Oskar Klein's paper, submitted on April 28, 1926, explores the connection between Kaluza's theory of electromagnetism and gravity and the quantum mechanical methods proposed by de Broglie and Schrödinger. Kaluza's theory introduces a fifth dimension to connect the ten Einsteinian gravitational potentials \( g_{ik} \) and the four electromagnetic potentials \( \varphi_i \) with the coefficients \( \gamma_{ik} \) of a line element in a Riemannian space. This leads to the equations of motion for electric particles taking the form of geodesic equations in electromagnetic fields, which can be interpreted as wave equations when matter is treated as wave-like. Solutions with a fifth dimension that is harmonically present with a period related to Planck's constant lead to the quantum mechanical methods. In the first section, Klein presents a brief overview of the five-dimensional relativity theory, which is similar to Kaluza's but differs in some key aspects. He considers a five-dimensional Riemannian line element and postulates an independent meaning for it. The coordinates \( x^0 \ldots x^4 \) represent the five dimensions of space, and the 15 components \( \gamma_{ik} \) form a symmetric tensor. To relate these to the usual four-dimensional theory, specific assumptions are made: four coordinates must always characterize the usual time, and \( \gamma_{ik} \) must not depend on the fifth coordinate \( x^0 \). This restricts the allowed coordinate transformations to a specific group. The invariance of \( \gamma_{00} \) and certain differential quantities under these transformations is demonstrated, leading to the conclusion that the \( \gamma_{0i} \) transform like the covariant components of a four-vector.Oskar Klein's paper, submitted on April 28, 1926, explores the connection between Kaluza's theory of electromagnetism and gravity and the quantum mechanical methods proposed by de Broglie and Schrödinger. Kaluza's theory introduces a fifth dimension to connect the ten Einsteinian gravitational potentials \( g_{ik} \) and the four electromagnetic potentials \( \varphi_i \) with the coefficients \( \gamma_{ik} \) of a line element in a Riemannian space. This leads to the equations of motion for electric particles taking the form of geodesic equations in electromagnetic fields, which can be interpreted as wave equations when matter is treated as wave-like. Solutions with a fifth dimension that is harmonically present with a period related to Planck's constant lead to the quantum mechanical methods. In the first section, Klein presents a brief overview of the five-dimensional relativity theory, which is similar to Kaluza's but differs in some key aspects. He considers a five-dimensional Riemannian line element and postulates an independent meaning for it. The coordinates \( x^0 \ldots x^4 \) represent the five dimensions of space, and the 15 components \( \gamma_{ik} \) form a symmetric tensor. To relate these to the usual four-dimensional theory, specific assumptions are made: four coordinates must always characterize the usual time, and \( \gamma_{ik} \) must not depend on the fifth coordinate \( x^0 \). This restricts the allowed coordinate transformations to a specific group. The invariance of \( \gamma_{00} \) and certain differential quantities under these transformations is demonstrated, leading to the conclusion that the \( \gamma_{0i} \) transform like the covariant components of a four-vector.