{\ 2\pi \ }}\ } E2−p2=m2 {\displaystyle \ E^{2}-\mathbf {p} ^{2}=m^{2}\ }
Four-vector form ( ◻+m2 )ψ=0 {\displaystyle \ \left(\ \Box +m^{2}\ \right)\psi =0\ } ψ( xμ )=∫e−i pμxμ ψ( pμ )  d4p (2π)4  {\displaystyle \ \psi (\ x^{\mu }\ )=\int e^{-i\ p_{\mu }x^{\mu }}\ \psi (\ p^{\mu }\ )\ {\frac {\ \mathrm {d} ^{4}p\ }{\;\left(2\pi \right)^{4}\ }}\ } pμ pμ=±m2 {\displaystyle \ p^{\mu }\ p_{\mu }=\pm m^{2}\ }

Unlike the Schrödinger equation, the Klein–Gordon equation admits two values ofω for eachk: 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

 [ ∇2− m2c2 ℏ2 ] ψ( r )=0 ,{\displaystyle \ \left[\ \nabla ^{2}-{\frac {\ m^{2}c^{2}}{\ \hbar ^{2}}}\ \right]\ \psi (\ \mathbf {r} \ )=0\ ,}

which is formally the same as the homogeneousscreened Poisson equation. In addition, the Klein–Gordon equation can also be represented as:[1]

 p^μ p^μ ψ=m2c2ψ {\displaystyle \ {\hat {p}}^{\mu }\ {\hat {p}}_{\mu }\ \psi =m^{2}c^{2}\psi \ }

where, the momentum operator is given as:

 p^μ=iℏ∂ ∂xμ =iℏ( ∂ ∂(ct) ,−∂ ∂x ,−∂ ∂y ,−∂ ∂z  )=( E^c,p^ ) .{\displaystyle \ {\hat {p}}^{\mu }=i\hbar {\frac {\partial }{\ \partial x_{\mu }\ }}=i\hbar \left(\ {\frac {\partial }{\ \partial (ct)\ }},-{\frac {\partial }{\ \partial x\ }},-{\frac {\partial }{\ \partial y\ }},-{\frac {\partial }{\ \partial z\ }}\ \right)=\left(\ {\frac {\hat {E}}{c}},\mathbf {\hat {p}} \ \right)~.}

Relevance

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The equation is to be understood first as a classical continuous scalar field equation that can be quantized. Thequantization process introduces then a quantum field whose quanta are spinless particles. Its theoretical relevance is similar to that of theDirac equation.[2] The equation solutions include ascalar or pseudoscalar field[clarification needed]. In the realm ofparticle physics electromagnetic interactions can be incorporated, forming the topic ofscalar electrodynamics, the practical utility for particles likepions is limited.[nb 1][3] There is a second version of the equation for a complex scalar field that is theoretically important being the equation of theHiggs Boson. In the realm ofcondensed matter it can be used for many approximations of quasi-particles without spin.[4][5][nb 2]

The equation can be put into the form of a Schrödinger equation. In this form it is expressed as two coupled differential equations, each of first order in time.[6] The solutions have two components, reflecting the charge degree of freedom in relativity.[6][7] It admits a conserved quantity, but this is not positive definite. The wave function cannot therefore be interpreted as aprobability amplitude. The conserved quantity is instead interpreted aselectric charge, and the norm squared of the wave function is interpreted as acharge 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 a consistent quantum relativisticone-particle theory, any relativistic theory implies creation and annihilation of particles beyond a certain energy threshold.[8][nb 3]

