Propagator for a free and stable particle[edit]
The propagator as a path integral[edit]
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:
![\langle B|A\rangle=\int\!\mathcal{DC}\,Z[\mathcal{C}:A\rightarrow B]](https://wikimedia.org/api/rest_v1/media/math/render/svg/560ba2e4f536aae1082a145b86cd3272a3d60266)
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 particle[edit]
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 particle[edit]
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:
![\int\!d^3r_B\left|\langle t_B,\mathbf{r}_B|t_A,\mathbf{r}_A\rangle\right|^2=
\int\!d^3r_B\left|\int\!\mathcal{DC}\,e^{z\,s[\mathcal{C}:A\rightarrow B]}\right|^2=1](https://wikimedia.org/api/rest_v1/media/math/render/svg/38f718f7da7c6d4d05653a32285c6728865345d9)
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 mass[edit]
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!):
![Z=e^{-ib\,s[\mathcal{C}]},](https://wikimedia.org/api/rest_v1/media/math/render/svg/76a7995ac71573a7048cc185d9fb70ee9115d848)
- 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): ![Z=e^{-i\,2\pi\,b\,s[\mathcal{C}]},](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e44157da1908243e02585b691660c746393b59f)
- to divide by Planck's constant
(so that
is measured in energy units and called the rest energy of the particle): ![Z=e^{-i(2\pi/h)\,b\,s[\mathcal{C}]}=e^{-(i/\hbar)\,b\,s[\mathcal{C}]},](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf41d385d7f74dd942da6bbb24779eacc88c2178)
- and to multiply by
(so that
is measured in mass units and called the particle's rest mass): ![Z=e^{-(i/\hbar)\,b\,c^2\,s[\mathcal{C}]}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5eca1b6ac00d70dbcb1c12dc4a7f29c4ae4fa48)
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.