Axiomatics of particle interactions
The author's aim is to present a connection between the Yang-Mills approach on gauge fields and the mathematical formalism developed by the author in the last forty years in the field of quantum mechanics. In section 2, the author reviews two of his discoveries which helped him ``to understand why the quantum mechanics of particles takes the form that it does''. The first of these results concerns the von Neumann formalism in which observables are associated with selfadjoint operators in a Hilbert space and pure states with unit vectors in the same Hilbert space. The author obtained the association between real-valued observables and selfadjoint operators by first obtaining an association between such observables and projection-valued measures on the real line and then using the spectral theorem of Hilbert to change this into an association between real observables and selfadjoint operators. It was proved that non-real valued observables do not correspond to selfadjoint operators [\textit{G. W. Mackey}, The mathematical foundations of quantum mechanics, W. A. Benjamin (1963; Zbl 0114.440)]. The other result was in some sense a justification of the famous Heisenberg commutation relations in the particular case of a single free particle using the imprimitivity theorem. In the next section (3), the author continues the exposition of his theory by generalizing the results quoted above to the case of a finite number of interacting particles. For such a system, the operators defining the velocity of the \(j\)-th particle are the commutators \(i[H,X_{1j}]\), \(i[H,X_{2j}]\) and \(i[H,X_{3j}]\), where \(H\) is a selfadjoint operator such that \(t \to e^{-iHt}\) is the one-parameter unitary group defining the time evolution of the system and \(X_{1j}\), \(X_{2j}\), \(X_{3j}\) are the usual multiplicative operators. Then the dynamical operator \(H\) reads \[ H = {1\over 2\mu} \sum_{j} \Biggl\{ {1\over i}\frac{\partial}{\partial X_{1j}} + A_ 1)^ 2 + ({1\over i}\frac {\partial}{\partial X_{2j}} + A_ 2)^ 2 + ({1\over i} \frac {\partial} {\partial X_{3j}} + A_ 3 )^{2}\Biggr \} + v \tag{1} \] where \(A_ 1, A_ 2, A_ 3\) and \(v\) determine the nature of interactions. In the last section of the paper (4), taking the classical limit, the author obtains the equations of motion of a particle of charge \(e\) and mass \(\mu/e\) in an electric field, \(\vec{\nabla} v\), and a magnetic field, \(\vec{\nabla} \times (A_ 1, A_ 2, A_ 3)\). Finally it is required that the physics defined by the operator (1) depends only on the unitary equivalence class of the quadruple \(H, X_ 1, X_ 2, X_ 3\). Then \(A\) transforms like \(A'_ j = uA_ j u^{-1} + i \frac{\partial u} {\partial X_ j}\) and \(v' = uvu^{-1}\), in the manner emphasized by Yang and Mills.
scientific article; zbMATH DE number 575477