This Quantum World/Feynman route/Free propagator

Propagator for a free and stable particleEdit

The propagator as a path integralEdit

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.

A free particleEdit

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  )

Thus for a free particle   With this, the multiplicativity of successive propagators tells us that

 

It follows that there is a complex number   such that   where the line integral   gives the time that passes on a clock as it travels from A to B via  

A free and stable particleEdit

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  .

Meaning of massEdit

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.