14 The meaning of mass

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.


spacetime path

Figure 2.13.1 A typ­ical space­time dia­gram dis­playing one space axis and the time axis. The space­time path shown is that of an object trav­eling along the x-​​axis with varying speed. Its speed equals the inverse of the slope of this path. The upward motion of the dashed line sug­gests time


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.