When we calculated the probability distributions (Figs. 1–4-2 and 1–4-3) that are observed in 2-slit experiments with electrons, we assumed that
- each of the amplitudes AL and AR is a product of two complex numbers called “propagators,”
- the magnitude of the propagator <B|A> — which is the amplitude associated with finding at B a particle last seen at A — is inverse proportional to the distance between A and B,
- the phase of <B|A> is proportional to this distance.
We are now better prepared to see why this is so.
The stability of atoms, we observed, requires the product Δx Δpx to have a positive lower limit. Hence if we make the idealizing and simplifying assumption that all position measurements are precise (Δx=0), then the probability p(C|B), with which an electron found at B (at the time tB) is subsequently detected at C, cannot depend (as it would if the classical theory were correct) on the electron’s momentum at tB, for this is as fuzzy as it gets (Δpx=∞). Consequently, p(C|B) cannot depend on where the electron was previously detected. The electron’s propagation from A to B and its propagation from B to C are thus independent, and so are the corresponding probabilities. As a result, the probability p(C,B|A), with which an electron originally found at A is detected first at B and then at C, is the product of p(B|A) and p(C|B). And since quantum-mechanical probabilities are the squared magnitudes of amplitudes, the amplitude from which p(C,B|A) is obtained is the product of <B|A> and <C|B>.
Next imagine a sphere of radius R centered at A. Let pR be the probability with which a particle launched at A is found by a detector monitoring a unit area of the surface S of this sphere. Because the particle proceeds from A in no particular direction, pR is constant across S. Because the area of S is proportional to R2, and because the probability of detecting the particle anywhere on S (no matter where) equals unity, pR is inverse proportional to R2. Hence the corresponding amplitude is inverse proportional to R, and the magnitude of <B|A> is inverse proportional to the distance between A and B.
Lastly, because a freely propagating particle launched at the precise point A proceeds in no particular direction (or in all directions with equal probability), the phase of <B|A> can only depend on the distance between A and B. Suppose that this is the same as the distance between B and C. The phases of <C|B> and <B|A> are then equal, and since the phase of the product of complex numbers <C|B> <B|A> is the sum of the phases of the individual factors, the phase of the amplitude associated with finding first at B and then at C a particle launched at A will be twice that of <C|B> or <B|A>. Twice the distance, twice the phase. In other words, the phase of <B|A> will be proportional to the distance between A and B.
Let us now replace the plate of the original two-slit experiment by a plate with n holes. And let us replace this one plate by m such plates. At this point, <D|G> is the sum of the amplitudes of nm alternatives.
Let us drill more holes in each plate, and let us add more plates. And still more holes, and still more plates. What happens in the limit in which (i) there are infinitely many plates between G and D and (ii) we drill so many holes in each plate that the plates are no longer there?
What happens is that the sum of nm amplitudes gets replaced by
(2.13.1) <D|G> = ∫DC Z(G→C→D).
The symbol ∫DC indicates a summation over all paths (curves) that lead from G to D: every path C from G to D contributes a complex number Z(G→C→D). As it stands, this is no more than an idea of an idea, albeit a most fruitful one.
Let us take the further step of replacing each 2-dimensional plate “filled” with n holes by a 3-dimensional array of detectors monitoring n mutually disjoint (non-overlapping) regions of space. In other words, we interpose between the particle’s launch at G and its detection at D a succession of m position measurements made at regular intervals Δt (or not made if Rule B applies). What happens in the limit in which (i) the total region monitored becomes the whole of space, (ii) the volumes of the individual regions tend to zero, and (iii) Δt also tends to zero?
What happens is that instead of the sum (2.13.1) over paths in space leading from G to D we obtain a sum over paths in spacetime beginning at the point A (say) at the time tA and ending at the point B at the time tB.
Let’s focus on an infinitesimal path segment dC, its end points given by the coordinates (x,y,z,t) and (x+dx,y+dy,z+dz,t+dt). In general, the amplitude Z(dC) associated with it will be a function of x, y, z, t, dx, dy, dz, and dt. In the case of a freely propagating particle, however, Z(dC) can depend neither on the location of dC in spacetime (given by the coordinates x,y,z,t) nor on its orientation in spacetime (given by the ratios of dx, dy, dz, and dt). This amplitude can only depend on what for the time being we may call the “length” ds of dC — not an ordinary length in space but a length in spacetime. This will be a function of dx, dy, dz, and dt. Its precise form remains to be found.
What follows from what we have learned so far is that Z(dC) is given by ez ds, where e is Euler’s number 2.7182818284… and z is some complex number. If we are dealing with a freely propagating particle that is also stable (it doesn’t decay), then Z(dC) is given by the complex number [1: −b ds]. (The negative sign is a convention.) In this case the propagator takes the form
(2.13.2) <B,tB|A,tA> = ∫DC ΠC [1: −b ds] = ∫DC [1: −b∫C ds] = ∫DC [1: −b s(C)].
If the spacetime path C is divided into infinitesimal segments, the expression ΠC [1: −b ds] stands for a product to which each segment contributes a factor [1: −b ds], and the expression ∫C ds stands for a sum to which each segment contributes a term ds. This sum is nothing but s(C), the total “length” of C.
The important conclusion here is that the behavior of a free and stable particle is controlled by a single real number b. Except for the units in which it is conventionally measured, this number is what we mean by the particle’s mass.