¹ The important papers are: R. P. Feynman, Phys. Rev. 76, 749, 1949; Stückelberg, Helv. Phys. Acta 15, 23, 1942. A preliminary attack upon these problems which forms the background for the present theory is J. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157, 1945.

² Physicists, being conservative in their habits of speech, sometimes refer to K as the “probability amplitude” for motion. It is my desire here to relieve the innocent word amplitude from one of the many duties it is forced to perform in modern physics. Tendencies can be negative, indeed complex.

³ This paper is not concerned with the older aspects of time reversibility, which are hardly problematic any longer and are well understood. Nevertheless, a brief summary of the “classical” theory of time reversibility may be helpful here.

Distance and time, although they appear conjoined in the theory of relativity (as long as it neglects phenomena involving entropy), do not have identical empirical attributes. Distance can be traversed in either direction simultaneously by two bodies, or at different times by one body. This property I have called “two-wayness” of distance in ref. 4; it might be symbolized by

Time, on the other hand, can flow only one way, and this way is called “into the future”. A formal demonstration of this asymmetry of space and time was given in ref. 4, p. 160. Hence the “one-wayness” of time. But while time, in our experience, flows in only one direction, the basic laws of mechanics (in so far as they do not contain odd derivatives or odd functions of t) are invariant with respect to time reversal. If t’ = − t is substituted in them, they remain unchanged. This means that to every solution for which t goes from 0 to r, there is another for which t goes from 0 to −τ. If a direction of t is not previously fixed, the only ordering relation which has meaning is the conjunction of times with distances, i.e.

for the first solution,

for the second solution. If we read the second series backwards, as we may because the direction of t is not fixed, we have a motion different from the first but equally possible. Both can be observed, yet never simultaneously. Hence we can choose the direction of time as we please so far as the laws of mechanics are concerned. The mechanistic character of time might be termed, perhaps awkwardly, “either-wayness” and symbolized by

Only the irreversible laws of nature allow the elimination of one of these arrows, for they represent tell-tale changes superposed on the indifferent mechanical laws. They force us to symbolize time in the one-way manner

Our intuitive experience seems to convey this final conclusion directly, probably because the “flow of consciousness” is intimately connected with irreversible organic phenomena.

⁴ H. Margenau, The Nature of Physical Reality. McGraw-Hill, 1950.