Solution for free particle

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Here, the Klein–Gordon equation in natural units,(◻+m2)ψ(x)=0{\displaystyle (\Box +m^{2})\psi (x)=0}, with the metric signatureημν=diag(+1,−1,−1,−1){\displaystyle \eta _{\mu \nu }={\text{diag}}(+1,-1,-1,-1)} is solved by Fourier transformation. Inserting the Fourier transformationψ(x)=∫d4p(2π)4e−ip⋅xψ(p){\displaystyle \psi (x)=\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}e^{-ip\cdot x}\psi (p)}and usingorthogonality of the complex exponentials gives the dispersion relationp2=(p0)2−p2=m2{\displaystyle p^{2}=(p^{0})^{2}-\mathbf {p} ^{2}=m^{2}}This restricts the momenta to those that lieon shell, giving positive and negative energy solutionsp0=±E(p)whereE(p)=p2+m2.{\displaystyle p^{0}=\pm E(\mathbf {p} )\quad {\text{where}}\quad E(\mathbf {p} )={\sqrt {\mathbf {p} ^{2}+m^{2}}}.}For a new set of constantsC(p){\displaystyle C(p)}, the solution then becomesψ(x)=∫d4p(2π)4eip⋅xC(p)δ((p0)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 is common to handle the positive and negative energy solutions by separating out the negative energies and work only with positivep0{\displaystyle p^{0}}:ψ(x)=∫d4p(2π)4δ((p0)2−E(p)2)(A(p)e−ip0x0+ipixi+B(p)e+ip0x0+ipixi)θ(p0)=∫d4p(2π)4δ((p0)2−E(p)2)(A(p)e−ip0x0+ipixi+B(−p)e+ip0x0−ipixi)θ(p0)→∫d4p(2π)4δ((p0)2−E(p)2)(A(p)e−ip⋅x+B(p)e+ip⋅x)θ(p0){\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 the last step,B(p)→B(−p){\displaystyle B(p)\rightarrow B(-p)} was renamed. Now we can perform thep0{\displaystyle p^{0}}-integration, picking up the positive frequency part from the delta function only:

ψ(x)=∫d4p(2π)4δ(p0−E(p))2E(p)(A(p)e−ip⋅x+B(p)e+ip⋅x)θ(p0)=∫d3p(2π)312E(p)(A(p)e−ip⋅x+B(p)e+ip⋅x)|p0=+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 is commonly taken as a general solution to the free Klein–Gordon equation. Note that because the initial Fourier transformation contained Lorentz invariant quantities likep⋅x=pμxμ{\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 the1/2E(p){\displaystyle 1/2E(\mathbf {p} )}-factor into the coefficientsA(p){\displaystyle A(p)} andB(p){\displaystyle B(p)}.

History

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The equation was named after the physicistsOskar Klein[9] andWalter Gordon,[10] who in 1926 proposed that it describes relativistic electrons.Vladimir Fock also discovered the equation independently in 1926 slightly after Klein's work,[11] 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 andFrans-H. van den Dungen, andLouis de Broglie. Although it turned out that modeling the electron's spin required the Dirac equation, the Klein–Gordon equation correctly describes the spinless relativisticcomposite particles, like thepion. On 4 July 2012, European Organization for Nuclear ResearchCERN announced the discovery of theHiggs boson. Since the Higgs boson is a spin-zero particle, it is the first observed ostensiblyelementary particle to be described by the Klein–Gordon equation. Further experimentation and analysis is required to discern whether the Higgs boson observed is that of theStandard Model or a more exotic, possibly composite, form.

The Klein–Gordon equation was first considered as a quantum wave equation byErwin Schrödinger in his search for an equation describingde 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 of⁠4n/2n − 1⁠ for then-th energy level. The Dirac equation relativistic spectrum is, however, easily recovered if the orbital-momentum quantum numberl is replaced by total angular-momentum quantum numberj.[12] In January 1926, Schrödinger submitted for publication insteadhis equation, a non-relativistic approximation that predicts the Bohr energy levels of hydrogen withoutfine structure.

In 1926, soon after the Schrödinger equation was introduced,Vladimir Fock wrote an article about its generalization for the case ofmagnetic fields, whereforces were dependent onvelocity, and independently derived this equation. Both Klein and Fock usedTheodor Kaluza and Klein's method. Fock also determined thegauge theory for thewave equation. The Klein–Gordon equation for afree particle has a simpleplane-wave solution.

Derivation

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

p22m=E.{\displaystyle {\frac {\mathbf {p} ^{2}}{2m}}=E.}

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

p^22mψ=E^ψ,{\displaystyle {\frac {\mathbf {\hat {p}} ^{2}}{2m}}\psi ={\hat {E}}\psi ,}

where

p^=−iℏ∇{\displaystyle \mathbf {\hat {p}} =-i\hbar \mathbf {\nabla } }

is themomentum operator (∇ being thedel operator), and

E^=iℏ∂∂t{\displaystyle {\hat {E}}=i\hbar {\frac {\partial }{\partial t}}}

is theenergy operator.

The Schrödinger equation suffers from not beingrelativistically invariant, meaning that it is inconsistent withspecial relativity.

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

p2c2+m2c4=E.{\displaystyle {\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

(−iℏ∇)2c2+m2c4ψ=iℏ∂∂tψ.{\displaystyle {\sqrt {(-i\hbar \mathbf {\nabla } )^{2}c^{2}+m^{2}c^{4}}}\,\psi =i\hbar {\frac {\partial }{\partial t}}\psi .}

The square root of a differential operator can be defined with the help ofFourier transformations, but due to the asymmetry of space and time derivatives, Dirac found it impossible to include external electromagnetic fields in 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, isnonlocal (see alsoIntroduction to nonlocal equations).

