# This Quantum World/Feynman route/Free propagator

## Propagator for a free and stable particle

### The propagator as a path integral

Suppose that we make m intermediate position measurements at fixed intervals of duration ${\displaystyle \Delta t.}$ Each of these measurements is made with the help of an array of detectors monitoring n mutually disjoint regions ${\displaystyle R_{k},}$ ${\displaystyle k=1,\dots ,n.}$ Under the conditions stipulated by Rule B, the propagator ${\displaystyle \langle B|A\rangle }$ now equals the sum of amplitudes

${\displaystyle \sum _{k_{1}=1}^{n}\cdots \sum _{k_{m}=1}^{n}\langle B|R_{k_{m}}\rangle \cdots \langle R_{k_{2}}|R_{k_{1}}\rangle \,\langle R_{k_{1}}|A\rangle .}$

It is not hard to see what happens in the double limit ${\displaystyle \Delta t\rightarrow 0}$ (which implies that ${\displaystyle m\rightarrow \infty }$) and ${\displaystyle n\rightarrow \infty .}$ The multiple sum ${\displaystyle \sum _{k_{1}=1}^{n}\cdots \sum _{k_{m}=1}^{n}}$ becomes an integral ${\displaystyle \int \!{\mathcal {DC}}}$ over continuous spacetime paths from A to B, and the amplitude ${\displaystyle \langle B|R_{k_{m}}\rangle \cdots \langle R_{k_{1}}|A\rangle }$ becomes a complex-valued functional ${\displaystyle Z[{\mathcal {C}}:A\rightarrow B]}$ — a complex function of continuous functions representing continuous spacetime paths from A to B:

${\displaystyle \langle B|A\rangle =\int \!{\mathcal {DC}}\,Z[{\mathcal {C}}:A\rightarrow B]}$

The integral ${\displaystyle \int \!{\mathcal {DC}}}$ is not your standard Riemann integral ${\displaystyle \int _{a}^{b}dx\,f(x),}$ to which each infinitesimal interval ${\displaystyle dx}$ makes a contribution proportional to the value that ${\displaystyle f(x)}$ takes inside the interval, but a functional or path integral, to which each "bundle" of paths of infinitesimal width ${\displaystyle {\mathcal {DC}}}$ makes a contribution proportional to the value that ${\displaystyle Z[{\mathcal {C}}]}$ takes inside the bundle.

As it stands, the path integral ${\displaystyle \int \!{\mathcal {DC}}}$ is just the idea of an idea. Appropriate evaluation methods have to be devised on a more or less case-by-case basis.

### A free particle

Now pick any path ${\displaystyle {\mathcal {C}}}$ from A to B, and then pick any infinitesimal segment ${\displaystyle d{\mathcal {C}}}$ of ${\displaystyle {\mathcal {C}}}$. Label the start and end points of ${\displaystyle d{\mathcal {C}}}$ by inertial coordinates ${\displaystyle t,x,y,z}$ and ${\displaystyle t+dt,x+dx,y+dy,z+dz,}$ respectively. In the general case, the amplitude ${\displaystyle Z(d{\mathcal {C}})}$ will be a function of ${\displaystyle t,x,y,z}$ and ${\displaystyle dt,dx,dy,dz.}$ In the case of a free particle, ${\displaystyle Z(d{\mathcal {C}})}$ depends neither on the position of ${\displaystyle d{\mathcal {C}}}$ in spacetime (given by ${\displaystyle t,x,y,z}$) nor on the spacetime orientiaton of ${\displaystyle d{\mathcal {C}}}$ (given by the four-velocity ${\displaystyle (c\,dt/ds,dx/ds,dy/ds,dz/ds)}$ but only on the proper time interval ${\displaystyle ds={\sqrt {dt^{2}-(dx^{2}+dy^{2}+dz^{2})/c^{2}}}.}$

(Because its norm equals the speed of light, the four-velocity depends on three rather than four independent parameters. Together with ${\displaystyle ds,}$ they contain the same information as the four independent numbers ${\displaystyle dt,dx,dy,dz.}$)

