# Klein-Gordon equation

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The relativistically-invariant quantum equation describing spinless scalar or pseudo-scalar particles, for example, $\pi$-, and $K$- mesons. The equation was established by O. Klein [1] and somewhat later by V.A. Fock [V.A. Fok] as a wave equation under the conditions of cyclicity in the fifth coordinate and was shortly afterwards deduced by several authors (for example, W. Gordon [2]) without this requirement on the fifth coordinate.

The subsequent application of the Klein–Gordon equation as a relativistic quantum equation proved possible in quantum field theory but not in quantum mechanics. In [3] an interpretation of the Klein–Gordon equation was given as an equation for fields of particles of zero spin. The Klein–Gordon equation is applied in the description of $\pi$- mesons and corresponding fields; it plays the role of one of the fundamental equations of quantum field theory.

The Klein–Gordon equation is a linear homogeneous second-order partial differential equation with constant coefficients:

$$\tag{1 } \left ( \frac{\partial ^ {2} }{\partial x ^ {2} } + \frac{\partial ^ {2} }{\partial y ^ {2} } + \frac{\partial ^ {2} }{\partial z ^ {2} } - \frac{\partial ^ {2} }{c ^ {2} \partial t ^ {2} } - \mu ^ {2} \right ) \phi = 0 ,$$

where $\phi ( \mathbf x , t )$ is a (pseudo-) scalar function, in the general case — complex, $\mu = m c / \hbar$ and $m$ is the rest mass of the particle. If $\phi$ is a real function, then the Klein–Gordon equation describes neutral (pseudo-) scalar particles, while when $\phi$ is complex it describes charged particles.

In the latter case equation (1) is supplemented by the equation for the complex-conjugate scalar function $\phi ^ {*}$:

$$\tag{2 } \left ( \frac{\partial ^ {2} }{\partial x ^ {2} } + \frac{\partial ^ {2} }{\partial y ^ {2} } + \frac{\partial ^ {2} }{\partial z ^ {2} } - \frac{\partial ^ {2} }{c ^ {2} \partial t ^ {2} } - \mu ^ {2} \right ) \phi ^ {*} = 0 .$$

The interaction of (pseudo-) scalar particles with the electromagnetic field is described by the minimal substitution $\partial / {\partial x ^ \alpha } \rightarrow ( \partial / {\partial x ^ \alpha } ) - i e A _ \alpha / \hbar$. Each component of the wave function of particles of any spin also satisfies the Klein–Gordon equation, but only for the case where the spin is 0 is the function invariant with respect to the Lorentz–Poincaré group.

The Klein–Gordon equation can be obtained by means of the relationship between the energy $E$ and the momentum $\mathbf p$ of the particle in special relativity theory,

$$\frac{1}{c ^ {2} } E ^ {2} - p _ {x} ^ {2} - p _ {y} ^ {2} - p _ {z} ^ {2} = \ m ^ {2} c ^ {2} ,$$

by replacing quantities by operators (see [4], [5]):

$$E \rightarrow - \frac \hbar {i} \frac \partial {\partial t } ,\ \ \mathbf p \rightarrow \frac \hbar {i} \frac \partial {\partial \mathbf x } .$$

As for all relativistic equations, the Klein–Gordon equation can be expressed in the form of the Dirac equation, that is, it can be reduced to a first-order linear equation:

$$\tag{3 } \left ( \Gamma ^ \alpha \frac \partial {\partial x ^ \alpha } - \mu \right ) \psi = 0 ,$$

where the coefficients $\Gamma ^ \alpha$ are matrices similar to the Dirac matrices $\gamma ^ \alpha$. In the case of the Klein–Gordon equation the matrices $\Gamma ^ \alpha$ satisfy the commutation relations:

$$\tag{4 } \Gamma _ \mu \Gamma _ \nu \Gamma _ \rho + \Gamma _ \rho \Gamma _ \nu \Gamma _ \mu = \ \eta _ {\mu \nu } \Gamma _ \rho + \eta _ {\rho \nu } \Gamma _ \mu .$$

For example, $( \Gamma _ \alpha ) ^ {3} = \eta _ {\alpha \alpha } \Gamma _ \alpha$( Kemmer–Duffin matrices). Here $\eta _ {\mu \nu }$ is the metric tensor of Minkowski space. All the $\Gamma ^ \alpha$ are singular matrices $( \mathop{\rm det} \Gamma _ \alpha = 0 )$. Hence they do not have inverses.

Apart from the trivial solution $\Gamma _ \alpha = 0$, $\psi = 0$ to (4) and a solution in the form of five-row matrices, describing the scalar field $\phi$ itself and the four components of its gradient, equation (4) has a further solution in the form of ten-row matrices. The corresponding ten-component function contains the four components of the potential $A _ \alpha$ and the six components of the stress $F _ {\alpha \beta } = 2 \partial _ {[ \alpha{} } A _ { {}\beta ] }$, that is, equations (3) and (4) can simultaneously give a representation for the Proca equation describing vector particles with spin 1; for $\mu = 0$ and real $\phi$ they give a representation of the Maxwell equations.

When taking into account the interaction of the (pseudo-) scalar particles with a gravity field in accordance with the general theory of relativity, the Klein–Gordon equation is generalized onto an arbitrary Riemannian space as:

$$\tag{5 } \frac{1}{\sqrt - g } \frac \partial {\partial x ^ \alpha } \left ( \sqrt - g g ^ {\alpha \beta } \frac{\partial \phi }{\partial x ^ \beta } \right ) - \mu ^ {2} \phi = 0 ,$$

where $g _ {\alpha \beta }$ is the metric tensor and $g$ is the determinant of the matrix $\| g _ {\alpha \beta } \|$. In equation (5) the term $R \phi / 6$ is frequently added, where $R$ is the scalar curvature, as a result of which, when $\mu = 0$, the general relativistic Klein–Gordon equation

$$\frac{1}{\sqrt - g } \frac \partial {\partial x ^ \alpha } \left ( \sqrt - g g ^ {\alpha \beta } \frac{\partial \phi }{\partial x ^ \beta } \right ) - \frac{R \phi }{6} = 0$$

becomes conformally invariant.

#### References

 [1] O. Klein, Z. Phys. , 37 (1926) pp. 895–906 [2] W. Gordon, Z. Phys. , 40 (1926–1927) pp. 117–133 [3] W. Pauli, V. Weisskopf, "Ueber die Quantisierung der skalaren relativistischen Wellengleichung" Helv. Phys. Acta , 7 (1934) pp. 709–731 [4] N.N. Bogolyubov, D.V. Shirkov, "Introduction to the theory of quantized fields" , Interscience (1959) (Translated from Russian) [5] S. Schweber, "An introduction to relativistic quantum field theory" , Harper & Row (1962)