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

p2c2+m2c4=E2,{\displaystyle \mathbf {p} ^{2}c^{2}+m^{2}c^{4}=E^{2},}

which, when quantized, gives

((−iℏ∇)2c2+m2c4)ψ=(iℏ∂∂t)2ψ,{\displaystyle \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

−ℏ2c2∇2ψ+m2c4ψ=−ℏ2∂2∂t2ψ.{\displaystyle -\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

1c2∂2∂t2ψ−∇2ψ+m2c2ℏ2ψ=0.{\displaystyle {\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 arereal-valued, as well as those that havecomplex values.

Rewriting the first two terms using the inverse of theMinkowski metricdiag(−c2, 1, 1, 1), and writing the Einstein summation convention explicitly we get

−ημν∂μ∂νψ≡∑μ=03∑ν=03−ημν∂μ∂νψ=1c2∂02ψ−∑ν=13∂ν∂νψ=1c2∂2∂t2ψ−∇2ψ.{\displaystyle -\eta ^{\mu \nu }\partial _{\mu }\,\partial _{\nu }\psi \equiv \sum _{\mu =0}^{3}\sum _{\nu =0}^{3}-\eta ^{\mu \nu }\partial _{\mu }\,\partial _{\nu }\psi ={\frac {1}{c^{2}}}\partial _{0}^{2}\psi -\sum _{\nu =1}^{3}\partial _{\nu }\,\partial _{\nu }\psi ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi -\mathbf {\nabla } ^{2}\psi .}

Thus the Klein–Gordon equation can be written in a covariant notation. This often means an abbreviation in the form of

(◻+μ2)ψ=0,{\displaystyle (\Box +\mu ^{2})\psi =0,}

where

μ=mcℏ,{\displaystyle \mu ={\frac {mc}{\hbar }},}

and

◻=1c2∂2∂t2−∇2.{\displaystyle \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}.}

This operator is called thewave operator.

Today this form is interpreted as the relativisticfield equation forspin-0 particles.[6] Furthermore, anycomponent of any solution to the free Dirac equation (for aspin-1/2 particle) is automatically a solution to the free Klein–Gordon equation. This generalizes to particles of any spin due to theBargmann–Wigner equations. Furthermore, in quantum field theory, every component of every quantum field must satisfy the free Klein–Gordon equation,[13] making the equation a generic expression of quantum fields.

Klein–Gordon equation in a potential

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The Klein–Gordon equation can be generalized to describe a field in some potentialV(ψ){\displaystyle V(\psi )} as[14]

◻ψ+∂V∂ψ=0.{\displaystyle \Box \psi +{\frac {\partial V}{\partial \psi }}=0.}

Then the Klein–Gordon equation is the caseV(ψ)=M2ψ¯ψ{\displaystyle V(\psi )=M^{2}{\bar {\psi }}\psi }.

Another common choice of potential which arises in interacting theories is theϕ4{\displaystyle \phi ^{4}} potential for a real scalar fieldϕ,{\displaystyle \phi ,}

V(ϕ)=12m2ϕ2+λϕ4.{\displaystyle V(\phi )={\frac {1}{2}}m^{2}\phi ^{2}+\lambda \phi ^{4}.}

Higgs sector

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See also:Higgs sector

The pure Higgs boson sector of the Standard model is modelled by a Klein–Gordon field with a potential, denotedH{\displaystyle H} for this section. The Standard model is a gauge theory and so while the field transforms trivially under theLorentz group, it transforms as aC2{\displaystyle \mathbb {C} ^{2}}-valued vector under the action of theSU(2){\displaystyle {\text{SU}}(2)} part of the gauge group. Therefore, while it is a vector fieldH:R1,3→C2{\displaystyle H:\mathbb {R} ^{1,3}\rightarrow \mathbb {C} ^{2}}, it 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

V(H)=−m2H†H+λ(H†H)2{\displaystyle V(H)=-m^{2}H^{\dagger }H+\lambda (H^{\dagger }H)^{2}},

which can be viewed as a generalization of theϕ4{\displaystyle \phi ^{4}} potential, but has an important difference: it has a circle of minima. This observation is an important one in the theory ofspontaneous symmetry breaking in the Standard model.