Thus for a free particle ${\displaystyle Z(d{\mathcal {C}})=Z(ds).}$ With this, the multiplicativity of successive propagators tells us that

${\displaystyle \prod _{j}Z(ds_{j})=Z{\Bigl (}\sum _{j}ds_{j}{\Bigr )}\longrightarrow Z{\Bigl (}\int _{\mathcal {C}}ds{\Bigr )}}$

It follows that there is a complex number ${\displaystyle z}$ such that ${\displaystyle Z[{\mathcal {C}}]=e^{z\,s[{\mathcal {C}}:A\rightarrow B]},}$ where the line integral ${\displaystyle s[{\mathcal {C}}:A\rightarrow B]=\int _{\mathcal {C}}ds}$ gives the time that passes on a clock as it travels from A to B via ${\displaystyle {\mathcal {C}}.}$

### A free and stable particle

By integrating ${\displaystyle {\bigl |}\langle B|A\rangle {\bigr |}^{2}}$ (as a function of ${\displaystyle \mathbf {r} _{B}}$) over the whole of space, we obtain the probability of finding that a particle launched at the spacetime point ${\displaystyle t_{A},\mathbf {r} _{A}}$ still exists at the time ${\displaystyle t_{B}.}$ For a stable particle this probability equals 1:

${\displaystyle \int \!d^{3}r_{B}\left|\langle t_{B},\mathbf {r} _{B}|t_{A},\mathbf {r} _{A}\rangle \right|^{2}=\int \!d^{3}r_{B}\left|\int \!{\mathcal {DC}}\,e^{z\,s[{\mathcal {C}}:A\rightarrow B]}\right|^{2}=1}$

If you contemplate this equation with a calm heart and an open mind, you will notice that if the complex number ${\displaystyle z=a+ib}$ had a real part ${\displaystyle a\neq 0,}$ then the integral between the two equal signs would either blow up ${\displaystyle (a>0)}$ or drop off ${\displaystyle (a<0)}$ exponentially as a function of ${\displaystyle t_{B}}$, due to the exponential factor ${\displaystyle e^{a\,s[{\mathcal {C}}]}}$.

### Meaning of mass

The propagator for a free and stable particle thus has a single "degree of freedom": it depends solely on the value of ${\displaystyle b.}$ If proper time is measured in seconds, then ${\displaystyle b}$ is measured in radians per second. We may think of ${\displaystyle e^{ib\,s},}$ with ${\displaystyle s}$ a proper-time parametrization of ${\displaystyle {\mathcal {C}},}$ as a clock carried by a particle that travels from A to B via ${\displaystyle {\mathcal {C}},}$ 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!): ${\displaystyle Z=e^{-ib\,s[{\mathcal {C}}]},}$
• to multiply by ${\displaystyle 2\pi }$ (so that we may think of ${\displaystyle b}$ as the rate at which the clock "ticks" — the number of cycles it completes each second): ${\displaystyle Z=e^{-i\,2\pi \,b\,s[{\mathcal {C}}]},}$
• to divide by Planck's constant ${\displaystyle h}$ (so that ${\displaystyle b}$ is measured in energy units and called the rest energy of the particle): ${\displaystyle Z=e^{-i(2\pi /h)\,b\,s[{\mathcal {C}}]}=e^{-(i/\hbar )\,b\,s[{\mathcal {C}}]},}$
• and to multiply by ${\displaystyle c^{2}}$ (so that ${\displaystyle b}$ is measured in mass units and called the particle's rest mass): ${\displaystyle Z=e^{-(i/\hbar )\,b\,c^{2}\,s[{\mathcal {C}}]}.}$

The purpose of using the same letter ${\displaystyle b}$ everywhere is to emphasize that it denotes the same physical quantity, merely measured in different units. If we use natural units in which ${\displaystyle \hbar =c=1,}$ rather than conventional ones, the identity of the various ${\displaystyle b}$'s is immediately obvious.