An Operator Calculus Having Applications in Quantum Electrodynamics

Abstract

An alteration in the notation used to indicate the order of operation of noncommuting quantities is suggested. Instead of the order being defined by the position on the paper, an ordering subscript is introduced so that ${A}_{s}{B}_{{s}^{\ensuremath{'}}}$ means $\mathrm{AB}$ or $\mathrm{BA}$ depending on whether $s$ exceeds ${s}^{\ensuremath{'}}$ or vice versa. Then ${A}_{s}$ can be handled as though it were an ordinary numerical function of $s$. An increase in ease of manipulating some operator expressions results. Connection to the theory of functionals is discussed in an appendix. Illustrative applications to quantum mechanics are made. In quantum electrodynamics it permits a simple formal understanding of the interrelation of the various present day theoretical formulations.The operator expression of the Dirac equation is related to the author's previous description of positrons. An attempt is made to interpret the operator ordering parameter in this case as a fifth coordinate variable in an extended Dirac equation. Fock's parametrization, discussed in an appendix, seems to be easier to interpret.In the last section a summary of the numerical constants appearing in formulas for transition probabilities is given.