Conserved U(1) current

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The Klein–Gordon equation (and action) for a complex fieldψ{\displaystyle \psi } admits aU(1){\displaystyle {\text{U}}(1)} symmetry. That is, under the transformations

ψ(x)↦eiθψ(x),{\displaystyle \psi (x)\mapsto e^{i\theta }\psi (x),}
ψ¯(x)↦e−iθψ¯(x),{\displaystyle {\bar {\psi }}(x)\mapsto e^{-i\theta }{\bar {\psi }}(x),}

the Klein–Gordon equation is invariant, as is the action (see below). ByNoether's theorem for fields, corresponding to this symmetry there is a currentJμ{\displaystyle J^{\mu }} defined as

Jμ(x)=e2m(ψ¯(x)∂μψ(x)−ψ(x)∂μψ¯(x)).{\displaystyle J^{\mu }(x)={\frac {e}{2m}}\left(\,{\bar {\psi }}(x)\partial ^{\mu }\psi (x)-\psi (x)\partial ^{\mu }{\bar {\psi }}(x)\,\right).}

which satisfies the conservation equation∂μJμ(x)=0.{\displaystyle \partial _{\mu }J^{\mu }(x)=0.}The form of theconserved current can be derived systematically by applying Noether's theorem to theU(1){\displaystyle {\text{U}}(1)} symmetry. We will not do so here, but simply verify that this current is conserved.

From the Klein–Gordon equation for a complex fieldψ(x){\displaystyle \psi (x)} of massM{\displaystyle M}, written in covariant notation andmostly plus signature,

(◻+m2)ψ(x)=0{\displaystyle (\square +m^{2})\psi (x)=0}

and its complex conjugate

(◻+m2)ψ¯(x)=0.{\displaystyle (\square +m^{2}){\bar {\psi }}(x)=0.}

Multiplying by the left respectively byψ¯(x){\displaystyle {\bar {\psi }}(x)} andψ(x){\displaystyle \psi (x)} (and omitting for brevity the explicitx{\displaystyle x} dependence),

ψ¯(◻+m2)ψ=0,{\displaystyle {\bar {\psi }}(\square +m^{2})\psi =0,}
ψ(◻+m2)ψ¯=0.{\displaystyle \psi (\square +m^{2}){\bar {\psi }}=0.}

Subtracting the former from the latter, we obtain

ψ¯◻ψ−ψ◻ψ¯=0,{\displaystyle {\bar {\psi }}\square \psi -\psi \square {\bar {\psi }}=0,}

or in index notation,

ψ¯∂μ∂μψ−ψ∂μ∂μψ¯=0.{\displaystyle {\bar {\psi }}\partial _{\mu }\partial ^{\mu }\psi -\psi \partial _{\mu }\partial ^{\mu }{\bar {\psi }}=0.}

Applying this to the derivative of the currentJμ(x)≡ψ∗(x)∂μψ(x)−ψ(x)∂μψ∗(x),{\displaystyle J^{\mu }(x)\equiv \psi ^{*}(x)\partial ^{\mu }\psi (x)-\psi (x)\partial ^{\mu }\psi ^{*}(x),} one finds

∂μJμ(x)=0.{\displaystyle \partial _{\mu }J^{\mu }(x)=0.}

ThisU(1){\displaystyle {\text{U}}(1)} symmetry is a global symmetry, but it can also be gauged to create a local or gauge symmetry: see below scalar QED. The name of gauge symmetry is somewhat misleading: it is really a redundancy, while the global symmetry is a genuine symmetry.

Lagrangian formulation

[edit]

The Klein–Gordon equation can also be derived by avariational method, arising as the Euler–Lagrange equation of the action

S=∫(−ℏ2ημν∂μψ¯∂νψ−M2c2ψ¯ψ)d4x,{\displaystyle {\mathcal {S}}=\int \left(-\hbar ^{2}\eta ^{\mu \nu }\partial _{\mu }{\bar {\psi }}\,\partial _{\nu }\psi -M^{2}c^{2}{\bar {\psi }}\psi \right)\mathrm {d} ^{4}x,}

In natural units, with signaturemostly minus, the actions take the simple form

Klein–Gordon action for a real scalar field

S=∫d4x(12∂μϕ∂μϕ−12m2ϕ2){\displaystyle S=\int d^{4}x\left({\frac {1}{2}}\partial ^{\mu }\phi \partial _{\mu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}\right)}

for a real scalar field of massm{\displaystyle m}, and

Klein–Gordon action for a complex scalar field

S=∫d4x(∂μψ∂μψ¯−M2ψψ¯){\displaystyle S=\int d^{4}x\left(\partial ^{\mu }\psi \partial _{\mu }{\bar {\psi }}-M^{2}\psi {\bar {\psi }}\right)}

for a complex scalar field of massM{\displaystyle M}.

