Reprinted in Selected Papers on Quantum Electrodynamics, editor J. Schwinger, Dover Publications, Inc., New York (1958) 321.
Non-relativistic quantum mechanics is formulated here in a different way. It is, however, mathematically equivalent to the familiar formulation. In quantum mechanics the probability of an event which can happen in several different ways is the absolute square of a sum of complex contributions, one from each alternative way. The probability that a particle will be found to have a path x(t) lying somewhere within a region of space time is the square of a sum of contributions, one from each path in
the region. The contribution from a single path is postulated to be an exponential whose (imaginary) phase is the classical action (in units of h) for the path in question. The total contribution from all paths reaching x, t from the past is the wave function (x,t). This is shown to satisfy Schrödinger's equation. The relation to matrix and operator algebra is discussed. Applications are indicated, in particular to eliminate the coordinates of the field
oscillators from the equations of quantum electrodynamics.
Related references See also P. A. M. Dirac, Physik Zeits. Sow. 3 (1933) 64;
P. A. M. Dirac, The Principles of Quantum Mechanics, the Clarendon Press, Oxford (1935);
E. Schrödinger, Annalen der Physik. Leipzig 79 (1926) 489;
W. Heisenberg, The Physical Principles of the Quantum Theory University of Chicago Press, Chicago (1930);
P. A. M. Dirac, Rev. of Mod. Phys. 17 (1945) 195;
Invention of the path integral formalism for quantum mechanics.