We now study the Lorentz invariance of the Klein-Gordon equation. The operator is invariant under a Lorentz transformation because it is a scalar product of 4-vectors, . Also, the mass, , is a scalar. Now consider a transformation from an unprimed system to a primed system. In the transformed primed system
Since and refer to the same space-time point and is a scalar, the Klein-Gordon equation is Lorentz invariant. Notice that and are different and are related by . Also and refer to two different points with coordinates in the old and new system, respectively.
Since the Klein-Gordon operator does not change under continuous Lorentz transformations, we can reason that the wave function is multiplied by a factor with absolute value of unity in these transformations. In the case of the coordinate transformation (where is the Lorentz operator) the transformation law of the wave function is
with . Since is real, must be real and hence .
If the Lorentz transformation is continuous (ie. rotation in 4-space), depends continuously on some variables, say . For all , we must have the identity transformation and holds. A wave function which does not change under spatial rotations can describe a particle with spin-0.
For space inversion, which is a discrete Lorentz transformation, and . Applying the space inversion operator twice leads to the identity transformation. Therefore or . We define two states:
for the case ,
for the case ,
Therefore solutions of the Klein-Gordon equation are scalar or pseudoscalar, ie. invariant under spatial rotations and proper Lorentz transformations, and are invariant (scalar) or change sign (pseudoscalar) under space inversion. The pi-meson (pion) is an example of a pseudoscalar meson that obeys the Klein-Gordon equation.