Applying the formula for thestress–energy tensor to the Lagrangian density (the quantity inside the integral), we can derive the stress–energy tensor of the scalar field. It is

Tμν=ℏ2(ημαηνβ+ημβηνα−ημνηαβ)∂αψ¯∂βψ−ημνM2c2ψ¯ψ.{\displaystyle T^{\mu \nu }=\hbar ^{2}\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^{2}c^{2}{\bar {\psi }}\psi .}

and in natural units,

Tμν=2∂μψ¯∂νψ−ημν(∂ρψ¯∂ρψ−M2ψ¯ψ){\displaystyle T^{\mu \nu }=2\partial ^{\mu }{\bar {\psi }}\partial ^{\nu }\psi -\eta ^{\mu \nu }(\partial ^{\rho }{\bar {\psi }}\partial _{\rho }\psi -M^{2}{\bar {\psi }}\psi )}

By integration of the time–time componentT00 over all space, one may show that both the positive- and negative-frequency plane-wave solutions can be physically associated with particles withpositive energy. This is not the case for the Dirac equation and its energy–momentum tensor.[6]

The stress energy tensor is the set of conserved currents corresponding to the invariance of the Klein–Gordon equation under space-time translationsxμ↦xμ+cμ{\displaystyle x^{\mu }\mapsto x^{\mu }+c^{\mu }}. Therefore, each component is conserved, that is,∂μTμν=0{\displaystyle \partial _{\mu }T^{\mu \nu }=0} (this holds onlyon-shell, that is, when the Klein–Gordon equations are satisfied). It follows that the integral ofT0ν{\displaystyle T^{0\nu }} over space is a conserved quantity for eachν{\displaystyle \nu }. These have the physical interpretation of total energy forν=0{\displaystyle \nu =0} and total momentum forν=i{\displaystyle \nu =i} withi∈{1,2,3}{\displaystyle i\in \{1,2,3\}}.

Non-relativistic limit

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Classical field

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Taking the non-relativistic limit (v ≪c) of a classical Klein–Gordon fieldψ(x,t) begins with the ansatz factoring the oscillatoryrest mass energy term,

ψ(x,t)=ϕ(x,t)e−iℏmc2twhereϕ(x,t)=uE(x)e−iℏE′t.{\displaystyle \psi (\mathbb {x} ,t)=\phi (\mathbb {x} ,t)\,e^{-{\frac {i}{\hbar }}mc^{2}t}\quad {\textrm {where}}\quad \phi (\mathbb {x} ,t)=u_{E}(x)e^{-{\frac {i}{\hbar }}E't}.}

Defining the kinetic energyE′=E−mc2=m2c4+c2p2−mc2≈p22m{\displaystyle E'=E-mc^{2}={\sqrt {m^{2}c^{4}+c^{2}p^{2}}}-mc^{2}\approx {\frac {p^{2}}{2m}}},E′≪mc2{\displaystyle E'\ll mc^{2}} in the non-relativistic limitv=p/m≪c{\displaystyle v=p/m\ll c}, and hence

iℏ∂ϕ∂t=E′ϕ≪mc2ϕand(iℏ)2∂2ϕ∂t2=(E′)2ϕ≪(mc2)2ϕ.{\displaystyle i\hbar {\frac {\partial \phi }{\partial t}}=E'\phi \ll mc^{2}\phi \quad {\textrm {and}}\quad (i\hbar )^{2}{\frac {\partial ^{2}\phi }{\partial t^{2}}}=(E')^{2}\phi \ll (mc^{2})^{2}\phi .}

Applying this yields the non-relativistic limit of the second time derivative ofψ{\displaystyle \psi },

