A suggestion for unifying quantum theory and relativity

<jats:p> There seems to be a general conviction that the difficulties of our present theory of ultimate particles and nuclear phenomena (the infinite values of the self energy, the zero energy and other quantities) are connected with the problem of merging quantum theory and relativity into a consistent unit. Eddington’s book, “Relativity of the Proton and the Electron”, is an expression of this tendency; but his attempt to link the properties of the smallest particles to those of the whole universe contradicts strongly my physical intuition. Therefore I have considered the question whether there may exist other possibilities of unifying quantum theory and the principle of general invariance, which seems to me the essential thing, as gravitation by its order of magnitude is a molar effect and applies only to masses in bulk, not to the ultimate particles. I present here an idea which seems to be attractive by its simplicity and may lead to a satisfactory theory. 1. Reciprocity The Motion of a free particle in quantum theory is represented by a plane wave <jats:italic>A exp</jats:italic> [ <jats:italic>i</jats:italic> /ℏ <jats:italic>p</jats:italic> <jats:sub> <jats:italic>k</jats:italic> </jats:sub> <jats:italic>x</jats:italic> <jats:sup> <jats:italic>k</jats:italic> </jats:sup> ], where <jats:italic>x</jats:italic> <jats:sup>1</jats:sup> , <jats:italic>x</jats:italic> <jats:sup>2</jats:sup> , <jats:italic>x</jats:italic> <jats:sup>3</jats:sup> , <jats:italic>x</jats:italic> <jats:sup>4</jats:sup> are the co-ordinates of space-time <jats:italic>x</jats:italic> , <jats:italic>y</jats:italic> , <jats:italic>z</jats:italic> , <jats:italic>ct</jats:italic> , and <jats:italic>p</jats:italic> <jats:sup>1</jats:sup> , <jats:italic>p</jats:italic> <jats:sup>2</jats:sup> , <jats:italic>p</jats:italic> <jats:sup>3</jats:sup> , <jats:italic>p</jats:italic> <jats:sup>4</jats:sup> the components of momentum-energy <jats:italic>p</jats:italic> <jats:sub> <jats:italic>x</jats:italic> </jats:sub> , <jats:italic>p</jats:italic> <jats:sub> <jats:italic>y</jats:italic> </jats:sub> , <jats:italic>p</jats:italic> <jats:sub> <jats:italic>z</jats:italic> </jats:sub> , <jats:italic>E</jats:italic> . The expression is completely symmetric in the two 4-vectors <jats:italic>x</jats:italic> and <jats:italic>p</jats:italic> . The transformation theory of quantum mechanics extends this “reciprocity” systematically. In a representation of the operators <jats:italic>x</jats:italic> <jats:sup> <jats:italic>k</jats:italic> </jats:sup> , <jats:italic>p</jats:italic> <jats:sub> <jats:italic>k</jats:italic> </jats:sub> in the Hilbert space for which the <jats:italic>x</jats:italic> <jats:sup> <jats:italic>k</jats:italic> </jats:sup> are diagonal ( <jats:italic>δ</jats:italic> -funcions), the <jats:italic>p</jats:italic> <jats:sub> <jats:italic>k</jats:italic> </jats:sub> are given by ℏ/ <jats:italic>i</jats:italic> ∂/∂ <jats:italic>x</jats:italic> <jats:sub> <jats:italic>k</jats:italic> </jats:sub> ; and vice versa, if the <jats:italic>p</jats:italic> <jats:sub> <jats:italic>k</jats:italic> </jats:sub> are diagonal the <jats:italic>x</jats:italic> <jats:sup> <jats:italic>k</jats:italic> </jats:sup> are given by ­­­­—ℏ/ <jats:italic>i</jats:italic> ∂/∂ <jats:italic>p</jats:italic> <jats:sub> <jats:italic>k</jats:italic> </jats:sub> . Any wave equation in the <jats:italic>x</jats:italic> -space can be transformed into another equation in the <jats:italic>p</jats:italic> -space, by help of the transformation φ( <jats:italic>p</jats:italic> ) = ∫ <jats:italic>ψ</jats:italic> ( <jats:italic>x</jats:italic> ) exp [ <jats:italic>i</jats:italic> /ℏ <jats:italic>p</jats:italic> <jats:sub> <jats:italic>k</jats:italic> </jats:sub> <jats:italic>x</jats:italic> <jats:sup> <jats:italic>k</jats:italic> </jats:sup> ] <jats:italic>dx</jats:italic> . </jats:p>