Suppose that we make m intermediate position measurements at fixed intervals of duration Each of these measurements is made with the help of an array of detectors monitoring n mutually disjoint regions Under the conditions stipulated by Rule B, the propagator now equals the sum of amplitudes
It is not hard to see what happens in the double limit (which implies that ) and The multiple sum becomes an integral over continuous spacetime paths from A to B, and the amplitude becomes a complex-valued functional — a complex function of continuous functions representing continuous spacetime paths from A to B:
The integral is not your standard Riemann integral to which each infinitesimal interval makes a contribution proportional to the value that takes inside the interval, but a functional or path integral, to which each "bundle" of paths of infinitesimal width makes a contribution proportional to the value that takes inside the bundle.
As it stands, the path integral is just the idea of an idea. Appropriate evalutation methods have to be devised on a more or less case-by-case basis.
Now pick any path from A to B, and then pick any infinitesimal segment of . Label the start and end points of by inertial coordinates and respectively. In the general case, the amplitude will be a function of and In the case of a free particle, depends neither on the position of in spacetime (given by ) nor on the spacetime orientiaton of (given by the four-velocity but only on the proper time interval
(Because its norm equals the speed of light, the four-velocity depends on three rather than four independent parameters. Together with they contain the same information as the four independent numbers )
By integrating (as a function of ) over the whole of space, we obtain the probability of finding that a particle launched at the spacetime point still exists at the time For a stable particle this probability equals 1:
If you contemplate this equation with a calm heart and an open mind, you will notice that if the complex number had a real part then the integral between the two equal signs would either blow up or drop off exponentially as a function of , due to the exponential factor .
The propagator for a free and stable particle thus has a single "degree of freedom": it depends solely on the value of If proper time is measured in seconds, then is measured in radians per second. We may think of with a proper-time parametrization of as a clock carried by a particle that travels from A to B via provided we keep in mind that we are thinking of an aspect of the mathematical formalism of quantum mechanics rather than an aspect of the real world.
It is customary
to insert a minus (so the clock actually turns clockwise!):
to multiply by (so that we may think of as the rate at which the clock "ticks" — the number of cycles it completes each second):
to divide by Planck's constant (so that is measured in energy units and called the rest energy of the particle):
and to multiply by (so that is measured in mass units and called the particle's rest mass):
The purpose of using the same letter everywhere is to emphasize that it denotes the same physical quantity, merely measured in different units. If we use natural units in which rather than conventional ones, the identity of the various 's is immediately obvious.