Klein-Gordon equation

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)

Comments

Explicit formulas for the fundamental solutions of the Klein–Gordon equation (1) are derived in [a1], [a2]. For a derivation of the commutation relations (4) see also [a3].

References

[a1]	J. Hilgevoord, "Dispersion relations and causal description" , North-Holland (1960)
[a2]	E.M. de Jager, "The Lorentz-invariant solutions of the Klein–Gordon equation I-III" Indag. Math. , 25 : 4 (1963) pp. 515–531; 532–545; 546–558
[a3]	P. Roman, "Theory of elementary particles" , North-Holland (1960)
[a4]	J.D. Björken, S.D. Drell, "Relativistic quantum mechanics" , McGraw-Hill (1964)