∂ψ∂t=(−imc2ℏϕ+∂ϕ∂t)e−iℏmc2t≈−imc2ℏϕe−iℏmc2t{\displaystyle {\frac {\partial \psi }{\partial t}}=\left(-i{\frac {mc^{2}}{\hbar }}\phi +{\frac {\partial \phi }{\partial t}}\right)\,e^{-{\frac {i}{\hbar }}mc^{2}t}\approx -i{\frac {mc^{2}}{\hbar }}\phi \,e^{-{\frac {i}{\hbar }}mc^{2}t}}
∂2ψ∂t2=−(i2mc2ℏ∂ϕ∂t+(mc2ℏ)2ϕ−∂2ϕ∂t2)e−iℏmc2t≈−(i2mc2ℏ∂ϕ∂t+(mc2ℏ)2ϕ)e−iℏmc2t{\displaystyle {\frac {\partial ^{2}\psi }{\partial t^{2}}}=-\left(i{\frac {2mc^{2}}{\hbar }}{\frac {\partial \phi }{\partial t}}+\left({\frac {mc^{2}}{\hbar }}\right)^{2}\phi -{\frac {\partial ^{2}\phi }{\partial t^{2}}}\right)e^{-{\frac {i}{\hbar }}mc^{2}t}\approx -\left(i{\frac {2mc^{2}}{\hbar }}{\frac {\partial \phi }{\partial t}}+\left({\frac {mc^{2}}{\hbar }}\right)^{2}\phi \right)e^{-{\frac {i}{\hbar }}mc^{2}t}}

Substituting into the free Klein–Gordon equation,c−2∂t2ψ=∇2ψ−(mcℏ)2ψ{\displaystyle c^{-2}\partial _{t}^{2}\psi =\nabla ^{2}\psi -({\frac {mc}{\hbar }})^{2}\psi }, yields

−1c2(i2mc2ℏ∂ϕ∂t+(mc2ℏ)2ϕ)e−iℏmc2t≈(∇2−(mcℏ)2)ϕe−iℏmc2t{\displaystyle -{\frac {1}{c^{2}}}\left(i{\frac {2mc^{2}}{\hbar }}{\frac {\partial \phi }{\partial t}}+\left({\frac {mc^{2}}{\hbar }}\right)^{2}\phi \right)e^{-{\frac {i}{\hbar }}mc^{2}t}\approx \left(\nabla ^{2}-\left({\frac {mc}{\hbar }}\right)^{2}\right)\phi \,e^{-{\frac {i}{\hbar }}mc^{2}t}}

which (by dividing out the exponential and subtracting the mass term) simplifies to

iℏ∂ϕ∂t=−ℏ22m∇2ϕ.{\displaystyle i\hbar {\frac {\partial \phi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\phi .}

This is aclassicalSchrödinger field.

Quantum field

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The analogous limit of a quantum Klein–Gordon field is complicated by the non-commutativity of the field operator. In the limitv ≪c, thecreation and annihilation operators decouple and behave as independent quantumSchrödinger fields.

Scalar electrodynamics

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See also:Scalar electrodynamics

There is a way to make the complex Klein–Gordon fieldψ{\displaystyle \psi } interact with electromagnetism in agauge-invariant way. We can replace the (partial) derivative with the gauge-covariant derivative. Under a localU(1){\displaystyle {\text{U}}(1)} gauge transformation, the fields transform as

ψ↦ψ′=eiθ(x)ψ,{\displaystyle \psi \mapsto \psi '=e^{i\theta (x)}\psi ,}
ψ¯↦ψ¯′=e−iθ(x)ψ¯,{\displaystyle {\bar {\psi }}\mapsto {\bar {\psi }}'=e^{-i\theta (x)}{\bar {\psi }},}

whereθ(x)=θ(t,x){\displaystyle \theta (x)=\theta (t,{\textbf {x}})} is a function of spacetime, thus making it a local transformation, as opposed to a constant over all of spacetime, which would be a globalU(1){\displaystyle {\text{U}}(1)} transformation. A subtle point is that global transformations can arise as local ones, when the functionθ(x){\displaystyle \theta (x)} is taken to be a constant function.

A well-formulated theory should be invariant under such transformations. Precisely, this means that the equations of motion and action (see below) are invariant. To achieve this, ordinary derivatives∂μ{\displaystyle \partial _{\mu }} must be replaced by gauge-covariant derivativesDμ{\displaystyle D_{\mu }}, defined as

Dμψ=(∂μ−ieAμ)ψ{\displaystyle D_{\mu }\psi =(\partial _{\mu }-ieA_{\mu })\psi }
Dμψ¯=(∂μ+ieAμ)ψ¯{\displaystyle D_{\mu }{\bar {\psi }}=(\partial _{\mu }+ieA_{\mu }){\bar {\psi }}}

where the 4-potential or gauge fieldAμ{\displaystyle A_{\mu }} transforms under a gauge transformationθ{\displaystyle \theta } as

Aμ↦Aμ′=Aμ+1e∂μθ{\displaystyle A_{\mu }\mapsto A'_{\mu }=A_{\mu }+{\frac {1}{e}}\partial _{\mu }\theta }.

