By Paul Arthur Schilpp
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Additional resources for Albert Einstein: Philosopher-scientist
However, a relativistic wave mechanics built on such an assumption would be incapable of describing charged particles in electromagnetic fields (see Chap. V, § 1). The Lagrangian densities (2) and (3), from which the Klein-Gordon and the Schroedinger equation are deduced, have invariance properties which lead to some very important consequences. Thus the relativistic Lagrangian density is invariant under the transformations of the inhomogeneous Lorentz group, namely under linear transformations where aµ are constants characterizing space (a1, a2, a3) and time (a4) translations, while aµν are the coefficients of a homogeneous Lorentz transformation leaving x2 invariant (rotation in space-time).
5) becomes with Now we put thus distinguishing the variation in form from the variation due to the change of the argument in the transformation of ψi(x). ) The explicit expression for is Moreover we have Then, using the equations of motion the variation can be written in the form Therefore eq. (7) becomes Let us suppose that the coordinate transformation (4) depends on n parameters α1,α2 αn in the following way and that the wave functions are transformed according to For (9) and (10), (8) reads Since the δαj are arbitrary, this equation is equivalent to the vanishing of n four-divergences where As is well known, (11) is the differential expression of a conservation law.
Its transformation is ŠŠŠŠŠŠŠŠThe difference between ψ ′ (x′)= ψ(x) and ψ ′ (x) must be carefully noted. e. the form of the function ψ is not invariant. ψ(x) and ψ ′ (x′) refer to the same space-time point, ψ(x) and ψ ′ (x) to two different points having the coordinates x1, x2, x3, x4 with respect to the old and the new frame of reference, respectively. Let us now see how the wave function in k space is transformed under Lorentz transformations and space rotations. For this purpose consider the Fourier integral Here as has been shown in the Introduction (Note 4).
Albert Einstein: Philosopher-scientist by Paul Arthur Schilpp