With these definitions, the covariant derivative transforms as

Dμψ↦eiθDμψ{\displaystyle D_{\mu }\psi \mapsto e^{i\theta }D_{\mu }\psi }

In natural units, the Klein–Gordon equation therefore becomes

DμDμψ−M2ψ=0.{\displaystyle D_{\mu }D^{\mu }\psi -M^{2}\psi =0.}

Since anungaugedU(1){\displaystyle {\text{U}}(1)} symmetry is only present in complex Klein–Gordon theory, this coupling and promotion to agaugedU(1){\displaystyle {\text{U}}(1)} symmetry is compatible only with complex Klein–Gordon theory and not real Klein–Gordon theory.

In natural units and mostly minus signature we have

Scalar QED action

S=∫d4x(−14FμνFμν+DμψDμψ¯−M2ψψ¯){\displaystyle S=\int d^{4}x\,\left(-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+D^{\mu }\psi D_{\mu }{\bar {\psi }}-M^{2}\psi {\bar {\psi }}\right)}

whereFμν=∂μAν−∂νAμ{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }} is known as the Maxwell tensor, field strength or curvature depending on viewpoint.

This theory is often known as scalar quantum electrodynamics or scalar QED, although all aspects we've discussed here are classical.

Scalar chromodynamics

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It is possible to extend this to a non-abelian gauge theory with a gauge groupG{\displaystyle G}, where we couple the scalar Klein–Gordon action to aYang–Mills Lagrangian. Here, the field is actually vector-valued, but is still described as a scalar field: the scalar describes its transformation underspace-time transformations, but not its transformation under the action of the gauge group.

For concreteness we fixG{\displaystyle G} to beSU(N){\displaystyle {\text{SU}}(N)}, thespecial unitary group for someN≥2{\displaystyle N\geq 2}. Under a gauge transformationU(x){\displaystyle U(x)}, which can be described as a functionU:R1,3→SU(N),{\displaystyle U:\mathbb {R} ^{1,3}\rightarrow {\text{SU}}(N),} the scalar fieldψ{\displaystyle \psi } transforms as aCN{\displaystyle \mathbb {C} ^{N}} vector

ψ(x)↦U(x)ψ(x){\displaystyle \psi (x)\mapsto U(x)\psi (x)}
ψ†(x)↦ψ†(x)U†(x){\displaystyle \psi ^{\dagger }(x)\mapsto \psi ^{\dagger }(x)U^{\dagger }(x)}.

The covariant derivative is

Dμψ=∂μψ−igAμψ{\displaystyle D_{\mu }\psi =\partial _{\mu }\psi -igA_{\mu }\psi }
Dμψ†=∂μψ†+igψ†Aμ†{\displaystyle D_{\mu }\psi ^{\dagger }=\partial _{\mu }\psi ^{\dagger }+ig\psi ^{\dagger }A_{\mu }^{\dagger }}

where the gauge field or connection transforms as

Aμ↦UAμU−1−ig∂μUU−1.{\displaystyle A_{\mu }\mapsto UA_{\mu }U^{-1}-{\frac {i}{g}}\partial _{\mu }UU^{-1}.}

This field can be seen as a matrix valued field which acts on the vector spaceCN{\displaystyle \mathbb {C} ^{N}}.

Finally defining the chromomagnetic field strength or curvature,

Fμν=∂μAν−∂νAμ+g(AμAν−AνAμ),{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }+g(A_{\mu }A_{\nu }-A_{\nu }A_{\mu }),}

we can define the action.

Scalar QCD action

S=∫d4x(−14Tr(FμνFμν)+Dμψ†Dμψ−M2ψ†ψ){\displaystyle S=\int d^{4}x\,\left(-{\frac {1}{4}}{\text{Tr}}(F^{\mu \nu }F_{\mu \nu })+D^{\mu }\psi ^{\dagger }D_{\mu }\psi -M^{2}\psi ^{\dagger }\psi \right)}

Klein–Gordon on curved spacetime

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Ingeneral relativity, we include the effect of gravity by replacing partial derivatives withcovariant derivatives, and the Klein–Gordon equation becomes (in themostly pluses signature)[15]

0=−gμν∇μ∇νψ+m2c2ℏ2ψ=−gμν∇μ(∂νψ)+m2c2ℏ2ψ=−gμν∂μ∂νψ+gμνΓσμν∂σψ+m2c2ℏ2ψ,{\displaystyle {\begin{aligned}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{aligned}}}

or equivalently,

−1−g∂μ(gμν−g∂νψ)+m2c2ℏ2ψ=0,{\displaystyle {\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,}

wheregαβ is the inverse of themetric tensor that is the gravitational potential field,g is thedeterminant of the metric tensor,∇μ is thecovariant derivative, andΓσμν is theChristoffel symbol that is the gravitationalforce field.

With natural units this becomes

Klein–Gordon equation on curved spacetime for a real scalar field

∇a∇aΦ−m2Φ=0{\displaystyle \nabla ^{a}\nabla _{a}\Phi -m^{2}\Phi =0}

This also admits an action formulation on a spacetime (Lorentzian) manifoldM{\displaystyle M}. Usingabstract index notation and inmostly plus signature this is

Klein–Gordon action on curved spacetime for a real scalar field

S=∫Md4x−g(−12gab∇aΦ∇bΦ−12m2Φ2){\displaystyle S=\int _{M}d^{4}x\,{\sqrt {-g}}\left(-{\frac {1}{2}}g^{ab}\nabla _{a}\Phi \nabla _{b}\Phi -{\frac {1}{2}}m^{2}\Phi ^{2}\right)}

or

Klein–Gordon action on curved spacetime for a complex scalar field

S=∫Md4x−g(−gab∇aΨ∇bΨ¯−M2ΨΨ¯){\displaystyle S=\int _{M}d^{4}x\,{\sqrt {-g}}\left(-g^{ab}\nabla _{a}\Psi \nabla _{b}{\bar {\Psi }}-M^{2}\Psi {\bar {\Psi }}\right)}

See also

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Remarks

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  1. ^common spinless particles like the pions are unstable and also experience the strong interaction (with unknown interaction term in theHamiltonian)
  2. ^TheSine-Gordon equation is an important example of anIntegrable system
  3. ^To reconcile quantum mechanics with special relativity a multiple particle theory and thereforequantum field theory is needed, in which the Klein–Gordon equation reemerges as the equation obeyed by the components of all[clarification needed] free quantum fields.
    Steven Weinberg makes a point about this. He leaves out the treatment of relativistic wave mechanics altogether in his otherwise complete introduction to modern applications of quantum mechanics, explaining: "It seems to me that the way this is usually presented in books on quantum mechanics is profoundly misleading." (From the preface inLectures on Quantum Mechanics, referring to treatments of the Dirac equation in its original flavor.)
    Others, likeWalter Greiner does in his series on theoretical physics, give a full account of the historical development and view ofrelativistic 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. 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 fields by using complete sets (spanning sets of Hilbert space) of wave functions.

Notes

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  1. ^Greiner, Walter (2013-06-29).Relativistic Quantum Mechanics: Wave Equations. Springer Science & Business Media.ISBN 978-3-662-03425-5.
  2. ^Gross 1993.
  3. ^Greiner & Müller 1994.
  4. ^Bandyopadhyay, A. K.; Ray, P. C.; Gopalan, Venkatraman (2006)."An approach to the Klein–Gordon equation for a dynamic study in ferroelectric materials".Journal of Physics: Condensed Matter.18 (16):4093–4099.doi:10.1088/0953-8984/18/16/016.PMID 21690761.
  5. ^Varró, Sándor (2014). "A new class of exact solutions of the Klein–Gordon equation of a charged particle interacting with an electromagnetic plane wave in a medium".Laser Physics Letters.11 016001.arXiv:1306.0097.doi:10.1088/1612-2011/11/1/016001.
  6. ^abcdGreiner 2000, Ch. 1.
  7. ^Feshbach & Villars 1958.
  8. ^Weinberg, Steven. "Ch. I & II".The Quantum theory of fields I.
  9. ^O. Klein, ZS. f. Phys. 37, 895, 1926
  10. ^W. Gordon, Z. Phys., 40 (1926–1927) pp. 117–133
  11. ^V. Fock, ZS. f. Phys.39, 226, 1926
  12. ^SeeItzykson, C.; Zuber, J.-B. (1985).Quantum Field Theory. McGraw-Hill. pp. 73–74.ISBN 0-07-032071-3. Eq. 2.87 is identical to eq. 2.86, except that it featuresj instead ofl.
  13. ^Weinberg 2002, Ch. 5.
  14. ^Tong, David (2006)."Lectures on Quantum Field Theory, Lecture 1, Section 1.1.1". Retrieved2012-01-16.
  15. ^Fulling, S. A. (1996).Aspects of Quantum Field Theory in Curved Space–Time. Cambridge University Press. p. 117.ISBN 0-07-066353-X.

References

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